Now and again, there arise certain trends in science and technology which prove deleterious. Take, for instance, the carbon nanotube. It is, as of 2024, 33 years old, and millions of man-hours have gone into practical nanotube development projects. To say that the reward has not been commensurate with the effort would be far too generous -- just about nothing has come of those millions of hours. In hindsight, this should perhaps have been more obvious; the theoretical benefits of nanotubes hinge on the production of pristine submicron fiber-like (giant-) molecules, and those have always been somewhere over the horizon.
I feel that Cantor's theories are much the same way. They have severe logical shortcomings, which were highlighted over 100 years ago by the superior logician Skolem; namely that you can construct an uncountable set out of any countable set, and that every so-called uncountable set has a perfectly isomorphic countable model. Further, the diagonalization argument only works in the limit, with very generous use of ". . .", and the finitists have put together a number of very compelling arguments against it. People claim that Cantor's set theory might be a good foundation for mathematics, but it is at best a foundation made of sand. As with the nanotube, I feel that many researchers have spent countless hours -- millions, perhaps -- following an intellectual/scientific trend, and nothing good has come of it.
Löwenheim-Skolem gives you a countable elementarily equivalent submodel (assuming you're working in a theory in a countable language, otherwise it gives you an elementary substructure of the same cardinality of the language at best), but plenty of interesting properties of familiar mathematical objects cannot be captured by a first-order theory and are not preserved by elementary equivalence, completeness of the reals being the standard example
Yet the very notion of countability in ZFC, which is itself a first-order theory, is rendered completely relative by Löwenheim-Skolem. ZFC itself has a countable model.
If "plenty of interesting properties of familiar mathematical objects cannot be captured by a first-order theory" then that also undermines ZFC, which is a first-order theory.
ZFC was specifically designed to be immune to the classic paradoxes of naive set theory: Russell's paradox, the Burali-Forti paradox, and Cantor's paradox.
You are arguing that the ground moves to perfectly fit the shape of a puddle.
Zermelo was one of the first to reference "Cantor's theorem" in his papers.
These paradoxes do not occur in higher-order logic. You don't need ZFC or any first-order set theory for that. (Also, your comment doesn't address the sentence I quoted.)
I don't understand what you mean. Higher-order logic does have classical negation. First-order PA tries to approximate the (second-order) induction axiom by replacing it with an infinite axiom schema, but that doesn't rule out non-standard numbers.
Cantor's proofs showing that Z and Q are countable and R is uncountable.
Cantor's "diagonalization proof" showed that.
Turing extended to the computable numbers K, which can be conceptualized as a number where you can write a f(n) that returns the nth digit in a number.
The reals numbers are un-computable almost everywhere, this property holds for all real numbers in a set except a subset of measure zero, the computable reals K which is Aleph Zero, a countable infinity.
The set of computable reals is only as big as N, and can be mapped to N.
It is not 'non-standard numbers' that are inaccessible, it is most of the real line is inaccessible to any algorithm.
Can you elaborate? It all seems really straightforward to me. There is no bijection between a set and its power set, via diagonalization. Thus, there is no bijection from the natural numbers to the power set of natural numbers. By definition, that means the power set of natural numbers is uncountable.
Not the parent, but the argument uses quite a few assumptions (axioms) that may not be intuitive to everyone, but which are quite relevant when studying mathematics at the foundational level.
For example, why would one be able to create the diagonal set (those indices of the power set elements that do not contain that index as an element) and the enumeration of the power set (i.e. the entire list of possible sets of numbers) at the same time? The theorem proves that an enumeration of the power set cannot be made. Perhaps some sets cannot be constructed at will just by writing down its properties either?
In computer land, one would quickly run into self-referential problems when constructing sets like these. For mathematics of this kind, most people agree that this is all fine, and one can derive interesting things from it. But one can also reject the approach and still do some elementary fun stuff.
Then again, I might be completely misunderstanding all of this, and I love to be corrected.
I’m not sure there’s many axioms used. Given any set A, and a function from A to the power set, P(A), construct the set X = {a in A | a is not in f(a) }. Here all we’re using is the power set axiom to define the power set and the subset axiom schema to construct X. We claim there is no a such that f(a) = X. If there was such an a, is a in X? By construction, a is in X if and only if a is not in X, just by first order logic, which is a contradiction. Thus, X is not in the image of f, so f is not a bijection. Thus, there is no bijection from A to P(A). And that’s it. We don’t even need the axiom of choice
You can't prove such a thing in ZFC [1]. ZFC's "power set axiom" is a misnomer. It doesn't imply the existence of infinite power sets. It just says if any set S exists and is a subset of an infinite set A, then S is element of a set P (the supposed "power set"). But the axiom doesn't imply the existence of "all" subsets of A. There is no way for first-order logic to state "every possible combination of elements of A forms a set". And Cantor's theorem stating that there is no mapping f from A onto P merely means that the mapping f itself can't exist inside a model of ZFC. [2]
[1] Or any other theory of first-order logic.
[2] This is explained in more detail in Stewart Shapiro's book "Foundation without Foundationalism" p. 114f.
At the moment I don't remember enough of the axiom of choice to comment on it in detail, but we can infer from the Downward Löwenheim-Skolem theorem (roughly, every first-order theory, including ZFC, has a countable model) that the Choice axiom can't imply uncountable sets.
The main reason people like first-order logic is Gödel's completeness theorem. It says that for first-order logic there is a proof system such that it proves a sentence if and only if it is true in every model. That is, the proof system is both sound and complete. This doesn't work for second or higher order logic. If we want a second-order proof system to be sound (proves no false things), it must be incomplete, i.e. not prove some true statements.
On the other hand, the main advantage of second (and higher) order logic is that it allows for "categorical" theories. A theory, like second-order Peano arithmetic, is categorical when it only has one model up to isomorphism. For second-order PA this model is the natural numbers. First-order logic doesn't allow categorical theories, so the axiom of first-order PA can't rule out the existence of weird non-standard numbers. Categoricity of a theory means that the axioms of the theory "define" ("pin down") some model. First-order theories can't do that.
So naturally, first-order ZFC isn't categorical. It even allows for countable and uncountable models. It's interpretation (model) is highly indeterminate. However, second-order ZFC (unlike, say, second-order PA or second-order analysis) is also not categorical. It doesn't have a countable model anymore, but it is still has many non-isomorphic models. (Though I think it has some weaker property which means that it is at least "more categorical" in some sense.)
So second-order ZFC doesn't give us the main advantage that second-order logic allows (the possibility of categorical theories), while also not having the complete proof system of first-order logic.
However, we could simply not use ZFC at all and only use second (or rather: higher) order logic. Higher-order logic is powerful on its own to (often categorically) formalize mathematical theories, like PA, without the need of any set theory. Instead of sets we have properties ("being a real number" instead of the set of real numbers), relations, properties of properties etc.
Formal proof checkers like Isabelle actually use higher-order logic instead of first-order ZFC as a basis.
There is also some argument to be made that categoricity is more important than completeness, since completeness of a proof system turns out to be not a plausible property anymore once self-referential statements, like the Gödel sentence, get involved. But that would lead me too far afield, and it isn't the standard opinion, which favors completeness over categoricity.
I’m uncertain what your point is. Like I said, in ZFC, there is no bijection from A to its power set. I don’t think anything you’ve stated contradicts that. Are there alternative axiom systems in which you can construct such a bijection?
There is no bijection (surjection, to be more general) from A to P inside your ZFC model even if both A and P are assumed to be countable. The bijection we talk about here would be just another object inside the model. So you can't infer from the lack of such an object that P is uncountable.
Yeah, Cantors theorem is a theorem written in the language of ZFC set theory. In other axiom systems it may not be true. But you can also say that about literally every theorem beyond, like, modus ponens. Is that the point you are trying to make?
The point is simply that it doesn't imply the existence of uncountable sets in ZFC. Modus ponens is different. The semantic entailment relation ⊨ between (P→Q, P) and Q is valid iff there is no model such that the former is true and the latter is false. Which is a sentence in plain English. It doesn't require the existence of some object that is mapping the premises to the conclusion inside the model. It doesn't even require the existence of a syntactic deduction rule (⊢) that tells you from (P→Q, P) to infer Q.
It is a theorem of ZFC that uncountable sets exist and every model of ZFC will have a set that the model believes to be uncountable. It doesn't matter than the metatheory might believe that model to be countable (why should the metatheory have the correct notion of what it means to be countable anyway?).
> It is a theorem of ZFC that uncountable sets exist
This is simply false, as I already explained.
> and every model of ZFC will have a set that the model believes to be uncountable.
That is something else. (And I wouldn't use the nebulous term "believes" here, it's just that the model lacks an object which maps A to P.)
> It doesn't matter than the metatheory might believe that model to be countable (why should the metatheory have the correct notion of what it means to be countable anyway?).
"The meta theory" here is simply sentences expressed in natural language, or beliefs held by people expressing those sentences. It is the language in terms of which everything formal is ultimately defined. It's the only thing that ultimately matters.
It is absolutely not false! This is taught in every undergraduate set theory course. Please point to me the step in the above proof where there is an error.
Certainly not in every undergraduate class, though I don't doubt that the subtleties around these issues may often be taught wrong. I already did point you to the errors, and I included the reference to Shapiro's book.
You keep making vague references to concepts without showing how they even remotely contradict Cantors theorem. You explained to me the power set axiom, which, thanks, I guess? But I don’t understand what your point is. Are you claiming X is not in the power set of the natural numbers? X is, unambiguously, a set. And it is clearly a subset of A. Therefore, it is in the power set. If you don’t understand that, I think you need to review the power set axiom in ZFC. Then you said “Cantor's theorem stating that there is no mapping f from A onto P merely means that the mapping f itself can't exist inside a model of ZFC”. Which is literally identical to saying “under ZFC, there are uncountable sets”. You just don’t like it because that statement isn’t wrapped in eight layers of indirection with model theory.
I don't exactly understand your construction of X. But note that you are relying on the existence of ZFC's so-called "powerset", which, as we already know, can be countable. ZFC has no ability to talk about infinite powersets, since it can't and doesn't state that all subsets exist, and thus it doesn't imply the existence of a powerset. Accordingly, both f and X may not be what you want.
> Then you said “Cantor's theorem stating that there is no mapping f from A onto P merely means that the mapping f itself can't exist inside a model of ZFC”. Which is literally identical to saying “under ZFC, there are uncountable sets”.
No, it only means that ZFC can't contain a function f from A to P in its model, which doesn't make P uncountable. (Things can be true even if the theory itself can't express them. E.g. Gödel's second incompleteness theorem says that a theory can't prove its own consistency, but that doesn't mean that the theory is inconsistent.)
For any f and A, we can define X as { a in A | a is not in f(a) } . That set exists in ZFC by the axiom of separation, also known as the axiom of subsets or the axiom of comprehension.
I recommend you pick up a book on ZFC if you are interested in understanding set theory. I found Enderton’s “Elements of Set Theory” to be a really good introductory text.
I'm sorry but this is just wrong. Since you seem to like Shapiro's book more than traditional set theory books let me quote from page 144 that the existence of an uncountable set is a theorem of ZFC: "Let C be the statement of Cantor's theorem. It entails that the powerset
of the collection of finite ordinals is not countable. Since C is a theorem of
first-order ZFC..."
Also this is not how the metatheory is understood in mathematics, not even in Shapiro's book, who dedicates two whole chapters to the metatheory
> Since you seem to like Shapiro's book more than traditional set theory books let me quote from page 144 that the existence of an uncountable set is a theorem of ZFC: "Let C be the statement of Cantor's theorem. It entails that the powerset of the collection of finite ordinals is not countable. Since C is a theorem of first-order ZFC..."
You didn't finish reading the quote. It continues:
> "... Since C is a theorem of first-order ZFC, m ⊨ C, but, as just stated, m is itself countable and so are its elements. This, again, is the so-called Skolem paradox."
That is, he was making an (informal) contradictory statement in order to illustrate the paradox. But there is actually no contradiction (otherwise ZFC would have been proven inconsistent), so we already know the statement of the paradox must have been inaccurate. He then goes on to explain where the inaccuracy was. It turns out that it was mainly in the mistaken, but common, assumption that the "powerset axiom" implies the existence of a powerset:
> [The powerset axiom] is supposed[!] to assert the existence of the set of all subsets of each set. But the variables (like all first-order variables) range over the elements of the model. So the powerset axiom only guarantees the existence of a set of all subsets of (say) ω that are in the model. The subsets of ω that are 'guaranteed by the axioms' to exist in a given model m are those that are first-order m-definable, and only those. In some cases there are only countably many of them.
As I explained in a previous comment, you can't say in first-order logic "every possible combination of elements of this infinite set forms a set" (more precise expression of "all possible subsets exist"). ZFC's powerset axiom only states that all existing subsets are element of some set P, but it doesn't imply they exist in the first place. So it doesn't imply the existence of infinite powersets (finite powersets wouldn't require the axiom anyway). Indeed, only those subsets are implied to exist that are implied by the other axioms, which isn't very many. (See the Löwenheim-Skolem theorem.)
And Cantors theorem only states that inside the model no function f (which would be just another set) exists that maps A to its supposed powerset P, even if both A and P are countable. So there no contradiction between Cantor's theorem and the countability of P in ZFC.
> Also this is not how the metatheory is understood in mathematics, not even in Shapiro's book, who dedicates two whole chapters to the metatheory
See this (well-known) quote on page 254 where he talks about the metatheory of second-order logic:
> The language of set theory is employed, without apology, and no anti-realist interpretation or reduction its envisaged. Indeed, no explicit interpretation is envisaged at all. There is no perspective outside this language from which to discuss its interpretations, or its models, or at least none is contemplated. The set-theoretic universal quantifier reads 'for all sets' and the existential quantifier reads 'there is a set'. Thus sets are in the ontology of the background theory. If asked 'which sets?' or 'how many?', there is only one answer: 'all of them'. This is what it is to take the language literally.
Alternatively, one could already take the axioms of second-order logic as "rock bottom", insofar they are grounded in natural language (which he also offers arguments for). In any case, using other formal theories as a tower of meta- and meta-meta-theories only leads to an infinite regress. Everything bottoms out in natural language. Normal mathematicians don't bother with formal languages in the first place, it's only logicians which make this excursion, but even they resort to natural language on the (meta) meta level.
Weird multi-sized infinities pop up in physics in a few places. The number of physical positions you can be in in space is uncountably infinite, the number of protons in the universe appears to be countably infinite.
There are interesting physical differences between quantum systems whose spectra are discrete (countably infinite eigenvalues) and continuous (uncountably infinite spectrum) and even combinations of both.
> Actually, doesn't the notion of Planck length / Planck time
Nope, we have absolutely no evidence of that. Space-time may be either discrete or continuous. Based on our current understanding we have no evidence either way.
Aleph 1 turns up when you try to (very) formally deal with probability theory, specifically when dealing with probability measures over the reals. This is sort of useful for physics for example when trying to be very careful about operators like position and momentum in quantum mechanics but it isn't really central. Its sort of nice to know you can do this stuff "properly" but physicists don't care much.
It's arguable that there's no such thing as a probability measure over the reals, because Solomonoff induction only works over computable programs, and the reals (in the sense needed) are not computable.
No, I mean even if you had (perfect, non-approximated) Solomonoff induction, you could only generate probabilities for computable "theories" (programs that predict all your past and future input), but I suppose it's possible that the impossibility proofs actually depend in some way on Aleph 1, so you would need it for consistency.
Great series. It really humanizes Cantor for me to see him trying to walk back a claim in those letters. I did my senior project on the continuum hypothesis, up through Paul Cohen's proof of it's independence (he wrote a short book about it that is very clear and accessible). Thanks for sharing, was very interesting to learn a little more about Cantor's motivations and the contemporary reaction to his ideas outside mathematics.
Nah, infinities don't exist. There are just recursive procedures that emit ever-increasing numbers of digits with no bounds checking.
The diagonal proof is just arguing that two nested while(true) loops will run for longer than one. (And then we define this as "bigger" just to confuse undergraduates)
I don't understand what you mean by this. What would it mean for an infinity or any number to "exist", does the number 2 "exist" somewhere in a way that infinities do not?
In any case our best physical models right now are full of infinities. Space looks like it is infinitely big to an absurd degree of precision. The spectrum of the hydrogen atom contains two different infinities! A countable infinity of bound states and a continuum of free electron states at higher energies.
Well, I have two apples on my desk. I can't have an infinity of anything. In that sense 2 exists more tangibly than infinity.
Our best physical models are just that - models. All models are wrong, some are useful. Infinity in physics is usually a shorthand for "it's so big I don't have to care about the edges"
I see, so the belief is that improved theories of physics will remove all the infinities we appear to see around us?
I guess as religions go, that one is no worse than many, but there isn't really any evidence for it now. Space looks really infinite, the hydrogen atom spectra are very well described by infinite series.
> will remove all the infinities we appear to see around us?
Can you actually show us one of these apparent actual infinities?
All one can reasonably postulate is a potential infinity, but this is completely different from Cantors actual infinities.
One example would be the infinite number of ways there are for humans to form unsound conclusions. But of course, this is not "real" reality that can be tidily corralled with math/science so it doesn't count, but it's still interesting.
According to the Big Bang model the universe started with a super small volume. When did it get to infinity?
As far as I know, we can see/measure that the universe is about 93 billion light years across. We literally can't know what's outside, so we don't know how much bigger it is or if it's infinite.
This is a common misconception, in standard models of big-bang cosmology the universe was always infinite. It was hotter and denser earlier, and is cooler and less dense later but always infinite.
A decent mental model is to think about an infinitely long ruler or measuring tape, where I have positions marked every (e.g.) 1 meter interval. You can imagine stretching or shrinking the ruler which would move the marks further or closer to each other respectively.
If you think about a "cosmology" for this ruler where I keep stretching it further and further forever then the points I have marked will ger further and further away, but the ruler itself is always infinite.
That's not how physics works. We can't prove anything. All we do is come up with worse or better models, and the best model anyone has come up with so far has the universe being infinite. Finite universe models have some problems, since the global curvature of the universe is (as best we can measure) zero the universe is either flat or pretending to be flat. Finite volume flat spaces can be characterised and they're a bit weird (1).
The best we can say is that if the universe is not infinite it appears to be doing a very good job of pretending to be infinite.
(1) for a good idea of what "weird" means here imagine a 2d 1km by 1km square with geometry like Pac man lives in (toroidal), so if you go off the top of the square you appear at the bottom, and if you head off east you appear on the west. Now start at the middle and leave a rock at your current position. Head due east until you cycle round and hit your rock again. You'll have walked 1km (500m to the east edge and 500m from the west edge back to the middle). Now do the same experiment but walk north-east. You'll hit your rock after sqrt(2) km of walking (1/sqrt(2) takes you to the north east corner and the same to get back to your rock).
In other worlds the pac-man torus space is not isotropic some angles are special and more important than others. Essentially the same thing happens in other finite flat geometries you can invent.
I don't say it has to be isotropic, but observationally it looks very close to isotropic. There are quite clear signals we'd expect to see in the cosmic microwave background if the universe was anisotropic and they aren't there:
This study is (as far as I'm aware) the state of the art on this topic, and based on the CMB observations they use it appears the universe is incredibly close to isotropic. Note that because it is based on CMB data this sort of study is sensitive to what shape the universe was a long time ago when, if it is finite, it was a lot smaller.
Could you explain how a potentially infinite universe would be different to an actually infinite one? I'm not familiar with how you're using these terms.
I don't know what the difference would be. That's why I'm asking.
These terms regained popularity due to Cantor's work. The potential infinite, is a very old idea for a process that never terminates. Like counting, you can always find a bigger natural number, by addition.
Thus, the natural numbers are potentially infinite, because they grow forever, but it will never be finished.
However, Cantor popularized another notion of infinity, namely that you can treat the ever growing natural numbers as one finished big bag and started doing math with this "object". Cantor's theory requires that infinity actually exists, as a finished object. Then he starts doing interesting stuff, as in measuring how many things are in the bag, even though the bag technically grows without ever stopping.
Ok, well if you come up with a definition for what a potentially infinite universe means then let me know. Until then our best models have the universe being infinite in size in the normal sense of the word.
I apologise, since you're the one who brought up the idea that the universe was either potentially or actually infinite I thought you might have some idea what those terms meant.
By "normal" or "actual" here I meant how a physicist would use the term - an infinite universe is one which can contain objects of any finite volume (if you like, can contain onjects of any finite volume larger than some minimum). I presumed (perhaps incorrectly) this is how you were using the term acutal, but now I see I should not make such assuptions.
Potential infinity is the notion of infinity that is e.g. used in calculus. Potential infinities are compared by rates of growth relative to some common quantity, while actual infinities are compared by the possibility of defining a surjective mapping of objects in one to objects in the other (which is treated as ≤).
So when we talk about a universe that may be "infinite" in size, do we talk about a "potential" or an "actual" infinity?
I think it's clearly the former. If we talk about the size of the universe, we seem to talk about a geometric volume.
Now think of two "infinitely" tall towers standing on the ground. Tower A has a cross section of 100m², while tower B has 200m². Assume that the cross section of tower A has rectangular shape and of tower B square shape, such that you could fit exactly two of tower A into tower B if the latter was hollow.
This suggests a clear meaning in which tower B has "twice as much" volume as tower A, even though both have "infinite" volume. You can literally fill tower B exactly with two towers A. Here "twice as much" just means that as you go higher, the volume of tower B increases twice as fast as for tower A. Which is the rate-of-growth size-comparison from potential infinity.
But for actual infinity you can't say tower B has twice as much volume. You are forced to assume their volume is the same. But you don't even know whether their volume is "countably" or "uncountably" infinite, since you don't know whether space-time is quantized (countable) or continuous (uncountable). But that doesn't even matter for comparing the volume of the two towers:
It's not sensible to say the towers would gain volume by switching from a discrete universe to a continuous universe. Discrete vs continuous is only about how far space can be divided, which is independent of its volume. Otherwise we also would have to say that 1m³ in a continuous universe is more volume than 1000m³ in a discrete universe. Which would be plain wrong, 1m³ is less volume than 1000m³, no matter what the microstructure of space is like.
And if we say the universe is infinitely large, we apparently just talk about its volume (or hypervolume of space-time etc). Which would e.g. mean that an infinite universe with less dimensions would literally fit inside a universe with more dimensions, but not the other way round. (Assuming the physical laws are otherwise the same.)
So, it seems clear that an "infinite universe" assumes the notion of potential infinity, not of actual infinity.
I don't know anything about the supposed infinities present in a hydrogen atom, but I would guess that those probably are potential infinities, too. Which would suggest that the notion of an actual infinity is not backed by physics of the real world.
Ok I think we have a wildly different way of thinking and talking about things. When I talk about "volume" what I mean is what physicists usually discuss - the volume of the universe is the number of 1 cubic meter cubes you can hypothetically fill it with.
In other words what we want to do is divide up the universe into cubic meter boxes, then put those boxes into bijection with some mathematical objects to "count" them. Fairly obviously the number of 1m objects you can fill an "infinite" universe with is larger than any finite number, so we say it is infinite.
This doesn't look like any kind of "potential" process, the number of boxes is literally in bijection to the number of natural numbers, so we say it is infinite.
As an aside, calculus really doesn't deal with infinite quantities at all, at least as its formulated in (e.g.) a modern real anysis course or something. Sometimes we use infinity as a convenient shorthand/slang when describing limiting behavior of functions or whatever, but the actual formal constructions of calculus are entirely about finite numbers. You don't need "potential infinities" at all.
Incidentally the stuff about discrete/continuous space-time very strongly does not matter for this point. I'm dividing your tower up into a countable infinite set of unit volumes whether or not the underlying space-time is discrete or continuous.
You have ignored most of what I have written. I already explained why the potential notion of infinity makes more sense than the actual notion. The reason is that size comparisons of volumes make much more sense with the former.
Now you repeat the definition of actual infinity, which is highly irrelevant, because I already discussed size comparisons, a more advanced topic, which you ignore. Please read my post again.
> This doesn't look like any kind of "potential" process
There is no actual process in time anyway, there is just the property of some quantity being unbounded.
> As an aside, calculus really doesn't deal with infinite quantities at all, at least as its formulated in (e.g.) a modern real anysis course or something. Sometimes we use infinity as a convenient shorthand/slang when describing limiting behavior of functions or whatever, but the actual formal constructions of calculus are entirely about finite numbers. You don't need "potential infinities" at all.
That's completely wrong. You are operating under the assumption that potential infinity is a number. It's not, it's a property of being unbounded. Only actual infinity is a type of infinity that is a number. Or rather a collection of "transfinite numbers" of different size.
> Incidentally the stuff about discrete/continuous space-time very strongly does not matter for this point. I'm dividing your tower up into a countable infinite set of unit volumes whether or not the underlying space-time is discrete or continuous.
If you want to compare sizes under actual infinity, the discrete/continuous distinction matters. Consider the set described by the interval [0, 1]. Is it countable or uncountable? If you are dealing with real numbers, it is uncountable, even if you manage to divide it into only countably many sub-intervals. Arbitrarily dividing things is arbitrary, what matters for size is the underlying structure.
Otherwise the tower would have both countable and uncountable volume at the same time (which is a contradiction), because partitions satisfying either are possible.
I didn't ignore what you wrote, I disagreed with it. It may be clear to you that potential infinity is the way to assess the volume of the towers but I think it is incorrect.
I don't think that volume is the appropriate thing to use for the sort of size comparison you want to make. The argument you made is that we should consider one tower to have twice the volume of the other because we can fit it inside twice. I don't think this is a useful notion, since it is trivial to to fit one tower inside the other tower arbitrarily many times if you slice them up a bit.
I think the sense in which one tower is twice as big as the other is captured by another quantity you mentioned already - the cross-sectional area. You don't need this potential vs actual infinity stuff at all. You just talk about whichever of the volume and cross-section is relevant to you.
> That's completely wrong. You are operating under the assumption that potential infinity is a number.
No type of infinity is a number, but I was indeed working under the assumption that it could be used as a cardinality since you told me you could understand volume with it, and I told you I understand volume as the cardinality of a set of unit-volumes filling the universe. If potential infinity can not be understood as a cardinality it seems entirely impossible to talk about it being the volume of the universe.
> Consider the set described by the interval [0, 1]. Is it countable or uncountable?
Neither, it is finite. We are talking about volumes here (used as short hand for lengths/areas/whatever is appropriate for the dimension), not cardinalities. Its volume is 1. If the underlying space is continuous then its cardinality is going to be uncountable, if the space is discrete the cardinality will be finite (not countable but finite), but either way the volume (defined by the appropriate measure) will be 1.
> Otherwise the tower would have both countable and uncountable volume at the same time
Nope, by fairly standard arguments your tower can be partitioned up into an countable number of unit cubes, and it emphatically can't be partitioned up into an uncountable number of unit cubes. You can use essentially the same argument that says one can't partition the real line up into uncountably many unit intervals.
A potentially infinite universe is also potentially finite? I.E. There is not sufficient evidence to determine that it is infinite since we are only able to make observations of a finite distance?
Are you sure you have two apples on your desk? What happens if you take a small bite? Is it still two apples? When will said apple "cease" to exist as an unit while you're eating it?
If the Old Greeks [1] haven't been able to solve this problem I'm not sure we'll be able to do better than them.
I'm happy to accept that I have two apples of my desk if all my friends and the guy I bought them from agree with that definition. At some point some will start disagreeing, but with two big full apples, all of them will agree.
Or is it more helpful if I say I have 10^27 atoms in my body? Quarks? Gluons? Strings from string theory? Still finite number, not "biteable", not infinite.
> I'm happy to accept that I have two apples of my desk if all my friends and the guy I bought them from agree with that definition.
That’s useful, but very vague, and very movable criteria for “exists”. (Approx. Must be based in physical reality + enough people subjectively agree on it)
Do following finite numbers exist: -1, 0, 0.5, PI, 2^300 (more than particles in observable universe), sqrt(-1)? Do individual digits exist? If PI exist, how many digits does it have? Do models and algorithms in general exist? Do model existance depend on limits of your/somebodies capacity to understand them?
I would propose alternative, but useful way to look at this. “Two” and “infinity” are models. Both these models are useful, but “two” is just more common one. (Still, various infinities are useful for bunch of people)
Going by my (bad) analogy, in theory, yes, that is what that would mean.
More to the point, if we're not able to decide whenever it is exactly that an apple stops being an apple once we start eating it that could mean that the apple-ness of said apple has big chances of remaining intact (together with the underlying apple) irrespective of us eating it fully, and hence (theoretically) giving us an infinite supply of apples (or access to apple-ness, to be more exact).
But, going back to the Old Greeks, this is a very old question that we haven't been able to fully solve, I honestly think we'll never be able to solve it. Aristotle's focus on categories and especially his tertium non datur thing has allowed us to solve some practical problems (for example by allowing us to build a world that is based on techne, which among other things has allowed us to have this conversation here on the internet), but the bigger and more philosophical question of One vs. Infinity and everything in between remains un-solved.
Later edit: On Cantor vs. Aristotle, from here [1]
> Thus Cantor believed that Aristotle was quite mistaken in his analysis of the infinite, and that his authority was exceedingly detrimental
When it comes to the philosophy of mathematics and to philosophy itself I'm with Cantor on this. Could be that mainstream mathematics itself could benefit were we to ditch Aristotle and fully embrace Cantor, but I'm not a mathematician nor smart enough to say if that's in the realm of the possible.
Mathematics is just an exercise in symbol manipulation according to a particular rule set. Mathematics is a _game_. Park Place exists in the context of a game of Monopoly and also exists in Atlantic City. The tokens in a game of Reversi don't correspond to anything in the real world. You might be playing it on a physical board of a particular size, but you could just as easily play it on an imaginary board, or on a piece of paper, or a computer, or in some imaginary board of infinite size. Whether or not either exists in "reality" doesn't matter in the context of the game, though.
If the rules of mathematics allow for infinities, then they exist in the context of mathematics. Whether they exist in reality doesn't matter. You can still say things that are true about these imaginary objects in the context of the game that everyone is playing with these symbols, just like you can say things that are true about the game of Reversi, even one played entirely in the minds of two players on an imaginary, infinite board.
If you use "exist" to mean "is a symbol defined within some system" rather than the typical "has isomorphism with reality" then you can say literally anything exists. Unicorns exist because English has a word for them.
So be it, but it is a useless definition of exists then isn't it?
If you're playing dungeons and dragons, then unicorns "exist" within the context of that game, and there are rules for interacting with them. If you're reading a fantasy novel, then unicorns exist within the context of that fictional universe.
And because people enjoy playing Dungeons and Dragons and reading fantasy novels, that existence has real value both for those people and for the people creating those novels and games. And you can say that certain things are true about unicorns within the context of those games and fictional worlds. (For example, it is true that unicorns have horns).
If it makes you feel better to put infinity into the same category of reality as "unicorns", then I think that's fine? It doesn't necessarily need to be the case that anything in mathematics corresponds to something that is physically manifested, and probably _most_ of mathematics does not -- not just infinities.
Yes. Infinity doesn't exist in the same way unicorns don't exist. The difference is that some people religiously believe in the existence of infinities and try to defend their existence even though nobody has ever seen one: see other replies to this thread.
No, you haven't. You've seen some symbols that represent the ludicrous physical notion of a negative amount of an object. Look outside, count negative 3 sheep. You can't. It's a _fiction_ which happens to be useful for accounting for debt.
Sure I can. I start with the number of sheep I expect: 30, and then count backwards. When I count to -3 I know that I have too many sheep. There is no rule that says counting backwards isn't useful. Any positive number can be represented as a negative number in relation with a constant and remain a perfectly isomorphic model of reality.
Infinities are quite different, given that it is by definition impossible to measure whether something in reality is infinite or not unless you have infinite time.
> Infinities are quite different, given that it is by definition impossible to measure whether something in reality is infinite or not unless you have infinite time.
It's impossible to count a negative number of things. It's impossible to count an imaginary number of things, it's impossible to count a matrix of things. It's impossible to count an irrational number of things. These are all fictions that people use because they're useful. Infinity is exactly the same as any of those other concepts.
No, infinity is a different concept to all of those quite distinct concepts. There are also clear and observable uses for all of those things that make them hard to replace, whereas infinity is unique in that if you drop it as an axiom then everything still works and your mathematical system becomes more rigorous.
Also I clearly just demonstrated that you can count negative numbers easily so I don't understand why you find it impossible.
Your critique of the use of ”bigger” in your strawman applies equally to your own use of ”longer” - neither loop will ever terminate. So in what sense is either one longer?
The diagonal proof is arguing that there is no bijection from N to {0, 1}∞ , despite the fact that both are infinite. The sense in which the latter is ”bigger” is that there are always elements left over that are covered by N.
Neither. That's my point. Literally any definition of something infinite can always be reduced to a procedure that recursively transforms or observes some prior state. To say that one of these functions can produce more distinct states than another is pointless, because the procedure that produces the most states will always be the one that you ran the most times.
There is nothing observably infinite, since it would take infinite time to observe that any given thing was infinite. The only possible proof of infinity would be a machine that runs infinitely quickly. e.g. https://qntm.org/responsibility
You can define a procedure that reaches 2 very trivially. e.g. you generate all possible 1 digit numbers, then 2 digit numbers, then 3 digit numbers, etc.
This is how natural numbers work in the first place. You're just adding a decimal point to all of the possible places it could go.
Well, you have to ask yourself what is PI really? You've been taught it is a number with infinite decimal places, however, practically speaking when you use PI you actually round. Practically speaking this doesn't matter because it's not actually possible to have a shape with infinite points in reality so you only need to approximate to whatever fidelity suits you.
How is this? This is because PI is not actually a number. It's a procedure that generates digits for approximating things about circles.
This is the same with sqrt(2). The sqrt procedure emits digits just as the procedure to find all real numbers does.
You can't "reach" PI for the same reason that a natural number can't reach "f(x) => x + 1". That is, natural numbers aren't procedures or functions.
What about 1/3, 1/7, …?
Previously outlines recursive procedure doesn’t generate those.
But yeah, if you deny existance of irrational numbers, and redefine Real:=Rational, then you can generate these “real” numbers recursively and it does follow that all infinities have same cardinality here.
Btw. what is the diagonal of a unit square formed by 4 objects at the corners? I assume it is a rational number.
Btw2. If you take that answer and multiply by itself, what do you get?
1/3 is also a special kind of procedure. It's 1 divided by 3. Either you use the isomorphic 0.3 repeater procedure where you recursively add 3 digits until you get bored or you have enough, or you use division to generate the number sequence. The fun thing about fractions is that we've worked out some ways in which fractions can be multiplied with natural numbers and rational numbers to generate new fractions, and also some fractions conveniently are isomorphic with rational and natural numbers.
Important to note: When ggp asked for a recursive procedure to generate real numbers, they wanted that exactly same proceedure would generate all reals (not special procedure for each number)
If we have special procedure for each number, then procedure to generate 1/3 is just 1/3. …of course naively assuming notation of 1/3 is as valid as 0.33333…, and that base 10 is not the only possible base.
Once you drop infinity as an axiom you have to think about fractions and repeating decimals differently.
1/3 isn't a real, it's a fraction. Fractions can be used to generate reals, and they can be used in algebra along with reals.
0.(3) is also not a real. It's also just representative of a procedure that can generate reals.
Both 1/3 and 0.(3) can still be used in algebra in the same way as before. You don't lose any capability because you can't practically expand 0.(3) to infinite decimal places in the first place.
You can have as many different symbols as you like, so long as you can actually write them on paper. As soon as you tell me that one of those symbols is a number with infinite digits I will disagree with you, given that you are unable to tell me what those digits are before the universe ends.
I can tell you that the number is 0.1 in base 3, which defines the number precisely. Or I can tell you that it in base 10, it has an infinite representation, and all the digits are 3. There, I told you what they all are, and the universe has not ended yet.
> Or I can tell you that it in base 10, it has an infinite representation,
Does it really have an infinite representation? I can't imagine an infinite representation fitting on a page. I'm pretty sure you're representing it as 1/3 or 0.(3). Neither of those representations are infinite. They're only a few characters really.
It seems to me that this comes at the price that now in your geometry the diagonal of a unit square does not have a defined length. Only an approximate one of 1.41421, but what does this approximate?
What price? In an algebraic setting you are never going to convert sqrt(2) into a real number, and in any practical setting you're going to have to round because nothing real actually has infinite precision.
It would appear that sqrt(2) does not have a meaning now, as sqrt(-1) in R. So it does not seem to matter if you do not convert it in an algebraic setting. There is no such number that you can reach with the enumerative approach, which will only give you the rational numbers. There are lots of proofs that sqrt(2) is not rational.
But in all probability we are discussing the wrong thing here. Our difference it's likely at a deeper conceptual level than this.
Also, what has nested loops got to do with it? You can use one loop to generate the natural numbers, and a pair of nested loops to generate pairs of natural numbers (if you like, the rationals). But the diagonal proof doesn’t show that these will have ‘different’ run times — they’re in fact in bijective correspondence.
You need another loop for each real number. A "proper" real number is in itself not finite. If you want to actually physically create a real number, for almost all of them, the process will never terminate.
You need to define your terms more precisely to be able to make these kind of assertions. What do you mean by physically create? What do you mean by process? The answers depend on these very strongly.
Since you can't write down this number, how can you generate a second number using this number? You'll never finish generating the first number in order to start the second.
You can come up with a bunch of procedures that generate numbers indefinitely, and you can even define relationships between those procedures, but a procedure is not a number.
I learned about Cantor in university, and it blew my mind. I've always wondered if anyone's used his work in an applied field yet (like how non-euclidean geometry was just crazy maths for a while and now underpins most physics).
Was hoping this paper would tell me, but from what I've read its more if a (very nicely written) summary of his core work. Anyone know if there's any applications yet?
Of course. Theory of computation and formal logic are two examples of math built on top of Cantor. And theory of computation has one or two real-world applications or so I've heard :)
It's always fun to make fun of cranks. Thanks for linking that. The author really needs to find the right statement of what they call the Nested Interval Theorem. I cracked up at the complete misuse of it in the "Interval Argument for the Rationals" section
I learned about Cantor from this readable book:
Aczel, Amir D. "The Mystery of the Aleph Mathematics, the Kabbalah, and the Search for Infinity" Simon & Schuster (2001)
I don't understand most of Yuri Manin's mathematics, but I still find some of it interesting.
"Manin: I think that people engaged in research
in mathematics today are doing so the same way
it was done 200 years ago. This is partly because
we don’t choose mathematics as our profession,
but rather it chooses us. And it chooses a certain
type of person, of which there are no more than
several thousand in each generation, worldwide.
And they all carry the stamp of those sorts of
people mathematics has chosen."
