Gödel’s proof of the existence of God

_{News
May 10, 2020

Intelligent Design, Mathematics, Philosophy, theism

2}_{Categories
Intelligent Design
Mathematics
Philosophy
theism}

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You didn’t know, possibly, that when he thought we was dying, he showed the notebook to one of his colleagues, who copied out the proof:

In an unsanitized, politically incorrect (but factual) history, Selmer Bringsjord talks about how the tormented genius Kurt Gödel took up a quest that dated back a thousand years to prove the existence of God by formal logic. His original version didn’t quite work but his editor’s version passed an important logic test:
“When we go to Gödel, we skip over the modern advocates of this argument. It’s harsh—I’m just going to say it—from the standpoint of someone who’s reasonably well-versed in formal logic, I think it’s a bit of a doldrums, despite some of the attention, until Gödel does his thing.
Gödel does it formally and then some folks in Germany, doing automated reasoning, verified it a few years back. They verified the version that Dana Scott copied out of the notebook. That is, what they verify is that there is no doubt; it’s machine-verified proof. So now we’re left with just the truth of the premises and how we judge them.”
News, “Gödel and God: A surprising history” at Mind Matters News

Further reading:

Faith is the most fundamental of the mathematical tools: An early twentieth century clash of giants showed that even mathematics depends on some unprovable assumptions. (Daniel Andrés Díaz-Pachón)

and

God’s existence is proven by science. Arguments for God’s existence can be demonstrated by the ordinary method of scientific inference. (Michael Egnor)

Comments

Kairosfocus: It's *R by the way, not R*. I can't imagine you seeing anything of hyperreals except at University, maybe even graduate school, unless you just read it yourself. It's certainly not standard to make these kinds of arguments in undergraduate Analysis courses. I think we've exhausted this topic to be honest. Like I said: I am NOT saying there is an infinite past, I am not proposing an infinite past, I am saying that I didn't find your mathematical argument against it sound. That's it. I don't think continuing the discussion any further will substantially move things forward. Let's call it a day shall we?JVL_{May 16, 2020
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JVL (& attn DS), the hesitancy to go there speaks. There is something uncomfortable with the appeal to the structure of R mileposted by Z. That's where the problem I have been pointing to lies. Before going to that, I note on why of old the Steady State vision of a cosmos that had an undetectably low steady origin of matter etc was preferred to where the empirical evidence forced things in the '60's to 70's. It has no beginning, in effect the root of reality would be the cosmos, so reducing the first law of thermodynamics to an approximation was a good fit to the dominant preference. It just did not fit the data. The hard beginning at the singularity implies a contingent, caused world, given the challenges of a cosmos from utter non-being. It is unsurprising to see models that move to multiverses, quantum foams, brane collisions etc. All of which move beyond empirically controlled science and all of which are philosophical. Which on comparative difficulties cannot be provincial; factual adequacy is global, coherence is global, explanatory elegance and power are global. For instance, the logic of being is relevant, the moral government of our rationality is relevant [thus, the IS-OUGHT gap], the roots of functionally specific complex organisation and associated information is relevant, observed fine tuning is relevant, and more. Dressing up in the lab coat and posing on evolutionary materialistic scientism backed by selective hyperskepticism will not do. On many of these grounds, the evolutionary materialistic world picture falls apart as utterly incoherent, factually inadequate and forced into ever wider ad hoc patchwork. Coming back to the narrower question, Mathematics is about the logic of structure and quantity, i.e. it has deep roots in logic of being. In ontology, in short. Key abstract entities and structures are necessary beings, part of the framework for any possible world. Which is what gives it its power, answering Wigner's wonder. Now, let us look at Z as mileposting R. Clipping 146 above (and beyond):
We can, however capture Z and R in context and that makes a difference, through the hyperreals, R* mileposted by Z*. We then see that there are transfinite spans extending these numbers into the specifically infinite ranges. Mathematically, these transfinite spans cannot be traversed in finite stage successive steps, which is why step builder notation points to the completion of a potential infinity by using ellipses or the equivalent. Where also, the 1/x catapult say prof Carol Wood used allows us to see that from 0 to transfinites, we have a unified space and structure of quantities. Where, infinitesimals and transfinite hyperreals have as much claim to “reality” as Complex numbers, negative numbers, zero, real but irrational numbers such as pi or e etc. We may freely model: . . . . K’ –> K’+1 . . . . -2, -1, 0, 1, 2 . . . n . . . . w . . . . K-1. K, K+1 . . . ., where 4-dot ellipses denote specifically transfinite ellipses and the continuum is mileposted by numbers in Z* as shown [and where K + K' = 0, giving the negative range through additive inverses].
The span in the middle sets R and Z in wider context. Where, I remind, the 1/x catapult function between infinitesimals and transfinites secures a unified number space. There is no gold standard special threshold R and Z pass that R* and Z* don't, reducing them to second class and hyperskeptically suspect status. Recall, [0,1] has in it the catapulted image of every number from 1 on, with the infinitesimals near 0 being a cloud -- monad is a term used -- closer to 0 than any 1/z, z in N. The continuum is exceedingly strange. Those infinitesimals are a legitimate foundation of the Calculus, once suitably tamed. If infinitesimals like h are valid and part of the framework, so are transfinite hyperreals such as H = 1/h, period. Addition allows shifting the cloud of hyperreal infinitesimals, revealing that any r in R is surrounded by a similar cloud to 0. In effect, the number line is a vector space, once negative numbers give another direction and extension. Extension to C is instant, through the j* rotation operator. And of course, that cloud around an arbitrary real, is part of how calculus gains its power. The hyperreals pervade the reals and are everywhere incredibly dense. And by the similarity principle for R*, the same pattern extends to the transfinite realm. The continuum is exceedingly strange. This little exposition brings out part of what I think is going on. Paradigm shift. We are used to thinking in terms of N, Z, Q, R, maybe C. Now, I am arguing that R* pervades R everywhere with incredible density, giving fresh force to the idea of the continuum. This, we are decidedly not used to, the inextricable intertwining of R* and R, connected to the power of Calculus. The expression r + dr takes on new meaning once we see that dr is an image from the infinitesimal cloud around 0 vector shifted to r's close neighbourhood. Paradigm shift, r* is everywhere dense with infinitesimal-altered numbers in R and beyond of course. And not even mind bleach is going to take us back to the "safe" old days before the R* pandemic. R* is the new normal. Vouched for, by the power of Calculus. BTW, reducing an infinitesimally altered number in R* to a standard value in R looks uncommonly like taking a limit as dr --> 0; i.e, going to the core of the cloud around a particular real. A further intuitive bridge to familiar territory, here, the epsilon, delta close neighbourhood, limits picture as now influenced by the infinitesimally altered number picture. We need those pictures as plausibility structures. For sure, I needed them back in 4th form and arguably in 2nd form as we looked at asymptotes on graphs and on those pointing arrows on number lines. In College, the idea of an open terminus of an interval with no definable specific closest neighbour was also obviously connected. It also shows the echo of that transfinite ellipsis that I have chosen to highlight by using a 4-dot symbol not the usual 3. In a continuum, borders are fuzzy not crisp unless you insert a Dedekind cleavage, a schnitt. (I gather the word tends to a stronger suggested force in German.) Let us hold that fuzzy cloud in the small picture, vector shift it back to 0 and use the 1/x catapult to see its dual in the large. The border between the transfinites and the finites is . . . fuzzy, with no nearest definable finite neighbour to the transfinites and the converse. Oddly, there is a parallel in history. In the old days, borders of great empires were not crisp, surveyed lines on a map [formerly fought over by competing Kings and reflecting the emerging technology of precise survey], but graded zones of influence. Here, a review on a book on borders of the Roman Empire:
In the final third of the book Breeze reaches several conclusions. The most important of these is that the visible “defenses” had different purposes in different theatres, and often several at once. These included defensive lines, naturally, but also trip wires against raiders, population and migration control, customs enforcement, and more. The “barriers” in fact usually did not mark the actual limits of imperial authority, which tended to extend well beyond them in diminishing zones of influence; the neat lines found on maps defining Empire from Other did not actually exist on the ground.
Paradigm shift, fuzzy structures are real, and the reals are fuzzy, not just in the cloud around a safely finite real, but in the large, gradually and undefinably fading out as we come to the transfinites. This reflects the phenomena of the surreals -- isomorphic to certain developments of the hyperreals -- whereby we can go on stepwise to develop a complex structure of numbers around a given transfinite. Similarly, for hyperreals specifically. So, the idea of counting up in steps from K' to K'+1, etc is not unreasonable and the result that it is a supertask to try to span from there to finites in steps reflects the fuzzy zone. (Recall, K' is in the additive inverse region, negatives.) In that context, a further point of paradigm shift. Namely, that when we speak of z in Z as mileposting R then showing how any particular z can be exceeded R-ward z+1, or its additive inverse z' can be exceeded l-ward, we have crispified and so have yet to address the onward fuzzy zone to the transfinites. That is what the pointing arrows and ellipses are telling us, also the round braces marking open ends of intervals. Crispification does not mean the fuzzy zone is not there, it is pervasively present as we look to the zone where finites go on and on with the recognised transfinites on the other side. It is, in my view, the want of reckoning with that fuzziness that leads to the idea that crispification by count from zero or by simple algebraic symbolisation -- which brings in count up from zero so soon as one does substitution instance -- allows us to effectively disregard the implicit onward structure of the fuzzy zone. The ellipses etc are part of the structure of Z and R. They point to the fuzzy zone, and with the suffusing or pervading of R by infinitesimally altered reals, R and R* simply cannot be separated anyway. We have a unified, inherently transfinite space and structure of quantities, one embedded in the framework for any possible world. It is from this different paradigm that I am seeing. In this context, when we insist on crispifying durations n - p, we are inherently on the finite side of the fuzzy zone, we have made the infinite character implicit. Consequently, I suggest a modified way of thinking in light of the unified vector space: for any defined past instant or stage p that can be connected to n, now, here 2020, in a sequence of cumulative finite stage steps, n - p := d, duration since p, will be finite. However, the very act of crispification locks us in on this side of the fuzzy zone and the interconnexion of stepwise, finite stage access locks in that we are only warranted to talk of finitely remote p's. The idea that we can capture the fuzzy onward span L-ward by saying "any p" becomes a conceptual leap too far. In short, it seems that the very structure of R is locking us into talking of finitely remote points once we crispify, but the onward fuzzy zone is material. That is part of what is exercising me, the frustration of the receding fuzzy zone as we count outwards or as we crispify by symbol that represents a finite count. This is a key part of why I think the way we have seen discussed above fails to capture the challenge of an alleged transfinite past. I think crispification implicitly, perversely, subtly locks us on this side of the fuzzy zone. The fuzzy zone is material to the beyond crisp limits character that conveys the countable transfinite character of Z mileposting R. In this context, I further believe and argue for cause, that once we make the transfinite character of duration as claimed, we then are on the other side of the fuzzy zone and cannot cross it in finite stage steps to reach a finitely countable range of 0. The structure is frustrating the attempt, not a misconception. Hence, why I think we are talking on two sides of a paradigm shift. I believe that, implicitly, intuitively, inchoately, by 4th to 6th form, I was on this side of the paradigm. The bringing in of Surreals and Hyperreals simply gave me a better apparatus for expressing where I had been for decades. Maybe the best symbol for that was definite integrals with infinite limits. Those, were crying out for a number space that allowed us to make sense of that. R* and its cousin, the Surreals, have given me that space. From that perspective of an intellectually fulfilled hyperreals thinker on the structure and logic of quantity, I can make sense of the puzzles from those days, once I took on board the 1/x catapult and of course the vivid image of the hyperbola. We can christen it, the hyperbolic catapult. In addition, the concept of duration, d = n - p, allows me to see why I have been so uncomfortable with attempts to speak to a claimed infinite past with all symbolised past points p comfortably finitely remote. The fuzzy zone, perversely, is being inadvertently locked out by the implicit finitisation by algebraic symbol. Which then has consequences for how one interprets the matter at stake. Further, that inadvertent lockout is why I maintain that the reference to K' is legitimate, giving d = n - K' (defined on a stepwise count metric) an inherently transfinite character and pointing to why once the implicit lockout is recognised, we are not warranted to suggest a transfinite actual past that accumulates stepwise to now, in finite stages. All of this then becomes part of the worldviews exchanges. I do not expect agreement across paradigms, I do think there is a right to suggest that a "gold standard" lockout, however, is not warranted. KFkairosfocus_{May 16, 2020
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Kairosfocus: And, it says something about the evolutionary materialistic worldview that it is forced to commit itself to a past-infinite material world (including some pretty bizarre claims to make such seem half plausible), which should give pretty serious pause to would-be materialists and sympathisers. I am NOT committing myself or anyone to an infinite past, I am merely pointing out that you could have one an NOT be transversing the transfinite. I am disagreeing with your use of mathematics. I am happy to consider that time began with the big bang and leave it there.JVL_{May 15, 2020
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DS, so far as I can see, all you are doing is leaving the transfinite traverse implicit and possibly lurking under a contradiction or two. I can note that if all traverses between stages of time are finite, that says something regarding the global set, if that were meant literally. And, it says something about the evolutionary materialistic worldview that it is forced to commit itself to a past-infinite material world (including some pretty bizarre claims to make such seem half plausible), which should give pretty serious pause to would-be materialists and sympathisers. KF PS: It's been quite a day.kairosfocus_{May 15, 2020
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KF, Each interval between any two points in time is finite. The set of durations of these intervals is unbounded. It's just like the statements I posted above about intervals in the set of real numbers.daveS_{May 15, 2020
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DS, explain the actual duration. KF.kairosfocus_{May 15, 2020
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I’m afraid I have to award this round to JVL.Ed George_{May 15, 2020
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PS: To clarify, it does not imply ever being at a point at infinite remove from the present.daveS_{May 15, 2020
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KF, No. That is all.daveS_{May 15, 2020
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JVL, to propose an actual, infinite past is precisely to imply if not explicitly declare, spanning the transfinite in finite stage steps. As to why, that has been drawn out above and in onward linked. KFkairosfocus_{May 15, 2020
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Kairosfocus: As 164: I have put on the table the hyperreals which do make sense of transfinite values and infinitesimal ones. Only, they do not help the case of those who want a transfinite temporal causal past traversed in finite steps. I will say this again: you cannot move from the infinite to the finite in finite steps. No one is saying that. Any moment in the past is a finite number of steps away. You are never 'at' the infinite. That is what I am saying about the actual past as opposed to the abstracta of setting up R* with Z* as mileposts. I've said what I've said. If you consider the timeline like the number line and we are at zero on the number line then time stretches out to infinity before and behind us. At any given moment in time we are a finite number of steps away from today. You NEVER move into the transfinite or the hyper-reals or any of that. Which THE number line? The real number line. The hyperreals allow you to get to transfinitely removed values readily. Yes but that's not the point. Today is time zero on the time line. Tomorrow is 1, the day after is 2, etc. Yesterday is -1, the day before is -2, etc. The timeline stretches out infinitely far before us and after us. But every step along the way is a finite number of steps away from today. You do not need to talk about reaching H or whatever. You don't get to infinity. You can't. I't's not a point on the line. Take the 1/x catapult function, feed in a suitable h smaller than 1/z for any natural counting number, yes, an infinitesimal. 1/h –> H, a number in excess of any z in N. Where too, h is familiar for foundational calculus, Newtonian fluxions style: f'(x) = lim as h –> 0 of [f(x+h) – f(x)]/h What you cannot do, however, is to start at 0 and take steps, 1, 2, 3 etc and reach to H. And no one is saying you can!! That supertask cannot escape a finite span from 0 though it can point to a potential infinity, where having counted to any particular k, we can now go k+1, k+2 etc and that could be put 1-1 with 0,1,2 etc. We see that the naturals and reals do not have a crisp border but go on and on. However there is no definable specific u so u +1 = w, the first transfinite ordinal. That's right, you can't get to the infinite in a finite number of steps. There are no 'potential' infinities. They are or they aren't. Yes, as there are transfinitely many transfinite integers in Z* you cannot be at infinity, but you very well can be at a transfinite H which is at infinite remove from 0. No, not in our analogy. You cannot be at some infinitely distant point. You can only be at some finite point. You're trying too hard to defend your position and your muddling up the maths. Those are matters of structure and quantity bound up in the framework for any possible world. Necessary beings that shape and constrain what is possible in any world. And see, that concept is important, applying to the universality of core mathematics. It is not an arbitrary matter of opinion. I don't know if there is such a thing as a necessary being but I know your mathematical argument is not correct. The arrows on a number line drawing point to the transfinite realms. They point to continuing on ad infinitum. Forever. That is not a place or a being. In this context, time as considered in chaining finite stages [e.g. years] is causal-temporal and successive. 2019 –> 2020 –> 2021. We can label the stages and will see that the logic of being property of successive finite stage steps then imposes the challenge of the supertask of trying to span the transfinite in steps. You can't span the transfinite in steps! You can't be at infinity. I'm not sure we're getting anywhere here. I am not arguing about your philosophy or theology, just about your use of mathematics which I think is incorrect.JVL_{May 15, 2020
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JVL, 163:
Think of the number line, extending infinitely far in both directions. You never get to the end, to infinity, there is no end. Infinity is not a place or point. You can’t ‘be’ at infinity.
Which THE number line? The hyperreals allow you to get to transfinitely removed values readily. Take the 1/x catapult function, feed in a suitable h smaller than 1/z for any natural counting number, yes, an infinitesimal. 1/h --> H, a number in excess of any z in N. Where too, h is familiar for foundational calculus, Newtonian fluxions style: f'(x) = lim as h --> 0 of [f(x+h) - f(x)]/h What you cannot do, however, is to start at 0 and take steps, 1, 2, 3 etc and reach to H. That supertask cannot escape a finite span from 0 though it can point to a potential infinity, where having counted to any particular k, we can now go k+1, k+2 etc and that could be put 1-1 with 0,1,2 etc. We see that the naturals and reals do not have a crisp border but go on and on. However there is no definable specific u so u +1 = w, the first transfinite ordinal. Yes, as there are transfinitely many transfinite integers in Z* you cannot be at infinity, but you very well can be at a transfinite H which is at infinite remove from 0. Those are matters of structure and quantity bound up in the framework for any possible world. Necessary beings that shape and constrain what is possible in any world. And see, that concept is important, applying to the universality of core mathematics. It is not an arbitrary matter of opinion. The arrows on a number line drawing point to the transfinite realms. In this context, time as considered in chaining finite stages [e.g. years] is causal-temporal and successive. 2019 --> 2020 --> 2021. We can label the stages and will see that the logic of being property of successive finite stage steps then imposes the challenge of the supertask of trying to span the transfinite in steps. KFkairosfocus_{May 15, 2020
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JVL 161: Did you really mean to say:
any ‘now’ you specify is a particular point or time along the time line. It’s not infinity. It will always be some finite number of steps away. You cannot be infinitely far away, you can only be a finite number of steps away, so you never have to worry about getting from the infinite to the finite
That is what I am saying about the actual past as opposed to the abstracta of setting up R* with Z* as mileposts. KFkairosfocus_{May 15, 2020
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As 164: I have put on the table the hyperreals which do make sense of transfinite values and infinitesimal ones. Only, they do not help the case of those who want a transfinite temporal causal past traversed in finite steps. KFkairosfocus_{May 15, 2020
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"It’s used in mathematics all the time. " JVL, I know. Abstractly. Andrewasauber_{May 15, 2020
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Asauber: An idea of it is imaginable, I guess, like a square circle, but nothing that could ever resolve to any usable information. Again, details are finite. Like a point is. It's used in mathematics all the time. First place you'd most likely see it is in Calculus but it's really hammered out in Set Theory which gets pretty weird eventually.JVL_{May 15, 2020
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JVL, "I think it is comprehensible. " An idea of it is imaginable, I guess, like a square circle, but nothing that could ever resolve to any usable information. Again, details are finite. Like a point is. Andrewasauber_{May 15, 2020
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Asauber: This doesn’t help your problem. I think you missed the beginning of the conversation. All I'm doing is disputing Kairosfocus's mathematical reasoning why there cannot be an infinite past. Infinity isn’t comprehensible. It does nothing for anything you want to demonstrate. I think it is comprehensible. There are different sizes of infinity and some ways of handling them. Get a Set Theory textbook if you're interested. You just put it there for arguments sake. When you consider a point, you have changed the context to something comprehensible, which is finite. Of course, which is why I find Kairosfocus's reasoning incorrect. He says there is a past 'now' that crosses the line into the transfinite which is not correct.JVL_{May 15, 2020
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JVL "Think of the number line, extending infinitely far in both directions." This doesn't help your problem. Infinity isn't comprehensible. It does nothing for anything you want to demonstrate. You just put it there for arguments sake. When you consider a point, you have changed the context to something comprehensible, which is finite. Andrewasauber_{May 15, 2020
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Asauber: You are having conceptual problems. If you can be at a point, then you have switched to a finite context. Think of the number line, extending infinitely far in both directions. You never get to the end, to infinity, there is no end. Infinity is not a place or point. You can't 'be' at infinity. If you specify a particular point then you are specifying how many finite steps away from zero, the centre, you are. You cannot be infinitely far away at any given step even though there are infinitely many steps. I don't know if there is an infinite past but you can't dispute it using this kind of mathematical argument.JVL_{May 15, 2020
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JVL, You are having conceptual problems. If you can be at a point, then you have switched to a finite context. Andrewasauber_{May 15, 2020
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Kairosfocus: if the actual past is infinitely large, given causal temporal succession there were once nows from which the current now would have been remote beyond any number of stages n in N. Nope, any 'now' you specify is a particular point or time along the time line. It's not infinity. It will always be some finite number of steps away. You cannot be infinitely far away, you can only be a finite number of steps away, so you never have to worry about getting from the infinite to the finite. Otherwise the span from those past points p is finite and bounded by 2021 to come. No, because any limit or span can be exceeded very simply. There is no limit or span for an infinitely large set. That is the point after all. If you specify a now or time or value that's finite. There is no finite value that traipses into the infinite.JVL_{May 15, 2020
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ET: I didn’t say that you could. I understand that an infinite past in impossible. You were the one talking about an infinite past. There can be an infinite past but you can't be at infinity. Just like the number line: there are infinitely many integers and you can 'be' at any one of them. But you can't be at infinity. Not with an infinite past. That's how it works, just like the number line. Think of today as being zero on the number line. Tomorrow is 1, the next day is 2 and so on. Yesterday was -1, the day before was -2 and so on. You can be at 27 or 356 or -389994567473737 but you cannot 'be' at infinity. You don't 'get' from infinity to finite because you can't 'be' at infinity. It's not a place or value or time.JVL_{May 15, 2020
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DS, we can use whatever notations we will, the issue is to make coherent sense of a claimed infinite causal-temporal past without implying a span that cannot be completed in finite stage steps. See my just now to JVL. Gotta go out now. KFkairosfocus_{May 15, 2020
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JVL:
No, you can’t be at an infinite past.
I didn't say that you could. I understand that an infinite past in impossible. You were the one talking about an infinite past.
You can only be at some point along the way.
Not with an infinite past.ET_{May 15, 2020
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JVL, if the actual past is infinitely large, given causal temporal succession there were once nows from which the current now would have been remote beyond any number of stages n in N. Otherwise the span from those past points p is finite and bounded by 2021 to come. KFkairosfocus_{May 15, 2020
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Kairosfocus: If that now past is infinite there were once nows from which the current now would have been the transfinitely remote future; and I don’t care if there was an onward further infinity, That 'now past' could not be at infinity. Infinity is not a place or time or value. You can only 'be' at times a finite distance away from the present.JVL_{May 15, 2020
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KF,
DS, so, kindly define a metric or model of duration of actual time lapsed that does not require actual past points in the span to now.
Eh? I'm pretty sure that the IPPs (some cosmologists, I gather) do not propose such a thing. Of course there are actual past points in the, errm, past.
From this, I argue that a transfinite past requires actual past points beyond any finite past point.
All I can say is, it hasn't worked. Apparently people have decided that in reality, this is not true. I don't know what the state of the art is in infinite-past cosmology. All I can do is random googling. Here's the abstract from a document on arxiv.org which might help in understanding what these people are saying:
We discuss cosmological models for an eternal universe. Physical observables show no singularity from the infinite past to the infinite future. While the universe is evolving, there is no beginning and no end ---the universe exists forever. The early state of inflation is described in two different, but equivalent pictures. In the freeze frame the universe emerges from an almost static state with flat geometry. After entropy production it shrinks and "thaws" slowly from a "freeze state" with extremely low temperature. The field transformation to the second "big bang picture" (Einstein frame) is singular. This "field singularity" is responsible for an apparent singularity of the big bang. Furthermore, we argue that past-incomplete geodesics do not necessarily indicate a singularity or beginning of the universe. Proper time ceases to be a useful concept for physical time if particles become massless. We propose to define physical time by counting the number of zeros of a component of the wave function. This counting is independent of the choice of coordinates and frames, and applies to massive and massless particles alike.
daveS_{May 15, 2020
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AS, you raise a point, the issue is an actually traversed infinity, Some are trying to get that without the supertask of spanning the infinite in finite cumulative steps. They are trying to use R and Z to do it, I am pointing out the fuzziness -- transfinite fuzziness -- of the borders of those sets and how R* and Z* mileposting it lets us see more clearly. When that is there, duration since p to n = n - p brings out no ends of trouble. KFkairosfocus_{May 15, 2020
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ET & JVL, the actual past has to be what was actually once the now, succeeded incrementally and causally-temporally to the current now 2020 AD. If that now past is infinite there were once nows from which the current now would have been the transfinitely remote future; and I don't care if there was an onward further infinity, it is duration K' to today or p to today that spans the transfinite that have to be explained. KFkairosfocus_{May 15, 2020
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