Are some infinities bigger than others?
| February 15, 2018 | Posted by News under Cosmology, Intelligent Design, Mathematics, Philosophy |
infinity as snake swallowing tail
From theoretical physicist Paul Davies at Cosmos:
It turns out that the set of all points on a continuous line is a bigger infinity than the natural numbers; mathematicians say there is an uncountably infinite number of points on the line (and in three-dimensional space). You simply can’t match up each point on the line with the natural numbers in a one-to-one correspondence.
…
If it is continuous (and some physicists think it may not be) then it will contain an uncountably infinite number of points. But that doesn’t mean it has to go on forever. As Einstein discovered, it may be curved in on itself to form a finite volume. More.
One would have thought that once we get involved with infinities, we are skirting the boundaries of measurement—at best.
See also: God as a necessary, maximally great, endless being vs. the challenge to an actual infinity
Reader: Weirdness of infinity shows that the universe is not infinitely old
Durston and Craig on an infinite temporal past . . .
At Quanta: Is infinity real?
and
Cosmologist: In an infinite multiverse, physics loses its ability to make predictions. And that’s okay.
47 Responses to Are some infinities bigger than others?
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Someone seriously wrote this?
Andrew
There are always philosophical questions lurking in the background, but this is accepted as true by nearly everyone who has examined the issues seriously, regardless of their “worldview”.
Wow. A knee-jerk Appeal to Authority
or Size
Nearly Everyone is Pretty Big.
But is it as big as a larger infinity?
Guess the smart people will tell me.
Andrew
asauber,
Have you read Cantor’s proof that the real numbers cannot be put into 1-1 correspondence with the natural numbers?
DaveS,
Why would I bother? What in infinity could I would I find meaningful if I read it?
Andrew
asauber,
You might discover why people say that some infinities are bigger than others.
Gödel’s incompleteness theorem can be stated simply as such: “Anything you can draw a circle around cannot explain itself without referring to something outside the circle—something you have to assume but cannot prove”.
Even Stephen Hawking himself, who tried to put forth a mathematical theory of everything in his book ‘The Grand Design’, reluctantly admitted in that book that Kurt Gödel, with his incompleteness theorem. ‘halted the achievement of a unifying all-encompassing theory of everything’
Gödel’s incompleteness theorems were ultimately a cumulation of work that was originally started by Georg Cantor who lived from 1845 to 1918
Cantor was trying to bring a systematic understanding of infinity into mathematics. This endeavor was very much a theological quest for Cantor. Cantor, although he pioneered some very useful tools in mathematics, useful tools which were essential to Gödel in his work on incompleteness, Cantor, none-the less, ultimately failed in his endeavor to bring a systematic understanding of infinity into mathematics. This following video gives us a very interesting glimpse into how Gödel was able to pick up the pieces where Cantor had failed, and bring his incompleteness theorems for mathematics to fruition:
And although Cantor was ultimately proven wrong by Gödel in his quest to fully integrate a systematic understanding of infinity into mathematics,,,,
,,, none-the-less, I believe that both Cantor, and Gödel, (since they were both Christians), would be very pleased, since it is very comforting to overall Christian concerns, to see how infinity is now used, mathematically, in Quantum Mechanics and in modern day Quantum Electrodynamics,
From the preceding video, it is also very interesting to note that Quantum-Electrodynamics was the first attempt at unifying relativity and quantum mechanics. Specifically, in Quantum-Electrodynamics special relativity was merged with electromagnetism to produce what is held to be the ‘most precise theory of natural phenomena ever developed’:
Richard Feynman was only able to unify special relativity and quantum mechanics into Quantum Electrodynamics by quote unquote “brushing infinity under the rug” with a technique called Renormalization
In the following video, Richard Feynman rightly expresses his unease with “brushing infinity under the rug.” in Quantum-Electrodynamics:
I don’t know about Richard Feynman, but as for myself, as a Christian Theist, I find it rather comforting to know that it takes an ‘infinite amount of logic to figure out what one stinky tiny bit of space-time is going to do’:
,,, And whereas special relativity, by ‘brushing infinity under the rug’, has been successfully unified with quantum theory to produce Quantum Electrodynamics, no such mathematical ‘sleight of hand’ exists for unifying general relativity with quantum mechanics.
General relativity simply refuses to be mathematically unified with quantum mechanics in any acceptable way. In technical terms, Gravity has yet to be successfully included into a theory of everything since the infinities that crop up in that attempt are not renormalizable as they were in Quantum-Electrodynamics.
Some theoretical physicists have remarked that this failure to mathematically unify Quantum Mechanics and General Relativity into a mathematical “Theory of Everything” is ‘the collapse of physics as we know it’
At the 4:30 minute mark of the following video, Michio Kaku, after going through some calculations with the equation of General Relativity showing how it breaks down at black holes, states that, ‘It means the collapse of everything we know about the physical universe.’
Then in the same video at the 7:08 minute mark, Michio Kaku,, after going through some calculations trying to integrate Quantum Mechanics and Gravity at a black hole, goes on to state “And when you do this integral, you get something which makes no sense whatsoever. An infinity. Total nonsense. In fact, you get an infinite sequence of infinities. Infinitely worse than the divergences of Einstein’s original theory.”
As the preceding video clearly illustrated, the main conflict of reconciling General Relativity and Quantum Mechanics appears to arise from the inability of either theory to successfully deal with the Zero/Infinity conflict that crops up between each theory:
Yet, the unification, into a ‘theory of everything’, between what is in essence the ‘infinite Theistic world of Quantum Mechanics’ and the ‘finite Materialistic world of the space-time of General Relativity’ seems to be directly related to what Jesus apparently joined together with His resurrection, That is to say, it looks to be directly related to the unification of infinite God with finite man.
Dr. William Dembski in this following comment, although he was not addressing the Zero/Infinity conflict in General Relativity and Quantum Mechanics, offers this insight into what the ‘unification’ of infinite God and the finite man might look like mathematically:
Of personal note: I hold it to be fairly obvious that ‘growing large without measure’ is a lesser quality infinity than a fraction in which the denominator goes to zero.
Simply put, the reason why ‘growing large without measure’ is a lesser quality infinity than ‘a fraction in which the denominator goes to zero’ is because anything that begins to grow large without measure must necessarily have some sort of beginning in time and must also necessarily have some sort of preexistent space to grow into. Whereas, on the other hand, to form a fraction in which the denominator goes to zero is to force a finite object into a type of infinity that can have no discernible beginning in space or time. That is to say, since zero represents the absence of everything including the absence of space and time, then to form a fraction in which the denominator goes to zero is to force a finite object into a true infinity that is beyond space and time.
In more technical terms, I view growing large without measure to be a potential infinite and I view a fraction in which the denominator goes to zero to be an actual infinite.
Moreover, if we rightly let the Agent causality of God ‘back’ into the picture of modern physics, as the Christian founders of modern science originally envisioned, (Newton, Maxwell, Faraday, and Planck, to name a few), , then an empirically backed reconciliation between Quantum Mechanics and General Relativity, i.e. the ‘Theory of Everything’, readily pops out for us in Christ’s resurrection from the dead.
Specifically, the fact that Jesus Christ dealt with both general relativity and quantum mechanics in His resurrection from the dead is made evident by the Shroud of Turin.
Verses
This following video also touches on the ‘higher dimensional’ math behind Quantum Mechanics, Special Relativity and General Relativity
Albeit in a somewhat different context, William James, riffing on Bergson, once said this about the nature of mathematical mind:
‘When the mathematician,’ Bergson writes, ‘calculates the state of a system at the end of a time t, nothing need prevent him from supposing that betweenwhiles the universe vanishes, in order suddenly to appear again at the due moment in the new configuration. It is only the t-th moment that counts—that which flows throughout the intervals, namely real time, plays no part in his calculation…. In short, the world on which the mathematician operates is a world which dies and is born anew at every instant, like the world which Descartes thought of when he spoke of a continued creation.’ To know adequately what really happens we ought, Bergson insists, to see into the intervals, but the mathematician sees only their extremities. He fixes only a few results, he dots a curve and then interpolates, he substitutes a tracing for a reality.
http://www.philosophy-index.co.....rse/vi.php
Multiple infinities? Could all those countless “betweenwhiles” perhaps just be so many figments of the limited, dividing mind? I’d prefer to trust the perennial sages who say “One being, without a second”.
You’re digging the hole deeper.
Andrew
DaveS,
To be straightforward in letting you know where I’m coming from, someone is going to have to produce evidence that Infinity is more than just a entertaining mental exercise. I have only so much time and mental space to use, so I try to make all my exercising worth the while, whether it be physical excercising, mental or spiritual.
Andrew
Infinity is infinity. The alleged difference is how many elements each numerical infinity contain.
asauber@12 That’s an interesting question. As many have noted in the field of mathematics intuition from FINITE sets CANNOT answer questions about INFINITE sets. Thus when one studies infinities one comes up with things which are finite minds have no intuition for.
Thus it is easy to prove that two countable infinite sets have the EXACT same number of elements. Mathematicians do this by proving there is a 1-1 mapping from every element of one set to an element in the other set. So for example consider the natural numbers (counting numbers) starting from 1. ( 1, 2, 3, 4, 5..) And the set of counting numbers starting from 2. ( 2, 3, 4, 5…)
Using the intuition from finite sets one could reason that the first set has every element that the second set does AND the number 1 (one). Since the first set has every number the second set does + 1 more, finite reasoning (i.e. reasoning about finite sets) would conclude the first set has more elements. This finite reasoning would be dead wrong. Infinity as a concept does not become bigger by adding a finite number to it. There is a simple mapping ( n2 = n1 + 1) for each n1 in the first set that makes a 1-1 correspondence between the two sets. This is what is meant that they have equal cardinality or in layman’s terms are the same size infinity.
Cantor showed that there is no way to build a map between the counting numbers and the set of real numbers ( points on the number line ). So it is true that in mathematical reasoning the set of real numbers is somehow bigger.
The interesting question you raise is as to whether this is any more than a mathematical exercise.
I will leave the question of whether you may be persuaded that this has any meaning spiritual or physical to the ample list of links shared by BA77
JDH,
Infinity as Theology is an entirely different matter. But then you have to throw the numbers (as we can use them) away, if you go there. I don’t think mathematics apply. I think mathematics are a tool given to us finite creatures by God. You get mysteries like 3 and 1 at the same time when you think about God in a numerical context. No calculator has that function.
Andrew
News,
Davies is demonstrably right, C exceeds aleph-null, as can be shown. For that matter, the reciprocal of every counting number from 1 up lies in the interval (0,1], as a dust, these are not connected points. The vast bulk, in the close neighbourhood of zero — not a proof but an illustration. So a dual to the set of naturals is found in the continuum between 0 and 1. Astonishing, if you ponder it.
AS:
The numbers apply to different aspects. In orthodox Christian theology, God is unified as to being, diverse as to person; thus, triune. Infinity as applied to the Godhead has to do with being the greatest, beyond finite bound. But at the same time God cannot make a square circle, that is impossible of being. Likewise God will not violate his core character so will be truthful, etc.
KF
H’mm, I just thought of a way to drive the matter home conceptually. Not a proof, an illustration. Let any given reciprocal be 1/n, from 1, 2, 3 . . . then do the set for the interval (1,2] by adding 1 +1/n in succession, i.e. 1 +1, 1 + 1/2, 1 + 1/3 etc. The same dust reappears in (1,2]. Then, do it for 2 + 1/n, 3 + 1/n etc. Copies of the dust appear in the intervals between all counting numbers. And in no case does it begin to exhaust the interval (n, n+1]. If that does not astonish you, it should. KF
PS: try the von Neumann construction of the naturals:
{} –> 0
{0} –> 1
{0,1} –> 2
etc, endlessly.
Now recognise that we have a new class of quantity, the first transfinite, of cardinality aleph null. Then go on from there. There are transfinitely many transfinites as the surreals tell us.
Of related note:
asauber –
There are multiple ways to compare sizes of infinities. There is a difference between “ordinality” and “cardinality” of infinite numbers, but in each you have different sizes (ordinals can be ordered, but they aren’t necessarily “bigger” in an absolute sense).
The way that you can compare infinities is this:
1) If every member of set A can be matched to a unique member of set B, then B is at least as large as A
2) If every member of set B can be matched to a unique member of set A, then A is at least as large as B
3) If 1 & 2 can be shown, then A is the same size as B
Now, if (1) holds, but it can be proven that (2) does not hold (usually shown by demonstrating that (2) leads to a contradiction), then B is *larger* than A.
This demonstration has been done to show that the reals are a larger infinity than the integers. If you want the details, google Cantor and “diagonalization”. I can post it myself, but probably not until later.
Here’s two articles I’ve written on infinity:
How to Count to Infinity (basically what I’ve written here)
Seeing Infinity (the usage of infinity for practical applications, integrating scientific and theological outlooks)
This comment,,,
,,, Reminded me of this comment from Berlinski
And where Darwinian evolution is based on a materialistic view of reality which denies that anything beyond nature exists, on the other hand, Mathematics exists in a transcendent, beyond space and time, realm which is not reducible any possible material explanation. This transcendent mathematical realm has been referred to as a Platonic mathematical world.
As the following article points out, ‘Mathematical platonism enjoys widespread support and is frequently considered the default metaphysical position with respect to mathematics.,,, and that arguments for mathematical platonism typically assert,,, that mathematical entities are not constituents of the spatio-temporal realm.’
Simply put, Mathematics itself, contrary to the materialistic presuppositions of Darwinists, does not need the physical world in order to exist. And yet Darwinists, although they deny that anything beyond nature exists, need this transcendent world of mathematics in order for their theory to be considered scientific in the first place. The predicament that Darwinists find themselves in regards to denying the reality of this transcendent world of mathematics, and yet needing validation from this transcendent world of mathematics in order to be considered scientific, should be the very definition of self-refuting.
And although some mathematicians who are of the atheistic persuasion may be tempted to believe that this transcendent Platonic mathematical world can exist without God, yet, due to Gödel’s incompleteness theorem, they would be wrong to believe that.
,,, Kurt Gödel himself stated,,,,
The following article puts the implications of Gödel’s incompleteness theorem for mathematics as such,
Thus, although some people may be presupposed to think that the mathematical realm and the spiritual realm are not even in the same ballpark, the fact of the matter is that mathematics itself is very much a spiritual exercise.
In fact, when mathematicians are shown equations such as Euler’s identity or the Pythagorean identity the same area of the brain used to appreciate fine art or music lights up:
Moreover, besides mathematics being incomplete, and as Dr. Bruce Gordon points out in the following article, the evidence from cosmology and quantum mechanics now also proves that the physical universe is itself also causally incomplete.
And I would also argue that the mathematical infinities that are encountered at the Big Bang, Blackholes, and the quantum wave function, are some of the primary reasons, if not THE primary reason, why we know the physical universe is itself causally incomplete, i.e. neither self-originating nor self-sustaining.
Therefore, since infinity crops up in places that are of great interest in science, I would argue that a fairly accurate understanding of infinity would be of great benefit.
Of supplemental note is this quote from Newton:
I can think of one objection that a skeptic might raise.
If we claim that there are multiple “degrees of infinity”, then one might expect us to state precisely how many there are. Perhaps two? Aleph-null? Aleph-seventeen?
But no, a mathematician (working in ZFC, say) will mutter something about proper classes having no well-defined cardinality, and slam the door in your face. Some people will find that response unsatisfactory.
ba77,
I think this is true in the sense that anything could be a spiritual exercise. Everything we do can be a spiritual exercise if we offer it to the Lord.
Anyway, my beefs with infinity are perhaps what your comment touches on. It’s illogical to appeal to a 1) real infinity if materialism is your religion. There cant be an infinity, because we could never perceive or detect it with physical science. 2) Mathematical infinity is completely imaginary. Its realm is strictly the mind. Its just a way of arranging symbols.
Like I said, 3) infinity as theology is a whole ‘nother thing, and I get the feeling people want to smear the different notions of infinity together, which doesn’t make any sense to me.
Andrew
DS, it was you who pointed me to the surreals. There are transfinitely many transfinites — and the infinitesimals (all but zero quantities that appear in say the nonstandard analysis form of Calculus) are the same too. The relocation trick means that a transfinite cloud of infinitesimals surrounds any given specific number, forming an infinitely close neighbourhood, so to speak. KF
KF,
I would agree with that, or at least I’ve probably said similar things here myself.
For any natural number n, there are more than n different infinite cardinal numbers.
But when we say there are “transfinitely many” infinities, that naturally leads to the question of which degree of infinity “transfinitely many” refers to. And that is a question that is impossible or at least difficult to answer.
DS, that gets into things like W^W and worse, W standing for omega the order type of the natural counting numbers. Transfinitely many is clear, just what kind is not. KF
KF,
Yes, and I think a skeptic could find this state of affairs problematic.
By the way, the application of infinity as a quantity makes doing Calculus and related mathematics a *lot* easier. A huge barrier to learning Calculus has been from the attempt to purge infinities and infinitesimals from it. If you’ve ever had to do limit proofs, you know what I mean.
Infinitesimals make Calculus infinitely easier. They even make limits (which were supposed to replace them) infinitely easier.
johnnyb,
😮
I agree that the mechanics of calculus can be simplified by using infinitesimals, but would it really be a good idea to introduce them in freshman calculus, for example? Even an introductory analysis class?
Those students are really in no position to understand number systems with infinitesimals (granted, even the real number system can be challenging to them).
I guess I would be on the side of the purgers. Elementary calculus is a setting in which students build intuition about the topology of the real line, and I think infinitesimals would just muddy the waters.
What is the biggest infinity?
daveS –
Actually, I find that infinitesimals make it 1000x more intuitive. I teach homeschool Calculus co-ops every year. Infinitesimals are a lot easier to understand than, say, imaginary numbers. In fact, once you have the machinery to understand real numbers (i.e., the idea that fractions don’t give you all the numbers even though they can be super tiny), the extension to infinitesimals is very natural (just like there were numbers in-between the rationals, there are also numbers in-between the reals). You can also represent these algebraically with a “w^n” being the level of infinity, and w^(-n) (or e^n) being the level of infinitesimal. Then they work just as algebraic units, and you can just perform normal algebraic manipulations on them for answers.
If you are interested, I can send you the PDF of my calc book as it currently stands. Just email me at jonathan.bartlett@blythinstitute.org
Jon
johnnyb,
Thanks for the offer.
I suppose this is largely a matter of taste. I think of the real line, plane, etc. in terms of physical space and time, and as far as I know, infinitesimals don’t correspond to real spatial or temporal entities.
Two distinct points in space (or time) are always separated by a non-infinitesimal distance (or time interval). At least in a “basic” physics class, they are.
On the other hand, it’s always good to consider other approaches, and if they are successful at getting students interested in calculus and math in general, that’s a positive thing.
DS, there is an approach, non-standard analysis. It gives fairly obvious meaning to things like dy, dx, dt etc. There is a textbook online. KF
KF,
Of course, I’ve read about it. But AFAIK, there are no such things as distances/intervals of infinitesimal length in space or time, so for me, sticking with the real numbers works better. YMMV.
Fun exchanges: https://www.reddit.com/r/math/comments/5tppmy/nonstandard_analysis_in_physics/
Book: https://www.math.wisc.edu/%7Ekeisler/calc.html
KF & johnnyb,
One criticism of my own position, in light of the reddit thread that KF linked to:
I was saying that I prefer to think of calculus completely in terms of the real numbers (so no infinitesimals and no infinities) because that is more compatible with my physical intuition, including my belief that there are no “infinitesimals” in the physical world. I further suggest that as the “best” approach for elementary calculus education in general.
On the other hand, actual physicists often do use infinitesimal-style arguments, and presumably they know what’s most productive for them. So maybe my posts above simply reflect the fact that I was taught calculus in a particular way? And perhaps a lack of aptitude for physics… .
DS,infinitesimals are all over physics. So are vectors and matrices, complex numbers etc. Near as I figure, Math and Physics did not get really separated until C19. KF
KF,
Yes, at least they appear in arguments and calculations that physicists use. But I don’t think there are any physically real infinitesimal entities, which I find somewhat paradoxical.
DS, I was taught the old joke about the two drunks on a dark, rainy night. What are you looking for, down on your knees like that? My contact lenses. So down went the second to help. After some time, are you sure you lost them here? I actually lost them over there in the dark but this is where the light is, under this lamp-post. Sometimes, the difference between theory and model is a lot less than we are willing to acknowledge. KF
Yes, something to ponder…
Thinking back, I can remember a few times when reasoning about infinitesimals helped illuminate concepts for me, for example infinitesimal transformations in the context of Lie groups and algebras.
To answer my own question, is the set of all the different aleph infinities the biggest infinity?
@daveS, If physically unrealizable entities turn out to be extremely useful for understanding the physical world, does this mean there is more to existence than the physical world?
It seems physics contains many more physically problematic math concepts than just infinitesimals. Numbers, of course, cannot be physical. Zero is one of the original stumpers. Nothing doesn’t exist. Negative numbers is another. The discovery of irrational numbers got poor Hippasus killed. Imaginary numbers are another one.
Our daily reasoning and experience is filled with useful physical fictions. All of our decision making revolves around counterfactuals, which by definition do not physically exist. We try to figure out if something is true or false, but false things do not exist. Ideas do not have physical properties. The redness of an apple does not inhere in the physical/chemical sequence of lightwaves hitting rod cells and transmitting signals to the brain. And so on.
Without the blinding presupposition of physicalism, physicalism is obviously false. Self referentially so, since “physicalism is true” is not a physical object, otherwise I could step on it and physicalism would cease to be.
EricMH,
To answer this question:
At least in ZFC, the ‘collection’ of all cardinal numbers is not a set, so it doesn’t have a well-defined cardinality.
Furthermore, if S is a set, then the power set of S always has cardinality strictly greater than that of S, so there cannot be a largest cardinal number.
Perhaps so. I operate under the assumption that abstract things actually exist in the world, in any case.
Yes, and I’ve been thinking about some of these “useful” fictions lately, and whether sometimes they create more problems than they solve. We’ve had some discussions of potentially misleading notations & conventions in calculus, for example.
@daveS, thanks for your response. It is interesting there can be a listing of alephs, each of which is a set, but there is no set that can contain them all. Counter intuitive, since in my mind being able to say “them all” entails a collection. But if they cannot be a collection, what is the “all” I am referring to?
It is also interesting you say “abstract things exist in the world”. I do not understand what you mean. For example, the number 1 does not exist in the world, though our symbolic manipulation methodology can sometimes be embedded in a computer program. Saying the number 1 exists in the world suggests there is a correspondence between “1” and some physical object which grounds 1’s existence. In which case, if this physical object is destroyed, then 1 no longer exists. That sounds very counter intuitive to me.
EricMH,
I think some would call this “collection” a proper class. Sets have certain properties (they have elements, they have subsets, you can take the union of a set of sets, etc.), but these properties do not hold for arbitrary classes. I understand the notion of “class” is actually an informal concept. In ZFC, literally the only things that actually exist are sets.
I don’t really have a good answer for what it means for the number 1 to exist in the world. It certainly doesn’t exist physically, or correspond to any physical object which grounds it.
But I can talk about 1 and other integers with other people, and treat integers as real, almost concrete objects, so it seems there’s “something” there.
@daveS, much appreciated. The discussion of math and physicalism is fascinating, since in my mind math is one of the most concrete pieces of evidence that physicalism is false. Just about everything in math seems to not be physically realizable, while being both true and extremely useful in the physical realm.
Yet, it is almost as if these kinds of non-physical abstract concepts are the only things we know. We never know the material world in itself, every description of the material world is only true to a certain level, and fails to capture everything.
Yet, if we decided to do away entirely with abstraction due to its illusive nature, and try to live off of direct experience, we’d be no better than the animals around us, and set human progress back by millenia. Why does our very civilization depend on abstraction if it is an illusion?
So, with the correspondence mismatch between math (and abstraction in general) with the physical world, yet its extreme usefulness and truthfulness, it is hard to believe such a useful, concrete thing is not real in itself, as opposed to some sort of “ephemeral emergent pattern supervening on a material substrate.”
Mathematics really does seem to have an independent, non-physical existence which we can grasp with our minds. Which further implies our minds are not themselves purely physical things, but also have an independent, non-physical existence. In which case, engineering is essentially wizarding, where our arcane symbols and algorithmic incantations instantiate otherworldly entities into our physical world, and everyone can possess this magical ability in their mind by understanding math. Doing our taxes is a witching hex to keep away the IRS agents.