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Logic & First Principles, 17: Pondering the Hyperreals *R with Prof Carol Wood (including Infinitesimals)

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Dr Carol Wood of Wesleyan University (a student of Abraham Robinson who pioneered non-standard analysis 50+ years ago) has discussed the hyperreals in two Numberphile videos:

First:

Extended:

Wenmackers may also be helpful:

In effect, using Model Theory (thus a fair amount of protective hedging!) or other approaches, one may propose an “extension” of the Naturals and the Reals, often N* or R* — but we will use *N and *R as that is more conveniently “hyper-“.

Such a new logic model world — the hyperreals — gives us a way to handle transfinites in a way that is intimately connected to the Reals (with Naturals as regular “mileposts”). As one effect, we here circumvent the question, are there infinitely many naturals, all of which are finite by extending the zone. So now, which version of the counting numbers are we “really” speaking of, N or *N? In the latter, we may freely discuss a K that exceeds any particular k in N reached stepwise from 0. That is, some k = 1 + 1 + . . . 1, k times added and is followed by k+1 etc.

And BTW, the question of what a countable infinite means thus surfaces: beyond any finite bound. That is, for any finite k we represent in N, no matter how large, we may succeed it, k+1, k+2 etc, and in effect we may shift a tape starting 0,1,2 etc up to k, k+1, k+2 etc and see that the two “tapes” continue without end in 1:1 correspondence:

Extending, in a quasi-Peano universe, we can identify some K > any n in N approached by stepwise progression from 0. Where of course Q and R are interwoven with N, giving us the familiar reals as a continuum. What we will do is summarise how we may extend the idea of a continuum in interesting and useful ways — especially once we get to infinitesimals.

Clipping from the videos:

Pausing, let us refresh our memory on the structure and meaning of N considered as ordinals, courtesy the von Neumann construction:

  • {} –> 0
  • {0} –> 1
  • {0,1} –> 2
  • . . .
  • {0,1,2 . . . } –> w, omega the first transfinite ordinal

Popping over to the surreals for a moment, we may see how to build a whole interconnected world of numbers great and small (and yes the hyperreals fit in there . . . from Ehrlich, as Wiki summarises: “[i]t has also been shown (in Von Neumann–Bernays–Gödel set theory) that the maximal class hyperreal field is isomorphic to the maximal class surreal field”):

Now, we may also see in the extended R, *R, that 1/K = m, a newly minted number closer to zero than we can get by inverting any k in N or r in R, that is m is less than 1/k for any k in N and is less than 1/r for any r in R (as the reals are interwoven with the naturals):

and that as we have a range around K, K+1 etc and K-1 etc, even K/2 (an integer if K is even) m has company, forming an ultra-near cloud of similar infinitesimals in the extra-near neighbourhood of 0.

Of course, the reciprocal function is here serving as a catapult, capable of leaping over the bulk of the reals in *R, back and forth between transfinite hyperreals such as K and kin and infinitesimals such as m and kin ultra-near to 0.

Using the additive inverse approach, this extends to the negative side also.

Further, by extending addition, 0 plus its cloud (often called a Monad) can be added to any r in R or indeed K in *R, i.e. we may vector-shift the cloud anywhere in *R. That is, every number on the extended line has its own ultra-near cloud.

Where does all of this lead? First, to Calculus foundations, then to implications of a transfinite zone with stepwise succession, among other things; besides, we need concept space to think and reason about matters of relevance to ID, however remotely (or not so remotely). So, let me now clip a comment I made in the ongoing calculus notation thread of discussion:

KF, 86: >>Money shot comment by JB:

JB, 74: we have Arbogast’s D() notation that we could use, but we don’t. Why not? Because we want people to look at this like a fraction. If we didn’t, there are a ton of other ways to write the derivative. That we do it as a fraction is hugely suggestive, especially, as I mentioned, there exists a correct way to write it as a fraction.

This is pivotal: WHY do we want that ratio, that fraction?
WHY do we think in terms of a function y = f(x), which is continuous and “smooth” in a relevant domain, then for some h that somehow trends to 0 but never quite gets there — we cannot divide by zero — then evaluate:

dy/dx is
lim h –> 0
of
[f(x + h) – f(x)]/[(x + h) – x]

. . . save that, we are looking at the tangent value for the angle the tangent-line of the f(x) curve makes with the horizontal, taken as itself a function of x, f'(x) in Newton’s fluxion notation.

We may then consider f-prime, f'(x) as itself a function and seek its tangent-angle behaviour, getting to f”(x), the second order flow function. Then onwards.

But in all of this, we are spewing forth a veritable spate of infinitesimals and higher order infinitesimals, thus we need something that allows us to responsibly and reliably conceive of and handle them.

I suspect, the epsilon delta limits concept is more of a kludge work-around than we like to admit, a scaffolding that keeps us on safe grounds among the reals. After all, isn’t there no one closest real to any given real, i.e. there is a continuum

But then, is that not in turn something that implies infinitesimal, all but zero differences? Thus, numbers that are all but zero different from zero itself considered as a real? Or, should we be going all vector and considering a ring of the close in C?

In that context, I can see that it makes sense to consider some K that somehow “continues on” from the finite specific reals we can represent, let’s use lower case k, and confine ourselves to the counting numbers as mileposts on the line:

0 – 1 – 2 . . . k – k+1 – k+2 – . . . . – K – K+1 – K+2 . . .
{I used the four dot ellipsis to indicate specifically transfinite span}

We may then postulate a catapult function so 1/K –> m, where m is closer to 0 than ANY finite real or natural we can represent by any k can give.
Notice, K is preceded by a dash, meaning there is a continuum back to say K/2 (–> considering K as even) and beyond, descending and passing mileposts as we go:

K-> K-1 –> K-2 . . . K/2 – [K/2 – 1] etc,

but we cannot in finite successive steps bridge down to k thence to 1 and 0.

Where, of course, we can reflect in the 0 point, through posing additive inverses and we may do the rotation i*[k] to get the complex span.

Of course, all of this is to be hedged about with the usual non standard restrictions, but here is a rough first pass look at the hyperreals, with catapult between the transfinite and the infinitesimals that are all but zero. Where the latter clearly have a hierarchy such that m^2 is far closer to 0 than m.

And, this is also very close to the surreals pincer game, where after w steps we can constrict a continuum through in effect implying that a real is a power series sum that converges to a particular value, pi or e etc. then, go beyond, we are already in the domain of supertasks so just continue the logic to the transfinitely large domain, ending up with that grand class.

Coming back, DS we are here revisiting issues of three years past was it: step along mile posts back to the singularity as the zeroth stage, then beyond as conceived as a quasi-physical temporal causal domain with prior stages giving rise to successors. We may succeed in finite steps from any finitely remote -k to -1 to 0 and to some now n, but we have no warrant for descent from some [transefinite] hyperreal remote past stage – K as the descent in finite steps, unit steps, from there will never span to -k. That is, there is no warrant for a proposed transfinite quasi-physical, causal-temporal successive past of our observed cosmos and its causal antecedents.

Going back to the focus, if 0 is surrounded by an infinitesimal cloud closer than any k in R can give by taking 1/k, but which we may attain to by taking 1/K in *R, the hyperreals, then by simple vector transfer along the line, any real, r, will be similarly surrounded by such a cloud. For, (r + m) is in the extended continuum, but is closer than any (r + 1/k) can give where k is in R.

The concept, continuum is strange indeed, stranger than we can conceive of.

So, now, we may come back up to ponder the derivative.

If a valid, all but zero number or quantity exists, then — I am here exploring the logic of structure and quantity, I am not decreeing some imagined absolute conclusion as though I were omniscient and free of possibility of error — we may conceive of taking a ratio of two such quantities, called dy and dx, where this further implies an operation of approach to zero increment. The ratio dy/dx then is much as conceived and h = [(x +h) – x] is numerically dx.

But dx is at the same time a matter of an operation of difference as difference trends to zero, so it is not conceptually identical

Going to the numerator, with f(x), the difference dy is again an operation but is constrained by being bound to x, we must take the increment h in x to identify the increment in f(x), i.e. the functional relationship is thus bound into the expression. This is not a free procedure.

Going to a yet higher operation, we have now identified that a flow-function f ‘(x) is bound to the function f(x) and to x, all playing continuum games as we move in and out by some infinitesimal order increment h as h trends to zero. Obviously, f ‘(x) and f ”(x) can and do take definite values as f(x) also does, when x varies. So, we see operations as one aspect and we see functions as another, all bound together.

And of course the D-notation as extended also allows us to remember that operations accept pre-image functions and yield image functions. Down that road lies a different perspective on arithmetical, algebraic, analytical and many other operations including of course the vector-differential operations and energy-potential operations [Hamiltonian] that are so powerful in electromagnetism, fluid dynamics, q-mech etc.

Coming back, JB seems to be suggesting, that under x, y and other quasi-spatial variables lies another, tied to the temporal-causal domain, time. Classically, viewed as flowing somehow uniformly at a steady rate accessible all at once everywhere. dt/dt = 1 by definition. From this, we may conceive of a state space trajectory for some entity of interest p, p(x,y,z . . . t). At any given locus in the domain, we have a state and as t varies there is a trajectory. x and y etc are now dependent.

This brings out the force of JB’s onward remark to H:

if x *is* the independent variable, and there is no possibility of x being dependent on something else, then d^2x (i.e., d(d(x))) IS zero

Our simple picture breaks if x is no longer lord of all it surveys.

Ooooopsie . . .

Trouble.

As, going further, we now must reckon with spacetime and with warped spacetime due to presence of massive objects, indeed up to outright tearing the fabric at the event horizon of a black hole. Spacetime is complicated.

A space variable is now locked into a cluster of very hairy issues, with a classical limiting case. >>

The world just got a lot more complicated than hitherto we may have thought. END

F/N: I add (Apr 15) a link (HT: Wiki) on Faa di Bruno’s formula for the nth order derivative of a function of a function, which extends the well-known chain rule:

HT: Wikipedia

Hat tip MathWorld, here are the first few results:

NB, this is referenced by JB in his discussion.

PS: As one implication, let us go to Davies and Walker:

In physics, particularly in statistical mechanics, we base many of our calculations on the assumption of metric transitivity, which asserts that a system’s trajectory will eventually [–> given “enough time and search resources”] explore the entirety of its state space – thus everything that is phys-ically possible will eventually happen. It should then be trivially true that one could choose an arbitrary “final state” (e.g., a living organism) and “explain” it by evolving the system backwards in time choosing an appropriate state at some ’start’ time t_0 (fine-tuning the initial state). In the case of a chaotic system the initial state must be specified to arbitrarily high precision. But this account amounts to no more than saying that the world is as it is because it was as it was, and our current narrative therefore scarcely constitutes an explanation in the true scientific sense. We are left in a bit of a conundrum with respect to the problem of specifying the initial conditions necessary to explain our world. A key point is that if we require specialness in our initial state (such that we observe the current state of the world and not any other state) metric transitivity cannot hold true, as it blurs any dependency on initial conditions – that is, it makes little sense for us to single out any particular state as special by calling it the ’initial’ state. If we instead relax the assumption of metric transitivity (which seems more realistic for many real world physical systems – including life), then our phase space will consist of isolated pocket regions and it is not necessarily possible to get to any other physically possible state (see e.g. Fig. 1 for a cellular automata example).

[–> or, there may not be “enough” time and/or resources for the relevant exploration, i.e. we see the 500 – 1,000 bit complexity threshold at work vs 10^57 – 10^80 atoms with fast rxn rates at about 10^-13 to 10^-15 s leading to inability to explore more than a vanishingly small fraction on the gamut of Sol system or observed cosmos . . . the only actually, credibly observed cosmos]

Thus the initial state must be tuned to be in the region of phase space in which we find ourselves [–> notice, fine tuning], and there are regions of the configuration space our physical universe would be excluded from accessing, even if those states may be equally consistent and permissible under the microscopic laws of physics (starting from a different initial state). Thus according to the standard picture, we require special initial conditions to explain the complexity of the world, but also have a sense that we should not be on a particularly special trajectory to get here (or anywhere else) as it would be a sign of fine–tuning of the initial conditions. [ –> notice, the “loading”] Stated most simply, a potential problem with the way we currently formulate physics is that you can’t necessarily get everywhere from anywhere (see Walker [31] for discussion).

[“The “Hard Problem” of Life,” June 23, 2016, a discussion by Sara Imari Walker and Paul C.W. Davies at Arxiv.]

Yes, cosmological fine-tuning lurks under these considerations, given where statistical mechanics points.

F/N: J P Moreland comments:

250 Replies to “Logic & First Principles, 17: Pondering the Hyperreals *R with Prof Carol Wood (including Infinitesimals)

  1. 1
    kairosfocus says:

    Logic & First Principles, 17: Pondering the Hyperreals *R with Prof Carol Wood (including Infinitesimals)

  2. 2
    kairosfocus says:

    F/N: I have added to the discussion of hyperreals and derivatives, for reference, a clip on Faa di Bruno’s formula on the nth order derivative of a function of a function. Wikipedia often gives useful summaries on topics such as this. KF

    PS: Wolfram Mathworld: http://mathworld.wolfram.com/F.....rmula.html

    PPS: It seems the learned Padre taught a certain well-known Guiseppi Peano, of the well known axioms.

  3. 3
    kairosfocus says:

    It has been said, that philosophy begins with wonder (especially as we contemplate the sublime, the grand, the beautiful or sometimes, the puzzling, shocking or paradoxical). Here, we see the transfinite and the infinitesimal coupled, with zero coming onstage again as almost stealing the show; where we note that — following von Neumann — the grand scheme of structure and quantity spins out from zero. Strange as it seems, zero is here seen to be a pivotal quantity, it is not nothing. Nor should we overlook the wonder of the continuum, something we so often take for granted. KF

  4. 4
    kairosfocus says:

    F/N (and ATTN DS): The above gives rise to a question: which is more relevant to many mathematical considerations of interest, R (with N as mileposts in it) or *R (with *N as mileposts in it)? Of course, due hedges are appropriate.

    Especially, when by taking K in *N which is transfinite and catapulting through 1/K = m where m is less than 1/k for any identifiable finite value k in N, we now can conceptualise infinitesimals as connected to the transfinite extension. This re-opens a context in which we may responsibly — there are many hedges! — speak of infinitesimals and transfinites as legitimate numbers existing in a system of structured quantities we can address as hyperreals and may construct as surreals. Where, in maximalised forms we may legitimately identify an isomorphism.

    Moreover, as we already see that there is a transfinite order type w that succeeds the naturals and opens up the transfinites, we have good reason to hold that the hyperreal number space is credibly framework to any possible world W = {A|~A} in which A is some characteristic or feature that distinguishes W from near neighbours. This ties to the logic of being and possibility of being.

    The presence of infinitesimals of class m etc, then allows us to look at differentials and contemplate rates and accumulations in a “natural” way. Down this road obviously lies Calculus (and thence linked subjects).

    We can also now contemplate stepwise ascent from -K towards -k a finite negative integer and onward to -2, -1, 0 then upwards towards +k.

    This then entails interesting consequences: milepost stepwise descent from K cannot succeed in such cumulative successive increments of 1 to k, due to the transfinite span involved. The parallel result for -K and -k is patent by symmetry. Likewise ascent beyond +k in similar steps cannot attain to K precisely as this is transfinite also. This both allows us to mathematically conceive of a transfinite past of order K and indicates that descent from such in steps cannot be completed physically to some -k, taking the singularity as 0. This, due to the structure of the set and the implied supertask of spanning a transfinite range between -K and -k in milepost steps.

    But what about the negative mirror image of the naturals, the negative integers and zero?

    { . . . – k, . . . -2, -1, 0}

    Is there room in this for a transfinite actual past?

    Let us consider, -k stands here for an arbitrarily high magnitude finite value, and it is always bounded onward by -(k+1) etc. That is, any specifically identified negative integer is both finitely removed from 0 and is itself bounded by yet onward values. The ellipsis is a STRUCTURAL part of the set.

    So, any actual causal-temporal past stage we can count to or symbolise will be finitely remote from now (let’s call now n). We have no basis to speak of an actual identifiable past causal-temporal stage of the world that steps past the singularity at 0 to now n and which is transfinitely remote as a stage count from n.

    We can suggest indefinitely more stages than any particular past stage but cannot identify any such particular stage as transfinitely remote.

    This is subtle, but important: we cannot identify that any particular actualised past stage of the causal-temporal world is specifically transfinitely remote.

    Does the indefinite continuation beyond a particular k give room for saying or conceiving a transfinite past in the sense that indefinitely many finite, antecedent stages were already past at any given k?

    To this, I suggest, there is a difference between an abstract quantity and an actualised past stage that must through causal-temporal succession of countable finite stages, reach to n.

    This is the sense in which I would suggest that there is no warrant to speak of an actualised transfinite past.

    At least, that’s how it seems to me, revisiting an exchange from was it three years past.

    KF

    PS: One important point is the concept of standard vs non-standard numbers such that given say some r in R which has a hyperreals cloud around it, we may reduce values in the cloud to standard by going to the nearest standard number. The cloud around r is of course effectively the same as the cloud around 0, by simply making a vector transfer along the line, i.e. moving from 0 plus cloud, to r plus cloud shifted to r. Obviously for every m near 0, there is a -m as its additive inverse, i.e. the cloud is two-sided. Every member of the cloud around 0 is closer to 0 than any identifiable k in N transformed to 1/k.

    PPS: We may contemplate an orthogonal axis by using i* as a rotation operator and arriving at complex numbers as 2-dimensional vectors in the field so identified. Hyper-extensions would logically apply so we could do a *C with at least one hyperreal component.

  5. 5
    daveS says:

    KF,

    Does the indefinite continuation beyond a particular k give room for saying or conceiving a transfinite past in the sense that indefinitely many finite, antecedent stages were already past at any given k?

    This is a well-formed question, I believe, and does have a single correct answer (yes or no).

    To this, I suggest, there is a difference between an abstract quantity and an actualised past stage that must through causal-temporal succession of countable finite stages, reach to n.

    This is the sense in which I would suggest that there is no warrant to speak of an actualised transfinite past.

    There certainly is a difference between abstract quantities and events in the physical universe. However, we use abstract quantities to enumerate all sorts of concrete things. For example, we could meaningfully discuss an infinite arrangement of snooker balls or an infinitely long ladder, and whether either is mathematically or logically possible.

  6. 6
    hazel says:

    I don’t think we can assume that the number line is an accurate model of time as it applies to reality, especially as reality extends beyond our own universe, in whatever sense that might exist. Given all we have learned about time through both relativity and quantum mechanics, I think the old model of time as a linear background in respect to which events happens is very outdated. Therefore the question of whether an infinite past is possible isn’t an interesting question, because it’s based on an abstract model, the real number line, as a model for time that isn’t justified by either what we know about time in this universe nor by any evidence we have (which is none) about what time is like outside of, or before, this universe.

  7. 7
    ET says:

    An infinite past is not possible. We wouldn’t be here if there was an infinite past.

  8. 8
    daveS says:

    KF,

    ET presents the type of argument that I was claiming is faulty in those threads of a few years ago. For example, saying an infinite past is impossible because we would never be able to reach the present.

  9. 9
    ET says:

    LoL! Just saying it is faulty is a sure sign that it isn’t. But please do tell how we can reach the present from an infinite past. Then we can see the real faulty argument.

    But I am sure that no one will ever do so.

  10. 10
    daveS says:

    ET,

    First I have to see your actual argument (which I assumed exists).

  11. 11
    daveS says:

    hazel,

    I think you make a good point. The best we can do is ask whether these things are possible or not, given some assumptions about how time works and so on. Whether any of this reasoning actually tells us anything about the real world is not necessarily clear.

  12. 12
    ET says:

    LoL! You don’t have to see my argument in order to make your own.

    But please do tell how we can reach the present from an infinite past.

  13. 13
    daveS says:

    ET,

    You don’t have to see my argument in order to make your own. But please do tell how we can reach the present from an infinite past.

    I didn’t claim to have such an argument. Let’s have no burden-shifting in this thread, please. 😛

  14. 14
    kairosfocus says:

    H, note, I speak of successive, cumulative, temporal-causal stages of finite character (think years as an example but as we go beyond the singularity, that is not exactly applicable — I really have in mind phases of development of the cosmos and proto-cosmos whatever that is), as opposed to time as a continuum. KF

    PS: I have thermodynamics and grand cosmology in mind, such as is used to discuss the history of the cosmos.

  15. 15
    ET says:

    LoL! @ daves- Then please shut up about mine being a faulty argument. Clearly you cannot make that case. You ran your mouth now it is up to you to support it or retract it.

  16. 16
    hazel says:

    kf, you write, ” I really have in mind phases of development of the cosmos and proto-cosmos whatever that is), as opposed to time as a continuum.”

    And how do you know the real number line, in any sense, is an accurate model for that? We have absolutely no evidence for what is outside our universe, or what happened “before” the Big Bang.

    Your argument about the impossibility of an infinite past is based upon modeling it as “successive, cumulative, temporal-causal stages of finite character”, but you then say “that is not exactly applicable.” Yes, in fact, it might not be applicable at all.

  17. 17
    ET says:

    Anyone who thinks there was an infinite past needs to show how we got to the present. Good luck with that…

  18. 18
    ET says:

    The little one has informed me that the “Magic Bus” is how you get from an infinite past to the present. But the little one was puzzled when I couldn’t show where the infinite past was on any of our maps or globes.

    Now we are discussing the “we came from history?” part…

  19. 19
    daveS says:

    A relevant passage by William Lane Craig, which lays out some technicalities that arise in this debate. (link):

    The argument I gave takes P for granted—granted, that is, by the proponents of the infinitude of the past. They assume that if the past series of events is infinite, then the order type of that series is the order type of the negative numbers, or in the usual symbolism, ω*. Logically, that isn’t necessary. The series of past events could have the order type ω* + ω*, the order type exemplified by . . . , −3, −2, −1, . . . , −3, −2, −1. But it’s obvious why proponents of the infinite past don’t adopt this view: for then there are past events which lie at an infinite distance from the present. But then one could never traverse the infinite distance from, say, the first −3 to the second −3. That’s why proponents of the infinite past always insist that the existence of an infinite past doesn’t entail an infinitely distant starting point. The Tristram Shandy paradox about a man writing his autobiography so slowly that it takes him a year to record the events of a single day challenges the assumption that an infinite past would have the order type ω*. The only hope for proponents of the infinite past is to insist that the series of past events has the order type ω* so that every event lies at only a finite distance from the present. In that way, forming an infinite past by successive addition doesn’t involve, they claim, traversing an infinite distance.

  20. 20
    hazel says:

    I am going to jump into this “infinite past” discussion, with the following disclaimers:

    1. I am discussing this as a purely mathematical issue.

    As I have explained in 6 and 16, this has ABSOLUTELY NOTHING TO DO WITH REALITY AS IT MIGHT EXTEND BEYOND OR BEFORE OUR UNIVERSE. Hope that’s clear.

    2. I am going to use the real number line, and in fact the integers only, as my model. This has nothing to do with hyperreals.

    3. Given a point P, I will call numbers to the right the “future” and numbers to the left the “past”, although, as explained in disclaimer 1, this is just a mathematical convenience and not meant to imply anything about reality.

    First, let’s talk about the natural numbers N. We say that the set of naturals is infinite because given any natural number k we can find a larger number k + 1. The set of naturals can be extended indefinitely, and thus we say the set of natural numbers is infinite. That is, given point P, we could say that the future is infinite because no matter how many steps into the future you go from P, you can always go farther.

    Now assume you are at point P = 1000, where P is the present. Person Q(1) starts at k = -1000 and walks to you: he started in the past and reached the Present in 2000 steps.

    Person Q(2) started at k = -1 billion, in your past, and walked to you in 1,000,001,000 steps. He started in the past and indeed reached the Present, eventually.

    Let’s be more abstract. Person Q(k) started at -k, and thus reaches the Present in k + 1000 steps.

    Is there any limit to k? No. No matter how far in the past person Q(k) might have started, person Q(k + 1) started father in the past.

    Therefore the points at which Q(k) might have started can be extended indefinitely into the past.

    Therefore, the past is infinite just as the future is infinite: There is no positive limit to how far in the future you might go, and there is no negative limit in the past from which you might have come.

    I think the confusion about all this involves the notion of “traversing” an infinite number of numbers. The future is infinite even though that is not a claim that anyone could completely visit all of them.
    ——-
    After seeing Dave’s post:

    I think the confusion about all this involves the notion of “traversing” an infinite number of numbers. The future (that is, the set of natural numbers starting at point P) is infinite even though that is not a claim that anyone could completely visit all of them. However, you can traverse as many as you need to get to any particular number in the future.

    Similarly, the past is infinite because there is no limit to how far back one might have started, but that is not a claim that for any starting place, one has traversed all the points to the left. It is just a claim that you could have started farther left, so there is no place so far in the past that you couldn’t have started there.

    “Infinity” is not a place where one could go in the future, nor one from where came from in the past. It is not a place at all. Every possible point Q in the past is a finite distance from 0, so no matter where you start, you only have to traverse a finite number of steps to get to the present.

  21. 21
    ET says:

    Oh my. Yes if you start from an arbitrary 0 then you can feign an infinite past.

  22. 22
    kairosfocus says:

    H, I am simply counting successive, causal-temporal eras or stages — with big bang as zero point (with room for stages antecedent to it as in there’s room for a super cosmos or multiverse here), not laying out a chronologically linear timeline, though that can be done (also see my discussion). Part of why I am counting stages, is that beyond the singularity — as is commonly suggested — there is not a simple basis of observations to calibrate in years by contrast with Hubble expansion and things like H-R diagrams for star clusters. See this discussion. That is strong enough to apply a numerical count as quantisation. The issues raised then apply. KF

    PS: Observe also the discussion appended to the OP, by Davies and Walker. Notice, how statistical mechanicsnaturally leads to a temporally-causally unfolding phase process in phase space as a very general result. Notice, the point on how fine tuning of original state comes out.

  23. 23
    kairosfocus says:

    F/N: For convenience, let me post the clip:

    In physics, particularly in statistical mechanics, we base many of our calculations on the assumption of metric transitivity, which asserts that a system’s trajectory will eventually [–> given “enough time and search resources”] explore the entirety of its state space – thus everything that is phys-ically possible will eventually happen. It should then be trivially true that one could choose an arbitrary “final state” (e.g., a living organism) and “explain” it by evolving the system backwards in time choosing an appropriate state at some ’start’ time t_0 (fine-tuning the initial state). In the case of a chaotic system the initial state must be specified to arbitrarily high precision. But this account amounts to no more than saying that the world is as it is because it was as it was, and our current narrative therefore scarcely constitutes an explanation in the true scientific sense.

    We are left in a bit of a conundrum with respect to the problem of specifying the initial conditions necessary to explain our world. A key point is that if we require specialness in our initial state (such that we observe the current state of the world and not any other state) metric transitivity cannot hold true, as it blurs any dependency on initial conditions – that is, it makes little sense for us to single out any particular state as special by calling it the ’initial’ state. If we instead relax the assumption of metric transitivity (which seems more realistic for many real world physical systems – including life), then our phase space will consist of isolated pocket regions and it is not necessarily possible to get to any other physically possible state (see e.g. Fig. 1 for a cellular automata example).

    [–> or, there may not be “enough” time and/or resources for the relevant exploration, i.e. we see the 500 – 1,000 bit complexity threshold at work vs 10^57 – 10^80 atoms with fast rxn rates at about 10^-13 to 10^-15 s leading to inability to explore more than a vanishingly small fraction on the gamut of Sol system or observed cosmos . . . the only actually, credibly observed cosmos]

    Thus the initial state must be tuned to be in the region of phase space in which we find ourselves [–> notice, fine tuning], and there are regions of the configuration space our physical universe would be excluded from accessing, even if those states may be equally consistent and permissible under the microscopic laws of physics (starting from a different initial state). Thus according to the standard picture, we require special initial conditions to explain the complexity of the world, but also have a sense that we should not be on a particularly special trajectory to get here (or anywhere else) as it would be a sign of fine–tuning of the initial conditions. [ –> notice, the “loading”] Stated most simply, a potential problem with the way we currently formulate physics is that you can’t necessarily get everywhere from anywhere (see Walker [31] for discussion). [“The “Hard Problem” of Life,” June 23, 2016, a discussion by Sara Imari Walker and Paul C.W. Davies at Arxiv.]

    KF

  24. 24
    kairosfocus says:

    H, 20:

    “Infinity” is not a place where one could go in the future, nor one from where came from in the past. It is not a place at all. Every possible point Q in the past is a finite distance from 0, so no matter where you start, you only have to traverse a finite number of steps to get to the present.

    See the problem?

    Also, if there is a metric that effectively counts causally successive steps, if every actually past event or stage Q is finitely remote, we have no warrant to infer transfinitely remote times, or a cumulatively transfinite duration descending to the present.

    The challenge of traversing the transfinite since any particular past event or stage poses a supertask.

    There is no actual past Q that does not involve durations to now that must causally, stepwise succeed to now.

    If one claims a transfinite past, it is very hard to avoid the comment that one implies actual past stages Q that are remote in the sense of -K as opposed to – k as was discussed above. (Which does use the hyperreals to provide a conceptual context.)

    KF

  25. 25
    daveS says:

    KF,

    Also, if there is a metric that effectively counts causally successive steps, if every actually past event or stage Q is finitely remote, we have no warrant to infer transfinitely remote times, or a cumulatively transfinite duration descending to the present.

    Whether this is true or not, no one here is inferring a cumulatively transfinite duration descending to the present.

    Various people have claimed that such is impossible, on the other hand. To which I say, let’s see the argument.

  26. 26
    hazel says:

    kf, are you using the words “transfinite” and “infinite” as synonyms?

  27. 27
    kairosfocus says:

    DS, the claim infinite actual past pretty much entails a cumulatively transfinite succession of past stages to the present, which implies appropriate durations since past points and now. If the overall process is transfinite it implies transfinitely remote past stages with the transfinite durations since them to now. There is no duration of the whole above and beyond durations to actual specific, distinct, once present, now succeeded past stages. Where the structure of the temporal-causal order is such that there is a cumulative, causally successive process that yields the present out of what is now the past. I do not think that we can appeal to there being an indefinitely continued set such as the set of numbers provides so beyond any k there are k+1 etc, mirrored on the negative side.The span from ANY finitely remote actual past point to now is just that, finite. To have a transfinite span, it looks much like needing a specific past stage which is transfinitely remote or else we may be conflating abstract set properties to very concrete durations since very concrete causally successive past stages. KF

  28. 28
    daveS says:

    KF,

    No one is inferring an actual infinite past either. Several have claimed that an infinite past is impossible, however. That’s how this whole thing got started. (see Laszlo Bencze).

  29. 29
    kairosfocus says:

    H, transfinite is a description of a property, beyond any finite limit. KF

    PS: Wiki notes:

    Transfinite numbers are numbers that are “infinite” in the sense that they are larger than all finite numbers, yet not necessarily absolutely infinite. The term transfinite was coined by Georg Cantor, who wished to avoid some of the implications of the word infinite in connection with these objects, which were, nevertheless, not finite. Few contemporary writers share these qualms; it is now accepted usage to refer to transfinite cardinals and ordinals as “infinite”. However, the term “transfinite” also remains in use.

  30. 30
    hazel says:

    I read Wikipedia also. My question is, given that I at least have restricted this discussion to the real numbers, are you using the words “transfinite” and “infinite” as synonyms?

  31. 31
    kairosfocus says:

    H, I stated the property I am speaking of, beyond any finite bound. I use “infinite” in the same sense as a general rule, but strongly prefer the more precise mathematically specific term, transfinite. Contrast, the Athanasian Creed, Latin form: immensus in the sense of beyond measure i.e. specific finite limit, often rendered infinite in English. Do you know of contexts where there is a relevant difference between the two terms in studying the logic of structure and quantity? KF

  32. 32
    kairosfocus says:

    DS, 28:

    No one is inferring an actual infinite past either

    I presume you mean, that no one accepts that there was an actual past stage that we can apply some duration metric to and conclude it was remote of order K such that the duration to K from n, now, is transfinite per a count metric or any other relevant metric.

    That’s an important agreement, if this is what you mean.

    The next issue is then: does the claim that there was an infinite or transfinite past that actually occurred IMPLY that there was a past time remote of order K?

    Applied to this, I note that duration between times or stages s1 and s2 is specific to the stages, it does not extend beyond the stages.

    So, arguably, if there is a past that is cumulatively transfinite in character, it then has in it time points s1 such that duration to now s2 = n, will be of order K on some relevant metric (one that uses finite stages or units).

    Notice, I have put forward that if we have warrant only to speak of finitely remote past stages then we have no warrant to speak of a past of transfinite duration to now.

    KF

  33. 33
    hazel says:

    P = The set of natural numbers is infinite.
    Q = The set of natural numbers is transfinite.

    Do those two sentences mean exactly the same thing?

  34. 34
    daveS says:

    KF,

    I presume you mean, that no one accepts that there was an actual past stage that we can apply some duration metric to and conclude it was remote of order K such that the duration to K from n, now, is transfinite per a count metric or any other relevant metric.

    I actually meant that no one here has inferred that the past is indeed infinite. I’m just commenting on proposed arguments purporting to show an infinite past is impossible. I’ll remain neutral on the above statement, which WLC discusses above.

    The next issue is then: does the claim that there was an infinite or transfinite past that actually occurred IMPLY that there was a past time remote of order K?

    Not that I am aware of.

  35. 35
    hazel says:

    kf, is K a hyperreal as per the video? If so, since I’m limiting my discussion to the reals, I’ll quit trying to understand what you’re saying.

  36. 36
    kairosfocus says:

    H, observe the von Neumann construction I noted in the OP (and compare the diagram of the surreals constructuion process also in the OP). You will see that I identify the unlimited succession of naturals, and that the order type of the set as a whole is omega (for convenience w). Thus, the cardinality of the set is transfinite, specifically aleph-null. Likewise, I took time to include an old illustration, where pink and blue tapes effectively print the naturals on a given pitch. Slide pink to some arbitrarily large, finite k on the blue. A one to one correspondence continues onward without limit as may be seen from the subscripts for k on. This entails that the set is transfinite or infinite in the sense commonly used by Mathematicians. I note that this includes that for any particular natural we can state (in decimal form) or represent, k, the number will be finite and bounded by k+1 etc. Thus, part of the STRUCTURE of the set is that it continues indefinitely, is not finitely bound as a whole, there is no last finite natural we may identify or represent, f, that is such that f+1 = w. Instead, w is the order type of the structured set including that property of k+1 etc continuation. Thus, in the sense that N is transfinite being the primary sense, that it is infinite (meaning here, the same thing) is a synonym, though infinite may be used in other ways with other connotations that may colour our perceptions even in Mathematics. Notice, in the OP and comments, I have pointed out that the hyperreals may be a more relevant context for our pondering, effectively serving as extensions of N, Z, Q, R that allow us to freely range to transfinite values like K that exceed any particular finite natural k such that, catapulting m = 1/K is less — is in a closer neighbourhood to 0 — than any n = 1/k.In this sense, m and its cloud of kindred infinitesimals is infinitely small, but of course not quite zero. Catapult back out again to the transfinites through 1/m = K being larger than any 1/n = k, k a particular typical LARGE finite, natural number. I do think the relationships being explored by case study help to clarify the sense of key terms. But then, I am a Constructivist in education-linked thought on the nature of understanding, was one due to Skemp decades before I knew of such a named school of thought. KF

  37. 37
    kairosfocus says:

    H, the specific context of discussion involves both reals and hyperreals, where Dr Wood used K to denote a typical transfinite hyperreal and I have followed that usage. I then used k to symbolise a typical finite natural of large size and r for a similar real, pegged to fit in with naturals. I note that the naturals then integers act as mileposts, and can be extended to hyperreals such as K. Infinitesimals such as m = 1/K, cannot be integers. If we loosen up k to the case k = 1, 1/k will be an integer, 1.

    Note, from OP:

    In effect, using Model Theory (thus a fair amount of protective hedging!) or other approaches, one may propose an “extension” of the Naturals and the Reals, often N* or R* — but we will use *N and *R as that is more conveniently “hyper-“.

    Such a new logic model world — the hyperreals — gives us a way to handle transfinites in a way that is intimately connected to the Reals (with Naturals as regular “mileposts”). As one effect, we here circumvent the question, are there infinitely many naturals, all of which are finite by extending the zone. So now, which version of the counting numbers are we “really” speaking of, N or *N? In the latter, we may freely discuss a K that exceeds any particular k in N reached stepwise from 0. That is, some k = 1 + 1 + . . . 1, k times added and is followed by k+1 etc.

    And BTW, the question of what a countable infinite means thus surfaces: beyond any finite bound. That is, for any finite k we represent in N, no matter how large, we may succeed it, k+1, k+2 etc, and in effect we may shift a tape starting 0,1,2 etc up to k, k+1, k+2 etc and see that the two “tapes” continue without end in 1:1 correspondence

    KF

    PS: I noted, that Dr Wood’s mentor, Abraham Robinson, seems to have shown that in a maximal sense hyperreals and Surreals are isomorphic, opening up all sorts of interesting numbers great and small.

  38. 38
    hazel says:

    Fine. Then, as I said, I’ll bow out if we aren’t limiting this discussion about “the past” to the reals.

  39. 39
    kairosfocus says:

    H, the hyperreals are part of the context relevant to Calculus and to the exchange on what it means to have a transfinite past. KF

  40. 40
    daveS says:

    It’s not clear why the hyperreal numbers have to be in this conversation. As WLC says:

    That’s why proponents of the infinite past always insist that the existence of an infinite past doesn’t entail an infinitely distant starting point.

    The only hope for proponents of the infinite past is to insist that the series of past events has the order type ω* so that every event lies at only a finite distance from the present.

  41. 41
    kairosfocus says:

    DS, the hyperreals were there all along (even way back), and are part of the wider focus of the OP, as tools for thinking. Notice, there is an appendix on nth order derivatives? KF

  42. 42
    hazel says:

    Good quote, Dave. I am absolutely not interested in whether this has anything to do with the real world (see posts 6 and 16), (nor with the hyperreals in this context), but the two sentences you quoted express my view, I think: just as all positive numbers in the future are a finite distance from now, even though the future is infinite, all negative numbers in the past are a finite distance away, even though the past is infinite.

    Above I wrote, “Every possible point Q in the past is a finite distance from 0, so no matter where you start, you only have to traverse a finite number of steps to get to the present.”

    Kf replied by bolding the word “start”, and said “See the problem?”

    But that misses the “no matter where” part. Since the starting point can always be moved back, the possible starting place for the past is infinite in that it can always be placed further back than any particular distance. If someone says, did the past start at k, one can say, “no it started at k -1”. Since the start of the past can’t be assigned to any particular number, and can also be considered further in the past than any particular number, it makes sense to say the past is infinite.

    It really boils down to the statement that there are an infinite number of integers, in either direction, and each one of them is a finite number.

  43. 43
    daveS says:

    KF,

    Yes, but you’ve been importing the hyperreals into the infinite past discussion. The ‘proponents’ of an infinite past do not speak of time coordinates/stages with infinite separation. Therefore if you want to critique their position, you should stick with a system where all coordinates/stages have finite separation. Such as the real number system, for example.

  44. 44
    kairosfocus says:

    H,

    just as all positive numbers in the future are a finite distance from now, even though the future is infinite, all negative numbers in the past are a finite distance away, even though the past is infinite

    This is the precise problem.

    Going forward from now (or even since the singularity) actual future stages are always, ever, finite. The “infinity” in view has to do with an open, continuous future, one that is strictly potentially infinite not actualised infinite.

    Why is that? Because precisely a finite stepwise causal-temporal accumulation never actually completes a transfinite span or duration.

    (And in fact, the heat death thesis would suggest an effective cessation of time; thermodynamically connected matter in an isolated cosmos faces ultimate degradation of energy until the reserves of a universe are used up. A potentially infinite future points to an open universe with an eternal source. But we may set that aside for now.)

    That such a traverse is not completed going forward from now also strongly indicates that it has not been up to now.

    Which is before we get to observationally supported physics which points to a finitely remote singularity as start for the only universe we actually observe.

    Going further, the actual (not conceptual) past is a causally connected cumulative causal chain. The once present gives rise to its successor down to now. Duration in time is only ever the span as measured between successive stages. This is what brings up the asymmetry that breaks your just as A, B argument.

    The past had to be actually completed stage by stage in the forward direction, and holds duration between stages measurable as the time between them. Were there a transfinite actual past, a transfinite span was obviously spanned, save, it cannot.

    What is left is to suggest that negative integers never have a transfinitely remote particular value. So, perhaps, the past infinity lies in an indefinite, limitless prior succession up to any given stage.

    The problem is, this must address actually completed stepwise successive stages and where duration to now from any particular stage will be finite. There is no warrant for a transfinite duration of the past succeeded in stages that are everywhere finitely remote.

    The warranted conclusion is that the traversed past to now is of finite duration and arguably the future is potentially infinite.

    Where, reminder, the actual observational base for an indefinitely expended past beyond the singularity is nil.

    KF

  45. 45
    kairosfocus says:

    DS, do they instead IMPLY such despite their denial, given that the span of time between ANY two definite finitely separated stages is finite? KF

  46. 46
    hazel says:

    Not interested in the real world hypothesis, kf. Please see 6 and 16.

  47. 47
    daveS says:

    KF,

    If I understand your question, no.

  48. 48
    kairosfocus says:

    F/N: The core issue being raised to WLC, which he is answering:

    (P) IF (i) the temporal series of all past events is actually infinite in its duration (as measured by equal temporal intervals), THEN (ii) there COULD be some mind/clock/counting machine/computer/angel/god which would SUCCESSIVELY pair all the past equal intervals (say, seconds) to all negative whole numbers in the corresponding order.

    Notice, this would imply that the past countable stages are of order w, where any particular definite stage -k will be a corresponding count of magnitude k in the past. Where of course, { . . . -k, . . . -2, -1, 0} is transfinite leftwards. Consequently, it is implied that at any -k finitely removed from us, the transfinite causal-temporal succession of stages up to -k in the past has already happened; as can be seen by taking a leftward mirror of the pink vs blue ribbon tapes in the OP. Craig goes on to summarise:

    The only hope for proponents of the infinite past is to insist that the series of past events has the order type ?* so that every event lies at only a finite distance from the present. In that way, forming an infinite past by successive addition doesn’t involve, they claim, traversing an infinite distance.

    Now, let us note his comment on a related question:

    https://www.reasonablefaith.org/writings/question-answer/is-a-beginningless-past-actually-infinite/

    . . . It might be said that at least there have been past events, and so they can be numbered. But by the same token there will be future events, so why can they not be numbered? Accordingly, one might be tempted to say that in an endless future there will be an actually infinite number of events, just as in a beginningless past there have been an actually infinite number of events. But in a sense that assertion is false; for there never will be an actually infinite number of events, since it is impossible to count to infinity. The only sense in which there will be an infinite number of events is that the series of events will go toward infinity as a limit.

    But that is the concept of a potential infinite, not an actual infinite. Here the objectivity of temporal becoming makes itself felt. For as a result of the arrow of time, the series of events later than any arbitrarily selected past event is properly to be regarded as potentially infinite, that is to say, finite but indefinitely increasing toward infinity as a limit. The situation, significantly, is not symmetrical: as we have seen, the series of events earlier than any arbitrarily selected future event cannot properly be regarded as potentially infinite. So when we say that the number of past events is infinite, we mean that prior to today ?0 events have elapsed. But when we say that the number of future events is infinite, we do not mean that ?0 events will elapse, for that is false. [emphasis mine]

    Similarly, another question poses:

    https://www.reasonablefaith.org/writings/question-answer/forming-an-actual-infinite-by-successive-addition/

    . . . I have a question concerning one of the philosophical arguments you offer in support of the view that the universe began to exist, namely the argument from the impossibility of forming an actual infinite by successive addition. You set up the argument as follows:

    1. A collection formed by successive addition cannot be actually infinite.
    2. The temporal series of past events is a collection formed by successive addition.
    3. Therefore, the temporal series of past events cannot be actually infinite.

    This argument exposes a feature in the notion of an infinite series of events that I find bewildering. To set the situation up, we’ll assume the past is infinite. By virtue of a tensed conception of time, every event in the infinite past up to the present was a real event that had to be “lived” through. But, if that’s the case, how could all those events have been lived through, one by one, up till now? Just how, exactly, could we reach the end of that beginningless series? How could the present event arrive if, before it could arrive, an infinite number of previous events had to arrive first?

    Like I said, this seems very puzzling. But I can’t quite put my finger on why. Is it simply that, on an intuitive level, I find the idea of traversing a beginningless series absurd? As you wrote in your reply to John Taylor, “The question is whether an infinite series of events, having no beginning and having an ending in the present, is metaphysically possible given a tensed view of time. Intuitively, this does not seem possible, for it seems that the present event could not arrive if its arrival had to be preceded by the successive arrival of an infinite number of prior events.” [“A Swift and Simple Refutation of the Kalam Cosmological Argument?” Religious Studies 35 (1999): 57-72. Footnote 26] This is exactly what impels me to accept the argument. But is there a way to analyze our intuition more deeply to find out exactly why such a traversal is impossible? Or is it somehow non-analyzable?

    The “traditional objection” to this argument is that it is only impossible to traverse infinity if one begins at some point. But, whatever this reply manages to do, it doesn’t seem to rebut our intuition or reduce the apparent absurdity engendered by the situation; after considering the objection, I’m still genuinely perplexed as to how such a traversal could happen.

    In his answer we find:

    . . . In the case of beginning with some finite quantity and adding finite quantities to it we can pinpoint the problem clearly: since any finite quantity plus another finite quantity is always a finite quantity, we shall never arrive at infinity even if we keep on adding forever. Infinity in this case serves merely as a limit which we never attain.

    What becomes truly puzzling, even mind-boggling, is the suggestion that we can, by adding only finite quantities, form an infinite quantity or collection–say, a certain collection of baseball cards–by never beginning but ending at some time! Here the impossibility cannot be analyzed as due to the impossibility of adding finite quantities to finite quantities and getting an infinite quantity, for in this case the quantity to which finite additions are being made is always and already infinite. We are successively adding finite quantities to an already infinite quantity, so of course the sum is an infinite quantity. Here infinity is not functioning as a mere limit but as a collection of concrete elements.

    Now notice that one still hasn’t explained how we are able to form our infinite collection of baseball cards by successive addition. For at any time in the past the collection is already infinite, and yet the total collection has not yet been formed . . . . Here’s the problem, it seems to me: in order for the collection [of an infinite number of successively added baseball cards] to be completed, we must have already enumerated, one at a time, an infinite number of previous cards. But before the final card could be added, the card immediately prior to it would have to be added; and before that card could be added, the card immediately prior to it would have to be added; and so on ad infinitum. So one gets driven back and back into the infinite past, making it impossible for any card to be added to the collection . . . .

    About the best that the critic of the argument can do at this point, I think, is to say that if one adds cards at a rate of, say, one card per second, then the collection can be completed because there has been an infinite number of seconds in the beginningless past. But clearly this response only pushes the problem back a notch: for the question then is, how can the infinite collection of past seconds be formed by successive addition? For before the present second could elapse, the one before it would have to elapse, and so on, as before. Because the problem is applicable to time itself, it cannot be resolved by appealing to infinite past time.

    Of course, proponents of a static or tenseless theory of time will deny that moments of time really do elapse, but then their objection is actually to premiss (2), not premiss (1).

    If one is not yet convinced by this argument, then I would offer a further defense of premiss (1) by arguing that if an actual infinite could be formed by succesive addition, then various absurditites would result . . . .

    Suppose we meet a man who claims to have been counting down from infinity and who is now finishing: . . ., -3, -2, -1, 0. We could ask, why didn’t he finish counting yesterday or the day before or the year before? By then an infinite time had already elapsed, so that he should already have finished. Thus, at no point in the infinite past could we ever find the man finishing his countdown, for by that point he should already be done! In fact, no matter how far back into the past we go, we can never find the man counting at all, for at any point we reach he will already have finished. But if at no point in the past do we find him counting, this contradicts the hypothesis that he has been counting from eternity. This shows again that the formation of an actual infinite by never beginning but reaching an end is as impossible as beginning at a point and trying to reach infinity.

    Thus, we see in effect a begging of the question by inferring that at any -k prior no now (set that n = 0), the succession involving the transfinite has already occurred. But, that was the problem, how is that so, how could it be feasible without a duration between some past actual stage Q and n being itself transfinite thus IMPLICITLY — as opposed to explicitly — requiring traversing a transfinite in successive steps? Saying that there is an infinity of finite succession, with the duration between any two events or stages t1 and t2 being finite only, seems to be dubious, even contradictory.

    I think instead, it is first reasonable to argue that we have no warrant to claim a transfinite actual past that would not involve an actually transfinite duration to now. Where, that would imply precisely the spanning the transfinite duration in successive stages that is a supertask. Instead, it seems to me, we are only warranted to speak of a finite span of succession between any two stages t1 and t2. This, implying that we are warranted only to speak of a finite actual past, and of course of a potentially infinite future (ignoring for the moment the heat death issue).

    KF

  49. 49
    kairosfocus says:

    PS: He notes in another answer (having pointed to the Blackwell book on Natural Theology) How:

    https://www.reasonablefaith.org/writings/question-answer/muslim-appreciation-of-the-kalam-argument/

    . . . I published a paper a few years ago entitled “Why Is It Now?” Ratio 18 (2000): 115-122, in which I argued the same thing! If the universe is past-eternal, then why out of all the moments in time is 2012 now? Why not some earlier or later moment? I was intrigued that in his response to my recent lecture at the Sheldonian Theatre in Oxford, the philosopher Stephen Priest was struggling to articulate this same question, calling it one of the most profound philosophical questions. He misstated the question, I think, by asking, “Why is it now now?” That question is trivial, for when else could it be now than now? Of course, it is now now! But the question is really, “Why is 2012 now?” Postulating a beginning to time at least enables us to answer that it is now 2012 because that is how much time has elapsed since the beginning, an answer not available if the universe is past-eternal.

  50. 50
    kairosfocus says:

    PPS: I should note on the “arrow of time” view, the thermodynamic, temporal-causal view of what temporal succession implies or at least requires. Here, an event requires a change and some transfer of energy (which in the typical case will render the energy in the cosmos less available or at least constant, i.e. energy is always gradually dissipated from its concentrations). That is, temporal-causal succession is an energy-driven transaction that on the whole gradually deteriorates the available energy to do work, impose forced ordered motion so dW = F*dx, F being the acting force which moves or affects some entity through dx along its line of action. This is the physical concept of work and is connected to thermal energy conceived as random molecular level motions. The famous second law boils down to, energy cannot spontaneously be wholly reduced from random to ordered motion. This extends to, succession of stages and lapse of time through caused change, are rooted in energy flows and degradation. Successive stages are stages unfolding as events play out cumulatively. Where, as certain events seem quite regular (e.g. oscillatory cycles in certain masers) we may designate certain structures and linked regular, countable or continuous processes as clocks, then reckon time from their cumulative change, taking some zero-point as a reference start. Time and energetic processes are inextricably intertwined, now compounded through Einstein’s energy-time form of uncertainty and the effects of relativity including mass concentrations and distortions of the spacetime fabric. In this sense, time has a ratchet, forcing a direction of natural progress, ultimately headed for heat death it seems, if left to itself. On this, the claim, no first event, is a claim that effectively implies infinite concentrations of energy in some quasi-physical domain, presumably with our observed cosmos as a temporary bubble that will end in local ultimate degradation thus no energy available to drive clocks. Of such a quasi-physical, effectively infinite energy reservoir grand cosmos, we have no observational evidence. This is philosophy (with mathematical apparatus) not science. At that level of discourse, it is reasonable to posit the eternal deity as the infinite behind the finite that we see.

  51. 51
    kairosfocus says:

    DS, I haven’t been arbitrarily importing an extraneous issue, I have been taking an opportunity to discuss a key context relating to foundational issues of logic and first principles. In that context, I pointed to Calculus and the infinitesimals and to the issue of the hyperreals which then leads to the question of time. Where, the structure of quantities dealing with time as suggested infinite logically fits into the structure of the transfinite that has been developed. In that context, I am simply not satisfied that the claim that at any -k we already have completed the transfinite past settles the matter and it seems to me that we do have to face the issue of what duration from the past to now means, other than having a span from an actual past stage to now as the present stage. That is, duration is not relative to an ellipsis but to specific actual stages of time, which if they are only finitely removed from one another, points to a finite duration. In that context, if the only durations we can warrant are finite, then we face the issue that there is no warrant to speak of a transfinite (and of course unobserved, untraced) past. Where, were we to propose that the actual past is transfinite, then it is at least a serious question that there will be actual specific past events at transfinite remove from now. Such are best represented using hyperreals, here, -K. On such a remove being on the table, we then see the challenge to span the transfinite in finite stage steps. Indeed the attempt to suggest that there is an infinity of finitely removed past events, each of only finite duration to now but jointly constituting a transfinite past, is precisely because of the implications of that supertask. So no, the hyperreals do belong in the discussion as the way we can represent a claimed duration from past to now events that is transfinite. In that context, I put on the table that if we can only ever have duration as between specific events, then if all durations are finite, the overall past is finite as it is the maximum span of durations to now. Surely, durations can be ranked and we can conceive of a maximum, or at least of a case where every duration is finite, thus implying that there are no transfinite durations therefore there is no room for a transfinite past. the overall duration may be indefinite to us, but it would be finite. This is then multiplied by the issue (on physicalist views) of energy concentrations to drive change thus temporal-causal succession and so also time as we may measure or count it. Time is a dynamic that is coming from somewhere and that where is causal and thermodynamic. KF

  52. 52
    daveS says:

    KF,

    In that context, I put on the table that if we can only ever have duration as between specific events, then if all durations are finite, the overall past is finite as it is the maximum span of durations to now. Surely, durations can be ranked and we can conceive of a maximum, or at least of a case where every duration is finite, thus implying that there are no transfinite durations therefore there is no room for a transfinite past. the overall duration may be indefinite to us, but it would be finite.

    Consider the interval S = (−∞, 0] in the set of real numbers.

    For any two elements x and y of S, we can calculate the distance between them, d(x, y) = |x – y|. Note this is always a finite number.

    Does this function d have a maximum value? No. Given any positive integer n, we can find two real numbers in S such that |x – y| > n.

    Does S have finite length?

  53. 53
    kairosfocus says:

    DS, length of a line is in this case transfinite (in terms of cardinality — with naturals as mile-posts, looks to be aleph null), and that is tied to its unlimited nature leftwards as is symbolised by the open transfinite side of the interval — the L-ward line extends beyond any finite value we may symbolise, – k. That is the precise problem, the ellipsis implies unlimited duration and that is additional to how when we identify or represent particular finite values (e.g. x and y) the difference x – y is unsurprisingly, finite. That is part of why we need to bear in mind *R not just R. In R*, k – 0 is finite but K – 0 is not. k to 0 can be traversed in successive unit steps, K to 0 cannot. KF

  54. 54
    daveS says:

    KF,

    That is part of why we need to bear in mind *R not just R.

    I don’t understand your rationale. The models that infinite-past proponents put forward never include time/stage coordinates which are infinite. I guess you’re free to propose any model you choose, but then you’re talking about something different than what WLC, the infinite-past proponents, and myself are talking about.

  55. 55
    hazel says:

    Dave, the answer to your question to kf is that if you include *R, then there is no way the “past” (in the purely mathematical sense) could have started at some K because it is separated from the real numbers by an unbridgeable gap. This allows kf to avoid discussing the arguments you and I and others are making about the finite nature of any negative number k in the reals.

  56. 56
    kairosfocus says:

    DS and H, I have repeatedly pointed out per distinct identity, that duration since any event Q at say q to now, N at say n, will be specific to the lapse between those particular events, one that can be “milestoned” and counted to give a countable quantity. This means that if for every particular Q, the gap q – n is finite, so we have no warrant at all to speak of a net overall transfinite lapse between any finitely remote particular Q and N. Repeat, we are warranted to speak of a finite lapse but not a transfinite one. Sure, the natural counting numbers [our mileposts] are such that they form a structured set where for any k there is a k+1 etc which can be readily extended to the integers and used to milepost the reals, but all this does is it says that we have a structure that is indefinitely large and beyond our ability to reckon, which we give the cardinality aleph null to the milestones considered as a collective. This is a transfinite value, building in the structure of endless continuation. The problem of those who insist that we have an infinite span of duration within the reals is that duration since to now is always between definite specific points Q and N, where if Q is such that q is a particular natural it can be bounded by q+1 and is invariably finite. Therefore, we have no warrant here to discuss a transfinite cumulative lapse, absent some particular actual past point say S, that is itself transfinitely remote so that the span from s down to q is then involved with a transfinite span. Which we all agree cannot be bridged in finite step, cumulative stages. It thus seems to me that those who speak of an infinite past imply a transfinite past. They do not want or mean or intend it to be so, but that is immaterial. We do not have a span to the ellipsis in { . . ., – k, . . . -2, -1, 0, 1, 2, . . . n} as the ellipsis is not a particular value but a set property of continuation. I think what we see is that we are not warranted to assert an actual transfinitely remote past that has succeeded in causal temporal,. finite steps, to now. KF

    PS: I am not AVOIDING discussion of negative integers or reals, I am pointing out, yet again, that there is always a span between definite stages to get a duration Q to N, and that so long as Q is at a finite past point, it will be only finitely remote. So, we have no warrant to claim a transfinite span to any given Q in the integers or reals. Duration is, repeat, between particular specific points Q and N, not to the ellipsis which is not a stage.

  57. 57
    daveS says:

    KF,

    Take a look at the definition of the diameter of a subset of a metric space. That illustrates how the infinite past proponents are thinking.

    In particular, calculate the diameter of the set (−∞, 0] in the real numbers.

  58. 58
    kairosfocus says:

    DS, so long as the duration for any particular Q to N is definite and finite, we have no warrant to claim it is non-finite; where we speak of actual past events proceeding stepwise to now — a process that for each given event Q is potentially infinite (given that now is itself being succeeded) but yields q – n as finite at any given now-time N. We have constraints on the actual past and sucession to now driving the dynamics; we are not dealing with an abstraction but the path of cosmological history that has causal-temporal succession in steps from the past to now. We may not definitively know the number of cases Q as a matter of fact, but we know the characteristic of every case: a finite span to now. This is one case where an in common property counts. KF

  59. 59
    hazel says:

    Can one say that the “past started” at some point? No, because there is always a point previous to any other point.

    Therefore, the past didn’t start. If the past didn’t start, then we say it is infinite, just as we say the future has no end.

    No one is claiming that an infinite numbers of steps have been traversed. We (or at least me) are just claiming that it is perfectly correct to say the past had no beginning.

  60. 60
    daveS says:

    KF,

    DS, so long as the duration for any particular Q to N is definite and finite, we have no warrant to claim it is non-finite;

    Note we’re talking about ontology here, not epistemology. The past, if correctly modeled by (−∞, 0], is objectively either finite or infinite. So the question is which, correct?

  61. 61
    kairosfocus says:

    H,

    the past never started is a bald assertion. What is clear is that

    [a] every actual past stage had to once be the “now”

    [b] such temporally, causally gave rise to its successor stage

    [c] this cumulatively, in finite stage steps gave rise to now.

    [d] thus, the span from EVERY actual past event to now must be spanned in successive steps to now.

    [e] where, making a transfinite span is impossible, a supertask

    [f] the process of causal succession dissipates energy concentrations towards what is called heat death (which for the observed cosmos is to occur in finite time)

    [g] thus, too, we have no warrant for claiming a succession that is transfinite.

    KF

  62. 62
    hazel says:

    Here is a proof. Please point out any flaws.

    The past (as modeled here in the mathematical sense by the real numbers) is either finite or infinite.

    Claim: it is infiniteHere is a proof. Please point out any flaws.

    The past (as modeled here in the mathematical sense by the real numbers) is either finite or infinite.

    Claim: it is infinite

    Proof by contradiction.

    Assume the past is finite

    Then there is some farthest point k in the past that can be considered the beginning of the past

    However, for any k, k-1 is farther from the present than k

    Therefore, k is not the beginning.

    Therefore, this contradiction shows that our assumption that the past is finite is wrong

    That is, therefore the past has no beginning

    Therefore the past is infinite.

  63. 63
    asauber says:

    Hazel, your flaw is right in here:

    “Then there is some farthest point k in the past that can be considered the beginning of the past”

    The nature of math is that you can always perform operations because you are working in abstraction.

    The nature of time is not the nature of math.

    Andrew

  64. 64
    kairosfocus says:

    H, you inserted the bare, question-begging assertion of a prior stage k-1 for any given past stage k. That is decidedly not a given nor a necessity of being for physical entities as opposed to mathematical sets where succession extends without limit; we are specifically dealing with a contingent, causally sucessive physical cosmos, where the act of succession of stages dissipates energy concentrations in stage Q giving rise to Q+1; e.g. this is why oscillating universe models run out of steam after about 100 cycles IIRC. A past-infinite physical cosmos requires infinite energy concentrations if it is not to go to exhaustion of energy concentrations, heat death. IIRC 10^25 y is estimated for white dwarfs to cool down — and they don’t have the sort of distribution (they are dead small stars cooling down) that marks a very old cosmos. The issue is NOT whether we can assert transfinite spans of negative numbers but that consonant with heat death not being yet, every particular actual past stage must give rise to a chain of successors spanned to reach now. Where, it is generally accepted that this cannot involve stepwise spanning of a transfinite range. We do not have a warrant to speak of a transfinite span from any stage in the chain Q to now. KF

  65. 65
    kairosfocus says:

    AS, correct, we are dealing with temporal-causal succession of physical systems, with accompanying energy concentration dissipation. Stars use up H, He, etc down to when they get to Fe, then they die one way or another based on size. Relatively small ones give white dwarfs which then cool down over v. long but finite times. Bigger ones, we are looking at neutron stars and maybe black holes etc. The observed cosmos lacks the density to re-collapse, and more. We can have multiple generations of stars, but not forever, heat death is an issue. Abstracta such as numbers, don’t have that sort of constraint. KF

  66. 66
    daveS says:

    KF,

    Consider the family of propositions {P_n}. For each n > 0, P_n states that the universe existed prior to n stages ago. Each P_n is either true or false.

    Then the past is infinite iff every P_n is true.

    Is this incorrect?

  67. 67
    hazel says:

    kf writes,

    , you inserted the bare, question-begging assertion of a prior stage k-1 for any given past stage k. That is decidedly not a given nor a necessity of being for physical entities as opposed to mathematical sets where succession extends without limit; we are specifically dealing with a contingent, causally sucessive physical cosmos, …

    kf, I have made it perfectly clear (see posts 6 and 16), as well as the disclaimer at line 2 of post 62 that I am NOT, repeat NOT, talking about “physical entities”, but a purely mathematical understanding of the number line. You may be “specifically dealing with a contingent, causally successive physical cosmos”, but we we are not.

    If you don’t want to respond to the purely mathematical arguments, fine, but when addressing me, please show some awareness that you are talking about different things than I am talking about.

    So what do you think about my argument at 62, from a mathematical point of view?

  68. 68
    hazel says:

    I see that post 62 has some duplicated lines. That’s my fault. Here’s a clean version:
    ===========
    Here is a proof. Please point out any flaws.

    The past (as modeled here in the mathematical sense by the real numbers) is either finite or infinite.

    Claim: it is infinite

    Proof by contradiction.

    Assume the past is finite

    Then there is some farthest point k in the past that can be considered the beginning of the past

    However, for any k, k-1 is farther from the present than k

    Therefore, k is not the beginning.

    Therefore, this contradiction shows that our assumption that the past is finite is wrong

    That is, therefore the past has no beginning

    Therefore the past is infinite.

  69. 69
    kairosfocus says:

    H, time is not a mathematical abstraction but a physical entity, connected to energy flows, dissipation of energy concentrations and more. To retain accuracy to reality, that nature of time is a factor. KF

  70. 70
    kairosfocus says:

    DS, an abstract logical alternative fails to come to grips with need for actuality and causal temporal stepwise succession to now. The issue is not directly, is there a first stage, but can we have an actual stage Q with stepwise succession to N, now, where that succession Q –> N is not a finite duration process? (That is the number of links exceeds any given finite n in N.) Every actual past stage Q must chain stepwise to now. Can that chain be transfinite? Or, are we only warranted to speak of finite durations. KF

  71. 71
    daveS says:

    KF,

    DS, an abstract logical alternative fails to come to grips with need for actuality and causal temporal stepwise succession to now. The issue is not directly, is there a first stage, but can we have an actual stage Q with stepwise succession to N, now, where that succession Q –> N is not a finite duration process?

    Hm, I didn’t ask about whether there is a first stage or not (and there wouldn’t be, in an infinite past, so I don’t know what to make of this.)

    The question is: Is my characterization of an infinite past in #66 correct or not? I’m simply trying to apply the logic of structure and quantity to the problem.

  72. 72
    daveS says:

    PS to my #71:

    but can we have an actual stage Q with stepwise succession to N, now, where that succession Q –> N is not a finite duration process?

    No, and the infinite-past proponents are not suggesting this.

  73. 73
    hazel says:

    At 69 kf writes, “H, time is not a mathematical abstraction but a physical entity, connected to energy flows, dissipation of energy concentrations and more.”

    Then time started about 15 billion years ago, and the past is finite.

    Conversation is over.

  74. 74
    kairosfocus says:

    H, the problem often raised is that there is an onward wider cosmos or multiverse. KF

  75. 75
    kairosfocus says:

    DS, note, this is an in-common property for any actual past stage Q. KF

    PS: On a hypothesised infinite past, we both know that for any past stage, there would be prior ones. Thus, a chain that is of transfinite character. That is where the in-common property as to duration-since Q being from an event Q to now applies.

  76. 76
    hazel says:

    kf, we know absolutely nothing about whether outside our universe there is something like time that is anything at all like what it appears to be inside our universe.

    Are you saying that you think outside our universe time is “a physical entity, connected to energy flows, dissipation of energy concentrations and more.” Why would you think that?

  77. 77
    hazel says:

    kf writes, ” On a hypothesised infinite past, we both know that for any past stage, there would be prior ones. Thus, a chain that is of transfinite character.”

    No. It is true that “for any past stage, there would be prior ones,” but each one of them would be a finite distance from the present. Saying that there would be an infinite number of such stages is different than saying any of them would be infinite in length. Say “that there would be an infinite number of such stages” just means that we can keep moving one stage further into the past indefinitely, but no stage ever becomes infinitely far from the present.

    You seem to be conflating two different understandings of “infinite” when you use the phrase “transfinite character.”

    The heart of the matter is this: in the real number system, there is an infinite number of negative integers, and all of them are finite numbers. Do you agree with that sentence?

  78. 78
    kairosfocus says:

    H,

    First, if it were generally accepted that on observation we have a credible start to the observed cosmos some 14 BYA at a singularity, the issue of an infinite quasi-physical cosmos would not be on the table. But, it is, through several speculative models, so we do need to speak to such, though arguably this is philosophical speculation dressed up in mathematical apparatus and physical terminology. In that context, once we have a micro scale, statistical thermodynamics will obtain (as it is a matter of statistical properties and behaviour of very large numbers of molecule scale particles). This is directly connected to the temporal-causal order.

    In that context, it makes sense to speak of causal connections and a linked temporal order though presumably there were no clocks around (though cyclical processes — the heart of clocks — could obtain). Usually, quantum and relativistic, cosmological scale patterns are also projected. For example there has been talk of fluctuations in a quantum foam leading to this cosmos and the like. All, very speculative.

    Second, I am well aware that in an infinite past world model each stage Q would have prior ones without limit, in the way that any negative integer will have L-ward onward values without limit. I am also aware that any particular stage we list, count to or even symbolically represent as say -k will have -(k+1) etc beyond it, allowing an L-ward mirror image of the two tapes thought exercise in the OP. That is therefore used to argue two things, one that in effect at any Q the transfinite past has already happened. (Notice, above how I pointed out how this begs the question of how such could come to be, how one ascends to Q from the L-ward, temporally-causally prior infinite.)

    Second, it is used in an argument that we have an infinity of mileposts {0, -1, -2 . . . -k, . . . }, each only a finite amount removed from 0. This is I believe a step too far. What is better warranted is to say that any particular -k we can represent will be at finite remove as it can be exceeded in an onward chain, so that part of the structure is that of the going onward represented by the ellipsis. It is through that structure that the transfinite character enters.

    From this, we see that we are only warranted to speak of a finite span to any particular deep past point Q, at point q L-ward from 0, emphasis on particularity, but recognising that we here have an in common property for points Q. At no point have we escaped finiteness to particular Q-points. In the context of past time, we have only got warrant to speak to particular, finite remove points Q where we can identify that q – n is finite. Where, only a finite span to N, now, can be bridged in successive cumulative finite stage sets.

    The extrapolation that there is an infinity of past time by virtue of L-ward finite stage countable extension so that to every Q we have only a finite duration but we have in effect aleph null cases giving the transfinite past does not follow.

    For, duration is between Q and N. We have the metric, |q – n| (I generally suppress the take the magnitude part as trivial). This is in common to every Q. The span is finite. And there is no span that is not between relevant points Q and N, there is no span of the set as a whole that takes in the ellipsis as though it were a particular value. We would need something on its other side, as in the hyperreal negative integer -K, which is forbidden as we then have the challenge of spanning a transfinite ascent from -K to 0, what was being implicitly avoided.

    This is the context in which I have said that we only have warrant to speak to finite past extensions, regardless of proposed indefinite L-ward extension. This is because finitude of the metric q – n is an in-common property for any Q at a negative integer point on the scale of the hypothetical past.

    Yes, I am aware of the claim, reverting to naturals, that we have {0,1,2 . . .} with every value finitely removed from 0 and a scale of infinity by virtue of onward continuation. Looking at say the surreals construction [cf. the diagram in the OP and onward links], that is tantamount to, at point w, stage/day w in the game, we have constructed the reals line between w and – w. This is indicated by the first vertical bar in the diagram above, the steps being counted out along the upper arc. That is, after w steps, the relevant power series to construct any r in R is complete, and the integers are mileposts along the R-line, which obviously has ellipses of endless extension leading to w and -w. We may go onwards, to fill out a maximal expansion.

    The significance of this, so far as I can see (I here speak of myself and my own estimation per the logic of structure and quantity as it plays out thanks to the Surreal construction), is that in so constructing R with Z as mileposts embedded within, we bring out a fuller picture, that R is inherently embedded in *R with Z in *Z so that the ellipses do have implied termini, w, -w and kin in the transfinite span. This involves that we can consider relevant hyperintegers K and -K as inherently connected, not as arbitrary and optional extensions to our numbers framework that can be confined to the reals.

    Coming back to reflect on time and metrics, what I see then is that when we posit a past involving indefinitely many cases Q — and notice my use of this particular term to denote the onward extension beyond any (repeat: ANY) specific value k or -k in Z — we are mapping to mileposts in R embedded in *R. Where the ellipses we see in R or Z do in fact have logic-model world termini, in *R and *Z. Where, too, the duration passing through the ellipsis has a terminus, some K or -K. BTW, this is why I am not just arbitrarily adding in hyperreals as an optional extra, I am looking at the wider structure of numbers great and small to set context.

    In that light, to me it makes sense to see that any Q is only at finite remove from N, q -n, and that to get to an actually infinite past one needs to cap the negative going ellipsis with some -K. For, duration must span particular actual past points. Only, due to the physical constraint on a logic model of time due to temporal causal stepwise ascent from any past value Q to now, no Q is transfinitely removed. To span the ellipsis we would have to go to some -K, but that violates the finite succession to N challenge, requiring a supertask that cannot be completed stepwise. This is of course the context in which we use 1/x as catapult between hyperreals and infinitesimals.

    I do not ask you to agree with my thoughts, just that you see why I hold them per that wider picture of the logic of structure and quantity.

    Going beyond, of course, we need to ponder the nature of time, which is intimately connected to temporal causal succession and linked thermodynamics, thus dissipation of energy concentrations. This is how it makes sense to speak of an age for the cosmos as a whole and of the singularity at is it currently 13.85 BYA. Also, of onward heat death. Where the observed flatness of our cosmos points that way.

    KF

  79. 79
    daveS says:

    I guess if we cannot all affirm that every integer has finite distance from 0, then we shouldn’t expect to get very far in this discussion.

  80. 80
    daveS says:

    KF,

    I know the following has been presented before here, but I’m wondering whether your views on this argument have changed:

    (1) Every nonempty subset of the natural numbers contains a least element.

    (2) If n – 1 and n are two (consecutive) natural numbers, then n – 1 lies at finite distance from 0 if and only if n does.

    (3) Let S be the set of natural numbers that lie at infinite distance from 0.

    (4) Assume S is nonempty. Then S contains a least element n.

    (5) Clearly this n is not 0, so n – 1 is also a natural number.

    (6) Since n was minimal in S, n – 1 lies at finite distance from 0.

    (7) This contradicts premise (2).

    Is this reasoning correct?

  81. 81
    kairosfocus says:

    DS, Integers include those in *Z not just Z, called hyperintegers. More to the point, every particular integer in Z we can write out (say as a decimal or binary) will be finite, and every one we can represent as a particular case k will be exceeded by k+1 etc. So, every integer in Z we can directly refer to will be finite, but in the context that the structure of the set includes indefinite extension as is symbolised by the ellipsis. That ellipsis on the positive side finds its terminus in w, in the wider ambit of surreals. My point has been that every actual past point stepwise removed from now by temporal-causal succession (which is countable) will be of finite duration to now, and that this warrants only discussion of finite durations once we bring up particular past points Q. If instead we point to the ellipsis of indefinite extension, that is not a specific value but a representation of the structure of Z involving (on the negative side) indefinite extension. That extension will have completion at -w or beyond it. Mathematically tractable but not physically realisable. KF

    PS: I will now add as a F/N a clip from J P Moreland on the subject of the proposed infinite past.

  82. 82
    daveS says:

    KF,

    DS, Integers include those in *Z not just Z.

    No. Let’s not throw out long-established notation/terminology, please. The word “integer”, used without qualification, means an element of Z.

    Edit:

    So, every integer in Z we can directly refer to will be finite…

    This appears to stop short of saying that every element of Z is finite.

  83. 83
    kairosfocus says:

    DS, you will observe that I have specifically distinguished hyperintegers in *Z and finite integers in Z. Once you represent a particular integer in Z+ (the naturals) it will be finite as bounded by n+1 also a natural per the von Neumann construction. Turning your argument around, let K be a hyperinteger (which for argument is even), and let us now descend: K, K-1, . . . K/2 . . . as Dr Wood does in the vids in the OP. What we have, so far as I see, is that the descent from K downwards never terminates in values ascending from 0 in similar steps, precisely due to the ellipsis: spanning the transfinite ellipsis ascending or descending in steps is a supertask. That is, there is no specific least hyperinteger h, there is no specific maximal finite integer f and there is no particular identifiable meeting point where f + 1 = h or the like. The power of the ellipsis of transfinite span unbridgeable in incremental +/-1 steps as a structural element has to be reckoned with. That seems to be structural to the Surreals, the wider construction that sets up these. In that context, I look at your proof as little more than showing that we see two domains separated by a transfinite span bridged by a seemingly simple ellipsis. Which, is anything but. KF

    PS: For even more fun, do the catapult through 1/x and go to 1/K = m, 2/K = 2m etc, i.e. we see the dual of the same effects down in the infinitesimal cloud around 0, where 1/n for any n in Z+ will not even approach the closeness to 0 of m and kin.

  84. 84
    kairosfocus says:

    DS, I am saying that the ellipsis imposes a fuzzy border on Z+, so that where we have hyperintegers appearing, they are not in Z+ and every z in Z+ we may represent (this is a variable) will be bound and exceeded by z+1 etc so will necessarily be finite. It is that fuzzy border that has got my attention and leads me to the particular phrasing I am using. Is z in Z+, open to any particular case (substitution instance)? If so, finite. However, structurally, the ellipsis is there also {0, 1, 2 . . . } and bridges somehow to the transfinite realm involving K that is also sufficiently connected to the reals (having the values of z in Z as mileposts) that 1/K can be assigned a value in the interval (0,1] between two integers. Which I have represented as m. Where is K precisely on the surreals upper arc, is irrelevant. KF

  85. 85
    hazel says:

    I notice that kf cannot discuss this subject in terms of just the reals. I’m sure that when people think about time progressing as modelled by a number line, as they have since the time of Newton, Descartes, et al, they have not had hyperreals in minds, as those were only proposed and developed in the last 70 years or so.

    I also notice that he has pointed out no flaw in my proof by contradiction at 68 that the past has no beginning.

  86. 86
    kairosfocus says:

    H, I have to move out now. I pointed out that R has the integers in it as mileposts. I have used Z to bring out that we are dealing with finite stage cumulative steps rather than infinitesimals converging on a finite value in the limit etc. We can count stages in relevant blocs, including beyond the singularity, which is a physical reason not to use the otherwise very convenient year or second etc. I also note (son pressing me to move now!) that the surreal construction shows the reals are inherently embedded in the wider domain of numbers including transfinites, hyperreals and infinitesimals. Whether such was fully understood or discovered or developed in the past is irrelevant though obviously they thought about infinitesimals a lot, erecting Calculus on them, with antecedents to Archimedes and beyond. KF

  87. 87
    hazel says:

    None of those remarks address my simple proof, and I don’t anticipate that any are forthcoming: infinitesimals, and the singularity, and hyppereals, and calculus, have nothing to do with a simple argument that the past, when modeled on the real number line, has no beginning.

    So there is really no need for you to keep responding, I don’t think.

  88. 88
    kairosfocus says:

    H, if you speak of the argument you made yesterday, its problems were laid out. I have shown the reason why a wider domain is relevant, KF

  89. 89
    daveS says:

    KF,

    every z in Z+ we may represent (this is a variable) will be bound and exceeded by z+1 etc so will necessarily be finite.

    I gather we still haven’t come to terms on this issue. This statement still includes a qualification about representation, which as far as I can tell, is unnecessary. You do apparently think it is necessary, which indicates we are not seeing eye-to-eye.

  90. 90
    hazel says:

    Claiming that a “wider domain” is relevant doesn’t show that the proof is wrong.

    Given the real number line, is there anything wrong with my proof?

  91. 91
    kairosfocus says:

    DS,

    I think it draws out a relevant consideration, as is further discussed above. I have a bit more so I add: the ellipsis marks a fuzzy border, there is something structural at work.

    H,

    Time is physical, the integers and the reals are abstract. there are both physical and logical reasons to challenge the argument you made.

    I also add to you, that on looking at the surreals construction R is naturally bound by w and – w then the rest of the transfinites, with the infinitesimals making key appearance around 0.

    ____

    Later

    KF

  92. 92
    hazel says:

    So now you switch back to arguments about reality. You have no evidence whatsoever that “time” before the singularity is “physical”.

    Tell me the logical reasons to challenge my proof, given the real number line as an abstract model for time.

  93. 93
    daveS says:

    KF,

    As a platonist, you believe that the elements of Z exist independently of human minds, correct? And that the proposition “all elements of Z are finite” is either true or false. That is, either all elements of Z are finite, or at least one is infinite. Do you think humans are not equipped to decide its truth value?

  94. 94
    Brother Brian says:

    1) An infinite past cannot exist.
    2) We don’t have to explain what caused God because God has always existed.

    Does anyone see the incompatibility of these two claims? Claims often made by the same people?

  95. 95
    ET says:

    Brother Brian:

    1) An infinite past cannot exist.
    2) We don’t have to explain what caused God because God has always existed.
    Does anyone see the incompatibility of these two claims? Claims often made by the same people?

    Umm, God started us. So we don’t have an infinite past. God started this universe. So this universe doesn’t have an infinite past.

    Clearly Brother Brian lacks the capability to think.

  96. 96
    kairosfocus says:

    H, recall, I am responding to those who promote a multiverse or underlying super cosmos or the like. In that context, causal succession would obtain as the observed cosmos credibly begins. That which begins, is caused. Beginning implies as well before and after, so we have a causal-temporal order in that quasiphysical domain, which also has to account for energy, energy flows and concentrations. From these, much would apply. I have already shown why we have no warrant to speak of a transfinite physical or quasi-physical causal-temporal order, including the logic of stepwise causal succession to now, including the issue of energy concentration dissipation [entropy] and ever increasing non availability to sustain causal trains, and including the logic of extension in Z such that a specific z in Z will be finite. KF

  97. 97
    hazel says:

    Fortunately, this has not been a theology discussion! 🙂

  98. 98
    kairosfocus says:

    DS, structure and quantity including numbers are implicit in the distinct identity of any logically possible world. It seems to me that the issue is really about borders and I already pointed out that the ellipsis shows that structurally there is a fuzzy border between finite integers and transfinite hyperintegers. We construct the naturals from zero and get the negative integers by additive inverse, in a stepwise process. Accordingly any particular z in Z so constructed or represented will be inherently finite as we succeed to z+1 etc, but we cannot exhaust the process. Coming the other way, hyper integers can descend stepwise from an arbitrary K, where 1/K = m, an infinitesimal. We find a situation reminiscent of the continuum, next to a given real r, there is no definable closest neighbouring real AND there is a hyperreal cloud closer to r in R than we can get with any real. This is best seen for r = 0. Coming up from 0 we speak of finite integers z, and coming down from K we speak of hyperintegers, approaching a fuzzy border represented by the ellipsis. Weird, maybe, but in the end it makes sense. KF

  99. 99
    kairosfocus says:

    BB,

    God is not a physical entity, indeed no physical entity (being composite) will be a necessary being.

    We can show on logic of being that there is a necessary being world root. Later.

    ET

    Go easy on BB, especially today. He has serious food for thought.

    KF

  100. 100
    daveS says:

    KF,

    I take it that you are not going to affirm the proposition “all elements of Z are finite”. Therefore I don’t believe it’s possible to make much progress on the current discussion.

  101. 101
    hazel says:

    Well, kf, it seems like you jump from one perspective to another, which is your privilege, but I’m not interested in, for reasons I have mentioned, in speculating about whatever lies before the singularity, or about what time might be, if anything. I don’t see anyone here discussing multiverses and a larger cosmos with you, though.

    Given what you say, I am going to assume that you are not interested in discussing the situation of the reals as an abstract mathematical model for time, nor that you have any logical problems with the argument I presented in 68. Since you want to talk about things that I don’t want to talk about, and vice versa, then perhaps we should leave it at that.

  102. 102
    Brother Brian says:

    KF@99, but this does not answer the question of whether God has always existed (ie, did not have a beginning or an origin). If he did not have a beginning (ie, has always existed) then is this not incompatible with the idea that there cannot be an infinite past.

    ET

    Go easy on BB, especially today. He has serious food for thought.

    Don’t worry. When ET says something worth responding to, as you often do, I will respond to him directly. However, those times are few and far between.

  103. 103
    kairosfocus says:

    BB, The issues turn on logic of being: possible vs impossible, contingent vs necessary. True nothingness is non-being. That having no causal powers were there ever utter nothing, such would forever obtain. So, something, of some nature, always was; given that a world exists. We can consider possible vs impossible being, the latter being like a square circle: contradictory core characteristics mean there is no possible world where such could exist. Of possible beings first we have contingent, like a fire, there are enabling causal conditions for such to be and a particular contingent entity would exist in some possible worlds but not others. Necessary beings are not causally dependent, are part of the framework for any world to exist, do not begin, cannot cease; try to imagine a world without two-ness or where that ceased. As distinct identity W = {A|~A} is integral to any particular world, duality is necessary. Such beings will not be built up by assembling independent parts, and so physical systems will not be necessary ones. As you ask about God, in ethical theism, God is necessary of being and the root of reality, enabler of all worlds, being eternal is an aspect of that necessity of being. Possible and necessary, not contingent and caused. Also, given morally governed creatures (us) the inherently good One who grounds ought and bridges is and ought. More, later. KF

  104. 104
    Brother Brian says:

    KF

    So, something, of some nature, always was;…

    Forgive me for excerpting just a small part of your comment, but I think it is critical. If something, of some nature, always existed, does this not mean that an infinite past also exists?

  105. 105
    kairosfocus says:

    BB, there is a difference between the eternality of necessary being (which is inherently non-physical, non-composite) and an infinite causal-temporal succession by stages as a physical or quasi-physical matter-energy space-time order. KF

  106. 106
    ET says:

    Brother Brian:

    If something, of some nature, always existed, does this not mean that an infinite past also exists?

    How can something without a beginning even have a past?

  107. 107
    ET says:

    kairosfocus:

    Go easy on BB, especially today. He has serious food for thought.

    I am going easy because I understand the handicap. And I don’t see any evidence that supports your second sentence. These matters have already been dealt with. Do you really think this is the first time this type of discussion has ever arose?

    Have to ever been to a Catholic High School where every boy is always trying to one-up the teachers of Faith and the Priests and Nuns who run the place? Those people seem to live for our rhetorical ploys, which only meant we weren’t the first, either.

  108. 108
    ET says:

    Another way of saying it is the past refers to time. And time didn’t exist until the/ a universe did.

  109. 109
    Brother Brian says:

    KF@105, so you know for certain that nothing existed before the Big Bang? All we know with a high level of certainty is that our universe didn’t exist before then.

    With regard to God, I assume that you are referring to the argument that he exists outside of time and, therefore, has always existed. However, this appears to me to be a rationalization to justify the existance of free will and the existance of a non-caused being.

  110. 110
    ET says:

    In the time before time, there wasn’t any time. 😎

    The uncaused being is an argument borne from the fact we cannot observe and test God/ Creator/ Designer. And the fact that any regression beyond what is necessary is better left to those who care.

  111. 111
    hazel says:

    re 97: Unfortunately, it has become one. 🙁

  112. 112
    Brother Brian says:

    Hazel

    re 97: Unfortunately, it has become one.

    The devil made me do it. 🙂

  113. 113
    ET says:

    hazel @ 111- Not really. The attempt to make it one has been shot down.

  114. 114
    kairosfocus says:

    BB, has anyone in-thread suggested that the singularity is the origin of physical reality? No. Worse, no-one has suggested that nothing was there before it either. Such would be non-being, which — having no causal powers — would be the case still were it ever the case. That a world is now, implies directly that something is a necessary being root of reality, capable of causally grounding a universe on the observed scale. And by its nature, such an entity neither begins nor could it cease from being, in short, still around. KF

    PS: Necessary being is not equal to God (notice how duality is given as an example — part of the world-framework existence of abstract structure and quantity), though in ethical theism, God is seen as a necessary being. The theological discussion of God being outside time is irrelevant to our logic of being concerns. Start with a distinct possible world has a distinct A that marks it out so W = {A|~A} thence nullity, unity (simple and complex), duality, thence by way of von Neumann etc, N, Z, Q, R, C (and BTW hyperextensions) as part of the built in structural logic of a world. As a world is, such structures and quantities always were and are part of the root of reality. They are also abstracta, of strongly rational character. And that sounds fairly suggestive, especially as the actual theological discussion balances immanence and transcendence: in Him we live and move and have our being, upholding all things by his word of power, in the beginning Reason himself was and without him was not anything made that was made — hint, not every entity is made.

  115. 115
    kairosfocus says:

    ET, I actually went to Catholic primary and secondary schools. Solid educationally, and I learned to respect Franciscan nuns and Jesuit priests, personally and educationally. KF

  116. 116
    kairosfocus says:

    F/N: Wiki on heat death:

    The heat death of the universe, also known as the Big Chill or Big Freeze,[1] is an idea of an ultimate fate of the universe in which the universe has evolved to a state of no thermodynamic free energy and therefore can no longer sustain processes that increase entropy. Heat death does not imply any particular absolute temperature; it only requires that temperature differences or other processes may no longer be exploited to perform work. In the language of physics, this is when the universe reaches thermodynamic equilibrium (maximum entropy).

    If the topology of the universe is open or flat, or if dark energy is a positive cosmological constant (both of which are consistent with current data), the universe will continue expanding forever, and a heat death is expected to occur,[2] with the universe cooling to approach equilibrium at a very low temperature after a very long time period.

    The hypothesis of heat death stems from the ideas of William Thomson, 1st Baron Kelvin (Lord Kelvin), who in the 1850s took the theory of heat as mechanical energy loss in nature (as embodied in the first two laws of thermodynamics) and extrapolated it to larger processes on a universal scale.

    Sounds like, time itself winds down like a clock whose main spring has unwound and lacks something to wind it back up.

    Here’s a thought: is time inherently thermodynamic?

    KF

  117. 117
    daveS says:

    KF,

    In that light, to me it makes sense to see that any Q is only at finite remove from N, q -n, and that to get to an actually infinite past one needs to cap the negative going ellipsis with some -K.

    Then your critique does not apply to the model that the infinite-past proponents put forward.

    They propose scenario A, and you talk about B.

  118. 118
    kairosfocus says:

    DS, I am saying that it looks a lot like something is being missed in the ellipsis when in effect an infinite past of finitely remote stages is proposed. Something in there looks very incoherent to me, hence among other things my comment that if we are discussing and can warrant discussing finitely remote cases only, then that is just the case: our warrant is to speak to the finitely remote. A duration is about a closed interval, requiring exactly two definite termini (e.g. [Q, N], [K, N]), so the suggestion of other further on cases all finite that somehow attains to the transfinite despite the point that stepwise we cannot span a transfinite interval — recall, definite termini — just does not sit right. KF

  119. 119
    hazel says:

    A common problem. We try to, dare I say, focus on scenario A and responses slide off to scenarios B, C, D … K … :-).

    Proponents of an infinite past, I am virtually certain, have done so from an abstract point of view involving the real number line. Even though kf says there are logical problems with that approach, he doesn’t address them: rather he claims that other issues (physical time, hyppereals) must be discussed.

    “They propose scenario A, and you talk about B.” Indeed.

  120. 120
    hazel says:

    kf, nothing “somehow attains to the transfinite”, and nothing is “missing in the ellipsis.” Those statements highlight what doesn’t “sit right” with you, I think: You just don’t accept the standard and well-accepted understanding of what the infinite nature of the real numbers means.

  121. 121
  122. 122
    daveS says:

    KF,
    Credit to Moreland Tisthammerw for at least including serious responses to the anti-infinite-past arguments.

    Note that he refers to the fact that one can prove (granted reasonable premises) that one cannot “traverse” the sequence (1, 2, 3, …) starting from 1.

    However, he doesn’t have a proof showing the impossibility of traversing (…, −3, −2, −1, 0).

  123. 123
    hazel says:

    Kf, this guy has the same confusions.

    One of the confusions is about what infinity means: the difference between meaning a process that can be continued without end (which is correct) and a “place”, “amount” or state that can be realized.

    Let me restate my argument in 68 without mentioning infinity:

    —–
    The past (as modeled in the mathematical sense by the real numbers) either has a beginning or it does not have a beginning.

    Claim: the past has no beginning.

    Proof by contradiction.

    Assume the past has a beginning

    Then there is some farthest point k in the past that is the beginning of the past

    However, for any k, k-1 is farther from the present than k

    Therefore, k is not the beginning.

    Therefore, this contradiction shows that our assumption that the past has a beginning is wrong

    Therefore the past has no beginning
    —–
    You said earlier that this argument had logical problems, but you never said what they were.

    Is this argument logically sound?

  124. 124
    hazel says:

    Dave, the page kf linked to wasn’t by Moreland, but by some guy named Tisthammerw. Maybe you didn’t know that, or maybe you did.

  125. 125
    daveS says:

    Thanks for the correction, hazel. I didn’t notice that.

  126. 126
    kairosfocus says:

    H,

    for cause I beg to disagree.

    First, the surreals and hyperreals (despite your previous attempts that suggested irrelevance brought in) provide a broad and patently relevant context to understand and discuss the finite and transfinite, including understanding the way in which N is transfinite, N providing mileposts in R. In that light, what we are having exchanges over is matters of scales, magnitudes and borders.

    We can definitely conceive of the order type of the naturals w and place it among the surreals, where it has known transfinite magnitude aleph null, cardinality. Second, by additive inverse we reflect N in 0 to get Z, and this allows us to address descent from the order of w in stages, where the hyperreal K shows that descent is possible from – K but if done in finite stage, +1 steps will never attain to some -k finitely remote from 0 and from n greater than 0. Those are conceivable and are more or less agreed.

    Where the debates have pivoted is on claimed temporal ascent from a beginningless past in countable, finite stages, which can therefore be addressed through the properties of Z in context of R which is in the further context *R. Where, the surreals construction allows us to picture the overall map, as is in the OP. Starting from 0, after w steps, R appears pinned between w and – w. Onward we construct transfinite spans of ever higher order.

    In addition, the nature of temporal causal succession, its link to free energy availability, gradual dissipation of energy concentrations are also relevant. These point to inherent limits on what time does and how successive events occur. A consequence is, given the observed world as a thermodynamically isolated going concern, time has a future effective terminus, heat death as energy concentrations dissipate. This immediately implies the contingency of the observed cosmos. Arguably, also, a beginningless past requires infinite energy reservoirs and infinite heat sinks to dissipate such. The more direct relevance is that succession of stages is due to energy-driven, entropy-expanding causal processes. Thus, we see temporal-causal connexions giving time its famed arrow.

    However, our focus is, given stepwise succession, is there a logic that grants or blocks a beginningless past, tied to structure and quantity?

    It is agreed that a succession -K –> -K + 1 –> . . . -k –> . . . -2 –> -1 –> 0 –> +1 –> . . . n is infeasible as the first ellipsis is a supertask. Instead we are invited to examine:

    . . . -k –> . . . -2 –> -1 –> 0 –> +1 –> . . . n,

    on interpretation that the ellipsis is never infinite, at every stage -q it is still finite within reach of -k etc and has already traversed an unlimited further past succession where every member is only finitely removed just as -q is.

    This is where I have serious questions. For one, this implies that we are only, ever discussing finite cases, of finite duration to now. We have given no warrant for an actual duration from specific past events down to now that is transfinite. If every particular past case to now is reachable through a finite succession, then we have only discussed finite cases. Where, too, at every q we have pushed back the actual traversal that would deserve the term infinite. This seems question begging by a sort of infinite regress.

    And in fact that is an issue: to get to -q we had to have been at -(q+1), repeat going further back. The process begs the question of antecedents to reach succession. Nor can this be brushed aside by saying it was always going. That is the very matter at stake.

    What seems to me is that cumulatively Z- does have a known span:- w to 0, i.e. the transfinite lurks implicitly in the ellipsis and requires the finite stage stepwise spanning of the transfinite. This is clearest if one gives a specific -K, but that leads to an obvious form of the problem. Instead, by leaving the ellipsis unclosed and pointing to succession of inherently finite cases, the matter seems solved from a certain perspective.

    Only, it isn’t.

    There is no warrant to speak of a transfinite actual past, even ignoring energy and entropy.

    KF

  127. 127
    daveS says:

    KF,

    There is no warrant to speak of a transfinite actual past, even ignoring energy and entropy.

    You should have a word with your colleagues, Laszlo Bencze, WLC, et al. They do speak of such, and claim to be able to demonstrate its impossibility.

    Further, we don’t need “warrant” to discuss arguments purporting to show something is impossible.

  128. 128
    hazel says:

    re 126, and to quote Dave: “They propose scenario A, and you talk about B.”

    I take it, kf, that you find no logical flaws in my argument at 123.

    You write, “We have given no warrant for an actual duration from specific past events down to now that is transfinite. If every particular past case to now is reachable through a finite succession, then we have only discussed finite cases.”

    That’s right: I have never said that the phrase “infinite past” means the same as an actual transfinite duration. It is you that are conflating the two.

    I’m wondering if you can provide a source of a proponent of an infinite past in the logical way we are discussing it that says an infinite duration has passed. Perhaps you are arguing against a strawman of your own making, and misunderstanding, or misrepresenting, the people you are arguing against.

    Can you link to some people that actually make the argument that you are arguing against?

  129. 129
    kairosfocus says:

    DS, the warrant I spoke to is that of establishing credible actuality. I am not arguing that one cannot imagine or conceive of such a past, but that once the supertask of actual stepwise descent from an actual acknowledged transfinite (and obviously from beyond it) is on the table, it leads to insuperable problems with appeal to beginningless descent that can be enumerated along the mileposts set up by Z-.Indeed, for every stage Q at -q in Z-, the actual transfinite traverse is implied as already accomplished. That’s an infinite regress version of question-begging. And while one cannot say it cannot get started, one can say, it has not been warranted just asserted. KF

  130. 130
    daveS says:

    KF,

    DS, the warrant I spoke to is that of establishing credible actuality.

    And no one here has claimed to have established credible actuality. That’s not even on the agenda. I am discussing whether the anti-infinite-past arguments succeed.

  131. 131
    kairosfocus says:

    H, at 123, durations can only be between actual real world states, one in the remote past the other obviously now. For instance from the singularity to now some 10^17 s have reportedly lapsed (as an order of magnitude). So, where a past state -K is off the table per supertask, and we point to indefinitely many already surpassed antecedents to any remote past stage Q where Q is finitely remote from now, that simply begs the question on the causal chain antecedent to Q. Looking in the wider perspective, we have only ever warranted finite durations Q to N. Yes, we talk of how say N is transfinite in cardinality as for any particular k in N we may exceed k+1, k+2 etc, but that simply points to the order type of N, w — per von Neumann construction, which in the surreals context emerges in how R is spanned by w and -w. The ellipsis points to that spanning. KF

  132. 132
    kairosfocus says:

    DS, on the balance of warrant, there has been no infinite past. This holds physically and at least arguably, logically given what is required for an infinite duration and for an infinite actual physical entity. BTW, absent actual past states, there are no actual durations since imagined states. We have no good reasons to accept as actual suggested remote temporal stages that have succeeded in a causal chain to now that would yield an infinite duration of the past to now. KF

  133. 133
    daveS says:

    KF,

    We have no good reasons to accept as actual suggested remote temporal stages that have succeeded in a causal chain to now that would yield an infinite duration of the past to now.

    Bencze claimed to have shown it’s literally mathematically/logically impossible in a brief blog post. He didn’t merely say we have no good reason to accept that an IP actually occurred. (And I do agree with this, btw).

    Once again, I’m talking about A, you’re talking about B.

  134. 134
    hazel says:

    Yes B not A with me also. It’s hopeless.

  135. 135
    kairosfocus says:

    H, the issue is not whether one may define infinite abstractly as a set that extends beyond any particular finite bound, but that a suggested infinite past has an actual duration going through a chain of actual stages from the remote past to now which would be infinite. Where, as this is speaking of a proposed actual past, as was shown above, we have stages or even states of the world that CAUSALLY give rise to succeeding ones in a stepwise causal-temporal chain down to the present. In that context, duration does require a measure or count on the interval between particular stages in order to be an actual value. Where Q to N makes sense but is inevitably finite once we see that a transfinite span cannot be traversed in steps of finite scale that cumulate to achieve a traverse. Further to this, at every stage Q that is such that we may succeed to now in steps, the proposers of an infinite duration past in effect imply or assert that the infinite traverse was already traversed, begging the question of how that could ever be through an infinite regress. KF

  136. 136
    hazel says:

    kf writes, “a suggested infinite past has an actual duration going through a chain of actual stages from the remote past to now which would be infinite.”

    And who has suggested this, kf? Or are you tilting at straw windmills that don’t actually exist?

    Can you quote someone, or link to some source, which that makes an argument that an infinite chain has actually been traversed.

  137. 137
    kairosfocus says:

    H, those who are making the further argument of impossibility of an actually infinite past are pointing to further issues. Such an argument may be outlined in brief but its full substantiation will be quite involved. For example, while transfinite abstract sets such as the evens, odds and naturals may be put in 1:1 correspondence, it becomes a completely different matter when one argues that a spatially extensive Hilbert hotel with physically extensive guests in it can be actualised. Consider, as an illustration, it is full but countably infinitely more actual guests arrive. By telling existing guests to go to room 2n, room opens up for all the new guests in the odd numbered ones; without building new rooms and where both the even and odd rooms were all occupied. But, per factual assertion, it was already full, so there were no empty rooms as a physical fact. Likewise if odd number guests check out the hotel is half empty but has the same number of guests as before in terms of cardinality. And more, supporting that actual physical instantiation of the infinite is impossible. Similarly, if there has been an infinite actual past then there were stages (infinitely many in fact) that are removed from the present by a number of stages which exceeds any finite value k. Arguably, on the premise that all n in Z are finite there is no room for these onward past actual stages among the range of values -k may take, which is all the values in Z-. KF

  138. 138
    hazel says:

    B, not A.

    Can you respond to 136, kf. Who is claiming the case which you are arguing against?

  139. 139
    daveS says:

    KF,

    For example, while transfinite abstract sets such as the evens, odds and naturals may be put in 1:1 correspondence, it becomes a completely different matter when one argues that a spatially extensive Hilbert hotel with physically extensive guests in it can be actualised. Consider, as an illustration, it is full but countably infinitely more actual guests arrive. By telling existing guests to go to room 2n, room opens up for all the new guests in the odd numbered ones; without building new rooms and where both the even and odd rooms were all occupied. But, per factual assertion, it was already full, so there were no empty rooms as a physical fact.

    Recall that David Snoke has shown that these illustrations do not achieve the desired effect.

    In the example where each guest n moves from room n to 2*n, this shifting process could never be completed in the real world. After the new guests arrive, it will never be the case that all guests are housed.

  140. 140
    hazel says:

    Do you know how long it takes to change rooms at a hotel? Of course, not everyone will ever be housed! 🙂

    But seriously, noticed the introduction of a new irrelevancy.

    And who argues that ” it becomes a completely different matter when one argues that a spatially extensive Hilbert hotel with physically extensive guests in it can be actualised. The Hilbert hotel is a metaphor to help think about infinity in an abstract sense: I can’t imagine that anyone has seriously discussed a real “spatially extensive Hilbert hotel with physically extensive guests”

    B not A, again.

  141. 141
    daveS says:

    hazel,

    I took a guess that KF intended post #137 to be directed at me (at least in part), in which case it would be on point (I believe he’s talking about the metaphor WLC used in a lecture). But perhaps I’m wrong.

    In any case, I’m having trouble believing that KF is arguing sincerely, given how easily Snoke disposed of the Hilbert Hotel scenarios. This looks more like a rope-a-dope strategy. 🙂

  142. 142
    kairosfocus says:

    H,

    I am not obligated to show you why it is relevant to an argument about an actual infinite, that proposed actual infinites are replete with antinomies showing their utter implausibility at minimum; though there are extended arguments as I indicated that draw out very similar problems with temporal-causal succession; I suggest ponder a walking tour of all the rooms requiring just one minute per room, from the extension of the hotel away from its front desk, such will never complete due to the problem of traversing the transfinite in finite stage steps. And, it does no good to assert at some given time the manager is in room Q finitely removed from the front desk, as that simply begs the question of having visited all the further remote rooms. I have already shown how there is no good warrant for claiming an actually infinitely remote past stage of the physical cosmos.

    DS,

    Present the case, kindly do not just cite a name, note that I just gave to H a simple way to transfer to time. Where, recall, the actual past had to be traversed to get to now step by step, stage by stage. Which traverse must in this case exceed any finite value.

    KF

  143. 143
    hazel says:

    kf writes, “I have already shown how there is no good warrant for claiming an actually infinitely remote past stage of the physical cosmos.”

    B, not A

  144. 144
    daveS says:

    KF,

    Present the case, kindly do not just cite a name

    Are you asking for Snoke’s argument?

    It’s from this paper which was discussed here at some point.

    What about the Hilbert hotel? Fans of the Kalaam argument, and fans of Hilbert’s “finite mathematics,” often try to create conundrums to prove that real infinities can’t exist. One example is the “Hilbert hotel”. In this scenario, there is a hotel with an infinite number of rooms. One night, each of the rooms is occupied. A new guest arrives and asks for a room, and is told they are all full. The guest suggests the following ingenious scheme: ask the occupant of room 1 to go to room 2 ask that occupant to move to the next room. The occupant of room 1 then sleeps in room 2, and the occupant of room 2 goes to room 3, displacing that occupant, and so on, until everyone has moved one room number higher. Since there is an infinite number of rooms, everyone will find a room, and the new guest can sleep in room 1.

    This seems to imply a contradiction, since all the rooms were occupied at the start, with no empty spaces, but an empty space was found. For a physicist, though, this scenario is easily dealt with by the principle of locality. It takes a finite time for an occupant to move from one room to the next. So really what has happened is that the new guest has set up a traveling wave in the chain of rooms. At all later points in time, there will be one guest walking from a one room to the next, while the other rooms are all occupied. There has not been a new room discovered, but rather a moving “excitation” (to use physics language).

  145. 145
    kairosfocus says:

    H, asserting irrelevancy again and again is different from showing it, especially when there is in the literature exactly a discussion applied to time that elaborates concerns on an actual infinite. For our simple case above, kindly explain how Hilbert’s Manager can complete a tour of the rooms in decending order,thus reaching Q which is finitely removed from the front desk. Where, each room inspection requires one minute, so we see the temporal connexion directly. Notice, the actual past played out as a succession of cumulative, causally linked stages down to the present. No, it will do no good to claim that at any stage Q he has already visited the further rooms that are antecedent to reaching Q. we know these rooms are q+1, q+2 etc In short, there is an implicit transfinite traverse in stages of time, causally successive, down to Q that would have to be traversed in stages to reach Q. I suggest, it cannot be traversed in stages or finite size, due to the implied supertask. Further, this is directly relevant to any actual past as it has to cumulatively occur step by step to reach now. Every actual past stage was once the now, and gave rise to its successor through temporal-causal link, then this repeated to now. KF

  146. 146
    kairosfocus says:

    H, I simply point you, per 136,

    kf writes, “a suggested infinite past has an actual duration going through a chain of actual stages from the remote past to now which would be infinite.”

    And who has suggested this, kf? Or are you tilting at straw windmills that don’t actually exist?

    Can you quote someone, or link to some source, which that makes an argument that an infinite chain has actually been traversed.

    to the timeline since the singularity to illustrate what is meant by causally and temporally linked successive stages down to now. If you do not recognise such in say one of NASA’s cosmological timeline diagrams, then that is not our problem but yours. There is no need whatsoever to appeal to authority beyond say understanding that one walks down a staircase in successive steps.

    As for the infinite nature of the chain, it is plainly intended that for every past stage of finite remove Q, no matter how high, there is an onward chain of antecedent steps removed to q+1, q+2 etc without limit.

    That is plainly transfinite.

    Or, do you wish to equivocate between finite and transfinite?

    KF

  147. 147
    kairosfocus says:

    DS, surely, that is an irrelevancy as cited; a real one. The issue is not that guests put from room n to 2n take however long travelling but the contradiction between a full physically completely occupied hotel where no further rooms are built and simply by moving guests — physical guests — from room n to 2n [effectively, together], infinitely many odd numbered rooms from 1, 3, 5 etc are now suddenly open to receive the same cardinal number of guests afresh as were there when it was full to the gills and are still there when it is now half empty. Bang, hotel filled again. Oops, another infinity of guests arrived, send the present occupants to rooms at 2n again, and refill. The citation from Snokes you gave does not answer. KF

    PS: Zeno’s paradoxes have to do with sums and ratios of infinitesimals where such can readily complete an abstract infinity in finite time. We are talking about countable, finite stages that do not conveniently converge to finite displacements in finite times. Of course, rhetorically, ordinary people will easily be lost in the midst of such discussions as they have no limits, sequences, series, L’Hopital’s rule etc to refer to.

  148. 148
    daveS says:

    KF,

    DS, surely, that is an irrelevancy as cited; a real one. The issue is not that guests put from room n to 2n take however long travelling but the contradiction between a full physically completely occupied hotel where no further rooms are built and simply by moving guests — physical guests — from room n to 2n [effectively, together], infinitely many odd numbered rooms from 1, 3, 5 etc are now suddenly open to receive the same cardinal number of guests afresh as were there when it was full to the gills and are still there when it is now half empty.

    This hotel would have to be infinite in size, so it would be impossible for every guest n to complete his/her move in a finite amount of time. If the rooms were numbered 1, 2, 3, etc, and were arranged linearly in the obvious way, guest 1 could complete her shift quickly, while guest 10^100 will have a long walk. The hotel will never be “half empty” (and thus “half full”) after the move begins.

  149. 149
    hazel says:

    kf writes, ” when there is in the literature exactly a discussion applied to time that elaborates concerns on an actual infinite. ”

    Where? Please point to some literature.

    kf write, “to the timeline since the singularity to illustrate what is meant by causally and temporally linked successive stages down to now. If you do not recognise such in say one of NASA’s cosmological timeline diagrams, then that is not our problem but yours.”

    That has nothing to do with my discussion. Of course, time since the singularity has been finite, with a relatively known beginning. But that is not all related to the argument about the number line and the reals.

    P.S. My proof that the past has no beginning stands.

  150. 150
    hazel says:

    Changed my mind about this post …

  151. 151
    ET says:

    hazel:

    P.S. My proof that the past has no beginning stands.

    But what does it mean? Whose past? What past? What proof?

  152. 152
    hazel says:

    ET:
    1) A abstract model of the past based on the number line

    2. Proof is at 68

  153. 153
    ET says:

    hazel- The number line has a starting point, namely zero. And this:

    “However, for any k, k-1 is farther from the present than k”, is only valid if and only if (IFF) k-1 exists.

  154. 154
    hazel says:

    ET writes, “The number line has a starting point, namely zero.”

    No, not at all. The non-negative numbers start at zero. The negative numbers have no starting point, which is the point of the proof

    ET writes, ““However, for any k, k-1 is farther from the present than k”, is only valid if and only if (IFF) k-1 exists.”

    Yes, but the whole theory of the natural numbers starts with the axiom that every number k has a successor k + 1.

    Wikipedia says, “Like the natural numbers, Z is closed under the operations of addition and multiplication, that is, the sum and product of any two integers is an integer. However, with the inclusion of the negative natural numbers, and, importantly, 0, Z (unlike the natural numbers) is also closed under subtraction.”

    That is, if k is an integer, then k -1 is an integer also.

  155. 155
    kairosfocus says:

    DS, the same effective problem obtains for move over one room to get in one fresh guest. The same point obtains of full vs by rearranging mass, suddenly room appears. KF

  156. 156
    ET says:

    hazel:

    The negative numbers have no starting point, which is the point of the proof.

    They also start at zero an go on, infinitely.

    Yes, but the whole theory of the natural numbers starts with the axiom that every number k has a successor k + 1.

    OK.Just don’t conflate natural numbers with time.

  157. 157
    kairosfocus says:

    H,

    First, try here for a start:

    A response to a Platonistic and to a set-theoretic
    objection to the Kalam cosmological argument
    J. P. MORELAND
    Talbot School of Theology, Biola University, 13800 Biola Avenue, La Mirada, CA 90639
    Abstract: The ?rst premise of the Kalam cosmological argument has come under
    ?re in the last few years. The premise states that the universe had a beginning, and
    one of two prominent arguments for it turns on the claim that an actual in?nite
    collection of entities cannot exist. After stating the Kalam cosmological argument
    and the two approaches to defending its ?rst premise, I respond to two objections
    against the notion that an actual in?nite collection is impossible: a Platonistic
    objection from abstract objects and a set-theoretic objection from an ambiguity in
    the de?nition of ‘=’ and ‘<’ as applied to sets. The thought-experiment involving
    Hilbert’s Hotel is central to the dialectic, and the discussion clari?es its use in
    supporting the Kalam cosmological argument.

    Next, in 136 you challenged chaining and raised a question that suggested that you disputed infinite chaining as being implicit in claiming an infinite past. I pointed out just what sort of stage by stage successive finite phase chaining was in mind using NASA, and took it that you know that advocates of an infinite past physical world extend such chains beyond the singularity, e.g. through fluctuations in a quantum foam or the model of in effect perpetually budding off subcosmi or oscillating models etc.

    Such chaining to stages beyond any finite stage q in N as could be counted, is infinite. What has been in effect argued is as though one can pack in infinitely many finite stages and not become transfinite in the sense that cumulative chain length count is w or the like.

    If for any stage Q that is in finite reach of now, there already was an indefinite further chain that implies directly that at any Q finitely remote from now [i.e. Q in Z- effectively] the transfinite succession has already happened. This begs the question.

    It also does not evade the force of the point that an onward chain of cardinality aleph null is transfinite, and therefore a suprertask to span. You never get to Q as at every step beyond Q you had to complete a prior supertask to get there.

    You are locked into spanning the transfinite in steps, begged as a question.

    KF

  158. 158
    hazel says:

    kf, do the negative integers have a beginning?

    Just yes or no.

  159. 159
    kairosfocus says:

    H, ET is right, you can extend numbers from zero in both directions, but this is not quite the same as descent in actual stages of time to now through cumulative causal succession. Yes, looked at ellipse in, Z- never begins: { . . . -2, -1, 0} but that’s the point. To get to -2 temporally [considering the past stages as labelled with numbers for convenience] you have to descend down the full beginningless span of that ellipse. Where for every stage Q finitely removed from -2, {. . . Q, . . . -2, -1, 0} you have had to descend to Q-2, Q-1, i,e, the chain replicates L-wards beyond Q in direct copy of to 0, i.e. the span is demonstrably infinite L-wards. And EVERY Q takes in the claim that the whole chain L-wards comprises finite values so that Q is bound by Q-1 for every Q in Z-. The chain is indeed beginningless L-wards in the sense of infinite. By trying to avoid descent from an acknowledged transfinite -K, which will never reach to Q due to the obvious supertask, the infeasible supertask does not go away, it is only harder to see. KF

    PS: This answers your tell me Y/N. With the implication that a claimed infinite past within Z- does not in fact evade the supertask. There is no warrant for claiming a transfinite task, we are only warranted to speak of a finite span from values Q that can descend in steps to 0 without trying to do a supertask. Abstract numbers do not face the constraint of successive, stepwise cumulative descent that time faces. And we are still not addressing heat death or provision of actually infinite heat reservoirs and heat sinks. The last actually brings in the problems of an actual physical infinity.

  160. 160
    daveS says:

    KF,

    DS, the same effective problem obtains for move over one room to get in one fresh guest. The same point obtains of full vs by rearranging mass, suddenly room appears. KF

    What exactly is the problem? I don’t see anything contradictory with this scenario.

    I’m assuming that all guests leave their rooms simultaneously (according to their synchronized clocks), shift down one room, then immediately reenter, thus avoiding the issue Snoke talks about.

  161. 161
    hazel says:

    OK, to be clear, if we think of the numbers as progressing from left to right, so each integer is the successor k + 1 to the previous integer k, do the integers have a beginning? I think we have all been taking the left to right progression for granted.

    If you want to think of the negative numbers as progressing to the left from zero, then the question becomes is there any “last moment” as you recede into the past – the point in the past that is farthest from zero.”

    Is there such a “last moment” that is the farthest in the past?

  162. 162
    kairosfocus says:

    DS, the hotel, a physical, spatially extended entity, with similarly physical guests, is full. That is each and every room without exception has a guest in it. Now, a further guest appears at the desk, having seen: hotel, no vacancy, fresh guests welcome. The manager instructs all guests in rooms n to go to room n+1. Suddenly (after what 5 minutes to move) room 1 is empty, even though there were the same number of guests and rooms as before. The fresh guest moves into room 1. Then, repeat m times as m fresh guests appear. The antinomies in a real world where abstract matching of sets with proper subsets 1:1 are not the only consideration, are patent. In a FULL hotel, every room — a physical entity — is occupied, you don’t create fresh empty rooms in a full hotel by moving guests once or m times, etc. KF

  163. 163
    hazel says:

    The Hilbert Hotel is a mathematical object. No one thinks it is a “physical, spatially extended entity, with similarly physical guests.” Anyone who thought that would think the whole discussion was stupid.

  164. 164
    daveS says:

    KF,

    In a FULL hotel, every room — a physical entity — is occupied, you don’t create fresh empty rooms in a full hotel by moving guests once or m times, etc.

    That’s because you’re thinking of a hotel with finitely many rooms, where this would not work. The Hilbert Hotel is infinite.

    And it behaves just as expected. In all these cases, the guests can be paired up one-to-one with rooms, both before and after the new guests arrive.

  165. 165
    kairosfocus says:

    H,

    your point has been repeatedly answered.

    The ellipsis means there is no definable last finite integer ranging out L-wards step by step from 0. Likewise, there is no definable least transfinite descending K, K-1, K-2, . . . K/2, . . . which is mirrored on the L-ward side, just harder to represent. We have a fuzzy border lurking under the seemingly simple ellipsis.

    Take the hyperbolic 1/x catapult down towards 0 and we have -m = 1/-K which is closer to zero than 1/-k for any -k in Z-. Shift the resulting infinitesimal cloud anywhere along the reals, to r in R. [Just add 0 + cloud to r.] There is no definable closest real to r, part of there being a continuum. Also, there are hyperreals yet closer to r than any neighbouring real. The fuzzy border phenomenon appears around every real.

    Coming back, I already wrote to you:

    Z- never begins: { . . . -2, -1, 0} but that’s the point. To get to -2 temporally [considering the past stages as labelled with numbers for convenience] you have to descend down the full beginningless span of that ellipse. Where for every stage Q finitely removed from -2, {. . . Q, . . . -2, -1, 0} you have had to descend to Q-2, Q-1, i,e, the chain replicates L-wards beyond Q in direct copy of to 0, i.e. the span is demonstrably infinite L-wards. And EVERY Q takes in the claim that the whole chain L-wards comprises finite values so that Q is bound by Q-1 for every Q in Z-. The chain is indeed beginningless L-wards in the sense of infinite.

    In short we can do an L-wards 1:1 for every Q in Z-:
    {. . . Q-2, Q-1, Q}
    { . . . -2, -1, 0}

    That is we see a self-similar copy of Z- appearing L-wards at every Q, i.e. the set is infinite L-wards. This is sledgehammer to peanut.

    There is no identifiable L-wards first element of Z-, that’s what we ALL know the ellipsis means, and that therefore every identifiable Q in Z- is bound by an indefinite onward extension L-wards.

    It is the consequence of this when we apply it to succession of temporal-causal stages approaching to Q from L-wards going R-wards that the significance appears. From Q L-wards, we have a copy L-shifted of from ) L-wards, i.e. the order type of the succession L-wards from Q is w, of cardinality aleph null. We have not got away from the supertask of descent down a transfinite span, we still face it. Such a span cannot be bridged going R-wards to approach Q no more than it can be spanned going R-wards from 0.

    This is the familiar supertask of spanning the transfinite in steps.

    Going further, an argument that in effect holds that at every Q in Z- that is in finite span of 0 [having been constructed in the set builder sense by mirror image of the von Neumann construction] is a stage where the L-wards transfinite has already been spanned so is not a problem, fails. Fails by begging the question at issue. Spanning an actual past of step by step successive stages from a “beginningless” past.

    We find here no warrant for begging or setting the question aside.

    We are only warranted to address finitely remote past points Q that do not involve an antecedent transfinite span to Q.

    Credibly, there was no beginningless past that advanced by successive cumulative stages to the present.

    Such a claim would require a supertask essentially similar to descent to now from a defined transfinitely remote stage, -K.

    KF

  166. 166
    kairosfocus says:

    DS, no, it is because I am thinking of a physical hotel. Full has an independent physical meaning: that each and every room has a guest in it, with no exceptions. Shifting guests from room n to room n+1 does not change that, regardless of the room numbers on the doors. The numbers are labels, they don’t change what it means for a room to be occupied and they don’t change what it means for ALL rooms to be occupied. Move over does not make an extra room appear R-ward so an absurdity results. KF

  167. 167
    hazel says:

    kf, you brought in the hyperreals again: B not A. You’re incorrigible, or something. If I have any brains, I’ll quit this discussion. (I imagine some lurkers think I should have done so ages ago.)

  168. 168
    daveS says:

    KF,

    DS, no, it is because I am thinking of a physical hotel. Full has an independent physical meaning: that each and every room has a guest in it, with no exceptions. Shifting guests from room n to room n+1 does not change that, regardless of the room numbers on the doors.

    If every guest leaves her room, shifts to the next-higher-numbered room and reenters, then the hotel is no longer full in this sense. Now there is an empty room, even though the exact same guests are still at the hotel.

    Where does the contradiction arise?

    If you can prove that this implies that some room is both occupied and not occupied, for example, then I would believe you have identified a problem.

  169. 169
    daveS says:

    hazel

    If I have any brains, I’ll quit this discussion. (I imagine some lurkers think I should have done so ages ago.)

    Does this sound familiar? 🙂

  170. 170
    kairosfocus says:

    H, an irrelevant appeal to irrelevancy. You or onlookers can easily glance at the OP and for example read Ehrlich. It will be clear that the surreals allow us to construct the full panoply of numbers, setting context. That will show that the reals fit into a context, spanned by w and -w. Onward lie the transfinites, and the infinitesimals. I showed why a hyperreal cloud also surrounds every real, in the context of there being no definable nearest real to any r, i.e. the reals lie on a continuum. Therefore, the phenomenon of a fuzzy border between the reals mileposted by integers and the transfinites is part of a PERVASIVE structural phenomenon, something we need to get used to. In that context, I was further able to answer your challenge (yet again) on a L-most negative integer: there is no definable last L-ward integer, just as there is no definable last R-ward negative hyperinteger. Going further, for every Q in Z-, finitely removed — in steps traceable to the von Neumann construction — from 0, there is an onward L-going effective copy of the negative integers. This means that Z- has the expected transfinite cardinality and that the Q onward segments likewise have the same cardinality, aleph null. Therefore, as we saw: a claimed, beginningless temporal-causal descent in stages to Q will be infinite, comprises an infinite span that cannot actually be completed in finite stage steps, it is a supertask. There is only warrant to speak of a finite past that descends to now. KF

  171. 171
    kairosfocus says:

    DS, every room was full, occupied. There simply are no extra rooms R-ward to be filled as guests move around. This is not reverse musical chairs with an extra chair there that is waiting, empty or that appears out of nothing. KF

  172. 172
    ET says:

    Is there are farthest point that you can get to the edge of a cliff before you fall off? Imagine that you are just a mathematical point. And with each step you would travel 1/2 the distance to that edge. Will you ever reach it? You shouldn’t but the edge is still there.

  173. 173
    ET says:

    hazel, we only know there is a (-1) because we started/ began @ 0. And in accordance with all accepted standards we start/ begin counting with the lowest number.

  174. 174
    hazel says:

    kf writes, “There is only warrant to speak of a finite past that descends to now.”

    I agree with that, and have agreed with that from the beginning.

    But there is no beginning to that past, so the past is infinite in the same way the future is: no matter how long ago you think the past started, it started before that.

    The set of negative integers is infinite, and all of them are finite.

  175. 175
    daveS says:

    KF,

    In the scenario I described, all the guests exit the hotel momentarily, so at one point it is empty.

    Let’s say the first guest decides to leave at this time. Then all the other guests shift to the next room down.

    Will the hotel still be full once the guests re-enter?

  176. 176
    hazel says:

    yes, ET, that’s why we don’t start counting at 0 when negative numbers are concerned.

  177. 177
    kairosfocus says:

    DS, if it was full and the decision to shift is cut off in midstream as you suggest, that is a return to status quo. However, this does not undo what has already been shown. KF

  178. 178
    kairosfocus says:

    H, sufficient has already been shown. KF

  179. 179
    daveS says:

    KF,

    To be clear, guest 1 leaves the hotel.

    All the remaining guests are standing outside their rooms.

    Then guest 2 enters room 1, guest 3 enters room 2, and so forth, all simultaneously.

    The hotel is again full.

    (This is my answer to the question).

  180. 180
    hazel says:

    re172: to ET. Depends how fast you go. If you can go 1/2 way in 1/2 hr, the next 1/4 in 1/4 hr, etc. you’ll fall off the edge in an hour.

    Or maybe the tortoise never does catch the hare?

    But, of course I know that if the steps each take the same amount of time, you won’t fall off, but you will come infinitely close to falling off, which can be rigorously defined as for any number e, no matter how small, some number of steps well get you closer than e to the edge.

  181. 181
    kairosfocus says:

    DS, yes, further oddities. They don’t change what we have already seen. We have good reason to reject the idea of instantiation of the actual countable infinite in physical reality. This of course does not affect the abstract realities of structure and quantity built into every possible world. For example, that an abstract continuum will so exist. KF

  182. 182
    kairosfocus says:

    ET & H: A converging sequence of partial sums marking stages of a trajectory will of course go to a convergence on a finite limit in many relevant cases, and this in finite duration. Where, convergence speaks to ever closer approach so that beyond the kth partial sum, every onward partial sum will be within some arbitrarily small d-neighbourhood of the limit L, i.e. the difference between S_k, S_k+1 on and L, e, will be within d of L, where as k rises d and e can get ever smaller. If an entity moves to a cliff at steady speed, every smaller increment will take a correspondingly shorter time so it actually simply steadily moves toward then exceeds the limit. In this case, over the cliff. What would be different is when the time to move to the next partial sum does not converge, i.e. there is a steady slowing. In that case, for suitable values, the limit will never be reached as the entity has slowed to an effective stop. KF

    PS: Where Achilles is racing against the tortoise, the best view is that even with a head-start, there is an overtake-point where the two trajectories converge and thereafter the faster is in the lead. We are familiar with this from how faster moving vehicles catch up to and overtake slower ones.

    PPS: This shows how the series-limit approach works, since mid C19 that is the standard analysis approach to Calculus and a gateway to much more. Infinitesimals and hyperreals etc are connected to non-standard analysis. It seems, both have a point and open up different vistas. For myself, I can say that the surreals give a panoramic view that is in itself illuminating.

  183. 183
    daveS says:

    KF,

    DS, yes, further oddities. They don’t change what we have already seen. We have good reason to reject the idea of instantiation of the actual countable infinite in physical reality.

    If we are allowed to appeal to empirical science (cosmology, e.g.), I agree.

    I’m not convinced that the scenario I described in #179 (or an analogous one, say involving the infinite array of snooker balls) is impossible in every possible world, FTR.

  184. 184
    ET says:

    hazel:

    yes, ET, that’s why we don’t start counting at 0 when negative numbers are concerned.

    Except we do start counting from zero when negative numbers are concerned. The first negative number is (-1)

  185. 185
    ET says:

    hazel:

    Depends how fast you go. If you can go 1/2 way in 1/2 hr, the next 1/4 in 1/4 hr, etc. you’ll fall off the edge in an hour.

    So infinite steps can be had in an hour? Really? Looks like hazel doesn’t understand infinity.

  186. 186
    kairosfocus says:

    ET, once we refer to a continuum and have a limiting process that converges to in effect infinitesimal space increments and time increments, yes, an infinite traverse of points in a trajectory can be completed in a finite time. Where, recall, integers are mileposts in the continuum of reals and rationals are on the line but next to any given real r there is no nearest neighbour. Move your hand to type, you complete this sort of infinity of abstraction in a finite time, Draw a line with a pencil, the same, etc. KF

  187. 187
    hazel says:

    re 185: 🙂 What kf said in his first sentence.

    If you draw a line 1 foot long in 1 second, you just went through an infinite number of points in a finite amount of time.

  188. 188
    hazel says:

    ET, when you drive down the highway at 60 mph, you drive 1/2 a mile in 1/2 an minute, the next 1/4 mile in 1/4 minute, the next 1/8 mile in 1/8 minute. and so on. Are you telling me you’ll never drive the whole mile!!!

  189. 189
    ET says:

    hazel, Stop changing my scenario. It only makes you look stupid an desperate.

  190. 190
    ET says:

    kairosfocus:

    ET, once we refer to a continuum and have a limiting process that converges to in effect infinitesimal space increments and time increments, yes, an infinite traverse of points in a trajectory can be completed in a finite time.

    I agree. But that has nothing to do with my scenario.

  191. 191
    hazel says:

    to ET, re 189: I was responding to what you said at 185: all I did was make it a specific distance (1 mile) and change hours to minutes. Otherwise, I wrote exactly what you wrote.

  192. 192
    ET says:

    Is the car the size of a mathematical point? If not then you don’t have one. Was the car traveling half the distance each time?

  193. 193
    hazel says:

    We’re thinking of the car as a point. And yes, the “car” is travelling 1/2 the distance each time, but it is taking 1/2 the time each time also, as I wrote.

  194. 194
    ET says:

    Then it will never travel a mile.

  195. 195
    hazel says:

    But, ET, you do that ever time you drive a mile at 60 mph: you drive 1/2 mile in 1/2 minute, then you drive 1/4 in a 1/4 minute, and so on, and obviously you do drive the whole mile. And then you repeat the next mile. What you are saying is that no one could ever get anyplace!

  196. 196
    daveS says:

    Then it will never travel a mile.

    😲

  197. 197
    ET says:

    No, hazel. If there are infinite steps then they will never be completed. It’s impossible.

    And people do not travel like that so your point is moot.

    If someone is going to a grocery store and can only travel half the distance to it, stop, and then travel half the remaining distance, and so on, they will never reach that store. But we don’t travel like that, do we?

  198. 198
    hazel says:

    Please read carefully, ET: you drive 1/2 mile in 1/2 minute, then you drive 1/4 in a 1/4 minute, then 1/8 mile in 1/8 minute

    If the steps were each taking the same amount of time, then you would never go a mile.

    But the steps in this situation are also taking less and less time in proportion to the distance.

    The situation I have described is just one where someone is driving 60 mph. Of course they
    travel a mile.

  199. 199
    hazel says:

    P.S ET: kf agrees with me that you are wrong, FTR.

  200. 200
    ET says:

    LoL! @ hazel- Of course if you change the scenario you can travel a mile. But that is cheating and cheating is losing.

  201. 201
    ET says:

    hazel- I am not wrong and I have explained why. You can ignore that explanation but that just proves my point.

  202. 202
    hazel says:

    From above:

    188
    Hazel April 21, 2019 at 8:34 am
    ET, when you drive down the highway at 60 mph, you drive 1/2 a mile in 1/2 a minute, the next 1/4 mile in 1/4 minute, the next 1/8 mile in 1/8 minute. and so on. Are you telling me you’ll never drive the whole mile!!!

    193
    Hazel April 21, 2019 at 8:58 am
    We’re thinking of the car as a point. And yes, the “car” is travelling 1/2 the distance each time, but it is taking 1/2 the time each time also, as I wrote.

    194
    ET April 21, 2019 at 9:02 am
    Then it will never travel a mile.

    This is what you said about the scenario I offered.

    I don’t think you understand the difference between steps of decreasing size taken in equal times and steps of decreasing size taken in proportionately decreasing times.

    In the first case, you’ll never reach one mile (or the edge of the cliff), but in the second case, to which you referred at 194, you will reach one mile in one minute.

  203. 203
    ET says:

    hazel:

    I don’t think you understand the difference between steps of decreasing size taken in equal times and steps of decreasing size taken in proportionately decreasing times.

    In the second scenario you run out of time before reaching the target.

    Again, any scenario requiring infinite steps can never be completed. If you change that scenario then I my claim does not pertain to it.

  204. 204
    ET says:

    ET, when you drive down the highway at 60 mph, you drive 1/2 a mile in 1/2 a minute, the next 1/4 mile in 1/4 minute, the next 1/8 mile in 1/8 minute. and so on. Are you telling me you’ll never drive the whole mile!!!
    193
    Hazel April 21, 2019 at 8:58 am
    We’re thinking of the car as a point. And yes, the “car” is travelling 1/2 the distance each time, but it is taking 1/2 the time each time also, as I wrote.

    If it reaches the mile then somewhere along the line it traveled more than half the distance left.

  205. 205
    daveS says:

    Which weighs more, a ton of feathers, or a ton of bricks? 🤔

  206. 206
    Axel says:

    How long is a piece of string ?

  207. 207
    kairosfocus says:

    DS, ton as mass or as weight (given the upthrust problem)? KF

  208. 208
    kairosfocus says:

    ET & H: on the scenario that say every minute an object moves half the remaining distance to a target, then the rate of approach is slowing down continually and the convergence to the limit becomes a supertask. KF

  209. 209
    hazel says:

    Kf, as I said at 202, if the steps take equal amount of time, such as one minute per step, then the limit will never be reached, as you say at 208.

    However, if the steps are taking proportionate amounts of time (1/2 mile takes 1/2 minute, 1/4 mile takes 1/4 minute, 1/8 mile takes 1/8 minute), then in a minute you will go a mile.

    I don’t think ET understands the difference between these two situations.

  210. 210
    ET says:

    hazel, Please stop it. They are TWO DIFFERENT SCENARIOs. If you can only travel half the distance each time you should never reach the target.

    How many halves are there between 0 and 1?

  211. 211
    hazel says:

    ET, are you saying you never reach the target in both scenarios?

  212. 212
    daveS says:

    KF,

    Weight, of course! 😛

  213. 213
    ET says:

    hazel- answer my question:

    How many halves are there between 0 and 1?

  214. 214
    hazel says:

    re 213: 2.

    Two halves make a whole. 🙂

  215. 215
    ET says:

    So half way from the first half is the remainder? First graders in my school district know better.

  216. 216
    hazel says:

    Here’s a old puzzle, ET. Do you agree with the conclusion, or not

    A rabbit is trying to catch a turtle. (These are mathematical animals: just points on the number line.) The rabbit can run 10 feet per second (fps) and the turtle can walk 1 fps. The turtle has a 10 foot headstart.

    In one second, the rabbit runs to where the turtle started, but of course the turtle has walked 1 foot from his starting point.

    In the next 1/10 second, the rabbit runs to where the turtle was, but the turtle has walked another 1/10 foot.

    In the next 1/100 second, the rabbit runs to where the turtle was, but the turtle has walked another 1/100 foot.

    This goes on forever. Whenever the rabbit get to where the turtle was, the turtle has moved a bit farther.

    Therefore the rabbit never catches the turtle.

    Mathematically,

    Rabbit goes 10 + 1 + 1/10 + 1/100 + 1/1000 + …., which approaches 11 1/9 feet as a limit

    Turtle goes 10 (his headstart) + (1 + 1/10 + 1/100 + 1/1000 + 1/10,000 + …) , which approaches 10 + 1 1/9 = 11 1/9 feet as a limit.

    Both sequences has an infinite number of terms. For a given number of terms n, the rabbit is always (1/10)^(n-1) ahead of the rabbit.

    Since you can’t actually go an infinite number of terms, the rabbit never catches the turtle.

    What do you think, ET?

  217. 217
    ET says:

    hazel- rabbits don’t chase turtles. And if they did, they would easily catch them.

    Zeno doesn’t play here. But I enjoy your desperation.

  218. 218
    hazel says:

    Desperately trying to explain to you! My bad.

  219. 219
    ET says:

    Explain what? That you have to change my scenario in order to show my conclusion from it is wrong?

    Your explanation doesn’t have anything to do with what I am saying.

  220. 220
    ET says:

    Your whole problem, hazel, is that you are trying to explain without reading and responding to what I post. That doesn’t work.

  221. 221
    hazel says:

    No, you are right about your first scenario, if we understand the idea of limit correctly.

    You are wrong about the second scenario.

    I haven’t confused the two situations.

  222. 222
    ET says:

    Your second scenario either misses the point or it doesn’t follow the rules. And it cuts to the reason why you avoided my question on the halves.

  223. 223
    kairosfocus says:

    H ( & attn ET): I note from your comment at 209 in response to my supertask remark at 2008:

    Kf, as I said at 202, if the steps take equal amount of time, such as one minute per step, then the limit will never be reached, as you say at 208.

    I had commented in 2008:

    on the scenario that say every minute an object moves half the remaining distance to a target, then the rate of approach is slowing down continually and the convergence to the limit becomes a supertask.

    It is plain that ET was speaking to this case, from the above. That settles the exchange with him on his main point; he is right.

    The further import of this, is that a succession in finite stage time steps comparable to the ordinals cannot be completed at any value we can count up to, k, even when there is a convergent series in space and even as we do approach the spatial point asymptotically. (Of course, descent to the converging infinitesimals of time as increments allows convergence in finite time. That solves the Zeno’s paradox-type challenges.)

    Previously, I have pointed out that this can be shown on one way to identify that we deal with a transfinite set: on attaining to some k, we continue in a way that can be matched 1:1 with going from 0, without definable upper limit:

    k, k+1, k+2 . . . k+k . . .
    0, 1, 2 . . . k . . .

    This is the same set if we reverse the direction of listing:

    . . . 2, 1, 0

    or if we then take the mirror image in 0, we just switch to the negative integers:

    . . . -2, -1, 0

    As a consequence, if it cannot be spanned in finite temporal stage steps going up, the same obtains for reversing the set and going down, likewise for the mirror in Z-. (Obtained through taking additive inverses with 0 as identity element.)

    Where, there is an additional factor. To reach to some Q finitely remote from 0 in Z-, a prior descent has to happen which can be matched to Z-:

    . . . Q-2, Q-1, Q
    . . . -2, -1, 0

    in the context of:

    . . . Q-2, Q-1, Q . . . -2, -1, 0

    Beyond Q is just as transfinite as beyond 0 in Z-, and is just as impossible of traverse across the span indicated by the ellipsis. [Truncating a finite segment to 0 in a coutably transfinite set has not eliminated its essential transfinite extension.] Where, Q is a general element of Z- finitely remote from 0. That is, we here see that the descent in proposed beginningless time will never reach a finite range from 0, never mind ascending thereafter from 0 to n, now.

    This brings home the point that we have no warrant to speak of traversing a transfinite, countable span in finite stage countable steps. Though of course, as the use of the hyperbolic catapult function 1/x shows, we may leap across it as a mathematical, single-step operation.

    KF

    PS: I point to the 1/x catapult, as it hammers home that *R and R are connected and even interwoven. K GRT any k in N is such that 1/K = m in (0,1], and the cloud of similar infinitesimals can be transferred to any r in R or indeed in *R by simple addition of 0 plus the cloud. For a given r, r + m is a hyperreal closer to r than any r + n where n = 1/k, k being a natural. And of course, as k may go without limit there is no definable pure nearest real to a given r in R.

  224. 224
    hazel says:

    kf writes, “That settles the exchange with him [ET} on his main point; he is right.”

    Yes, I agreed with him about scenario 1.

  225. 225
    ET says:

    Your second scenario either misses the point or it doesn’t follow the rules. And it cuts to the reason why you avoided my question on the halves.

  226. 226
    hazel says:

    What rules? 🙂

    And yes, I know I was flippantly literal when I answered your question “How many halves are there between 0 and 1?” As written, the answer is 2. But I know what you meant: you can keep cutting the remaining interval in half indefinitely. I have explained the idea of this approaching a limit multiple times.

  227. 227
    ET says:

    The rule of only going 1/2 the distance to the target.

    Look, obviously all you can do is change the scenario and disregard everything about it, in an effort to score imaginary interweb points.

    That doesn’t work here

  228. 228
    hazel says:

    My second scenario had you going half way to the target each step also. I didn’t change that.

  229. 229
    ET says:

    LoL! If that is true then you will never reach the target. So clearly you changed something.

  230. 230
    kairosfocus says:

    ET, in the limit of the sequence of partial cumulative sums, you do attain the target. In the case of steady speed, each successive increment takes less and less time and in the end an infinite series is completed through infinitesimal increments in infinitesimal times. The net result is why we drive up to and pass a given remote point on a road, or with the extension of overtaking a slower moving vehicle with a lead, it is why we can catch up and overtake. These are tied to foundations of Calculus. KF

  231. 231
    hazel says:

    Thanks, kf. You explained well what my scenario 2 was about, in which case you do cover a mile.

  232. 232
    ET says:

    Yes, kairosfocus, I understand that but that was never in doubt. The point is and always has been that if you have to take those infinite steps you will never reach the target. I have already stated that is NOT the way we travel. We skip over those infinite steps as if they weren’t there.

    As I said, clearly something changed between scenario 1 and 2. Meaning the following was untrue:

    My second scenario had you going half way to the target each step also. I didn’t change that.

  233. 233
    hazel says:

    But obviously you don’t understand kf, because my second scenario was just a car travelling at constant speed, which, as kf said, why you would cover a mile.

    You write, “clearly something changed between scenario 1 and 2. Meaning the following was untrue: ‘My second scenario had you going half way to the target each step also. I didn’t change that.'”

    As I have explained, what I changed was the time each step took. In scenario 1, the time for each step was constant, so the all the steps could never be completed, as you said. In scenario 2, the time for each step was getting less, so the car went a mile.

    If you really understand what kf said, you will understand that in scenario 2 you do cover all the steps, and you do go a mile.

  234. 234
    ET says:

    But obviously I do understand KF. Your scenario does NOT follow the rule of only taking half steps- ie going ONLY half the distance and therefore requiring infinite steps.

    You changed more than the time, obviously. You see, hazel, infinite steps can NEVER be completed- and that is regardless of the speed.

    If you really understand what kf said, you will understand that in scenario 2 you do cover all the steps, and you do go a mile.

    If you really only took half-way steps then you could never complete the mile. So clearly you didn’t.

  235. 235
    daveS says:

    ET,

    Suppose you walk the length of a football field in 1 minute, at constant speed.

    30 seconds in, you cross the 50-yard line (having walked half the distance).

    At 45 seconds, you are at the 25-yard line (having walked half the remaining distance).

    At 52.5 seconds, you are at the 12.5-yard line

    etc.

    In the process, you traverse infinitely many intervals.

  236. 236
    ET says:

    Yes, daves, I have already been over that. More than once, too.

  237. 237
    daveS says:

    ET,

    Are you thinking that you must *stop* at each halfway point?

  238. 238
    ET says:

    Does that matter? How?

  239. 239
    daveS says:

    ET,

    If you stop for 1 second at each halfway point, you would never finish.

    I’m not understanding what the “rule of only taking half steps” amounts to I guess. Could you describe traversing the football field:

    1) Following this rule

    and

    2) Not following this rule.

  240. 240
    hazel says:

    So, ET, do you walk across the whole football field, or not?

  241. 241
    ET says:

    daves- So if I am taking infinite steps without stopping, I can complete them? Is that what you are saying? Really?

  242. 242
    ET says:

    hazel, Anyone can just walk across a football field. That was NEVER in question. NEVER.

  243. 243
    daveS says:

    ET,

    No, I’m asking what the rule of “taking half steps only” amounts to. Can you answer #239?

  244. 244
    ET says:

    Unbelievable. It amounts to infinite steps. Just as I have been saying for days.

  245. 245
    daveS says:

    ET,

    Do you mean (once you get close enough to the goal line) literally moving your feet only a fraction of a millimeter at a time?

    Perhaps you can describe what videos of the two scenarios would look like

  246. 246
    ET says:

    daves:

    Do you mean (once you get close enough to the goal line) literally moving your feet only a fraction of a millimeter at a time?

    That is what it means, daves.

    Perhaps you can describe what videos of the two scenarios would look like

    Nope. I have wasted more than enough time on this already.

  247. 247
    daveS says:

    ET,

    That is what it means, daves.

    It would be hard to maintain a constant speed under these conditions.

  248. 248
    ET says:

    Yes, daves, it would. Which is why hazel changed more than just the time.

  249. 249
    kairosfocus says:

    ET, I think your point may be effectively summarised for the Mathematical minded as: if one takes successive, finite sized, finite duration steps, one cannot traverse a transfinite span or complete a transfinite process. In the overtaking case, that one completes an infinity of steps in finite time pivots on how the time per step and the distance per step converge through infinitesimal increments that approach a limit. For ordinary people, it is common sense to see that steps take appreciable time, and that at each step only half the remainder is bridged. Where obviously if dx is getting ever smaller but time per step remains large, at the nth step one is looking at dx_n/t, which is heading to zero, i.e. one is slowing down at every step and we can see why the gap will not be closed in finite time. Indeed, once dx_n is infinitesimal in the hyperreals sense (we won’t even get there!) one is closer to zero than any 1/n, where n is a natural number; here going 1/2, 1/4, 1/8 etc. We won’t even get to the infinitesimal cloud around zero in any natural number of steps. One cannot traverse the transfinite in cumulative finite stage steps. In this case, as one is ever slowing to a stop, the hare is becoming a tortoise and then even slower. KF

  250. 250
    daveS says:

    For ordinary people, it is common sense to see that steps take appreciable time, and that at each step only half the remainder is bridged.

    Are you calling ET “ordinary”? 😛

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