Georg Cantor and His Heritage(arxiv.org)
arxiv.org
Georg Cantor and His Heritage
https://arxiv.org/abs/math/0209244
116 comments
The diagonal argument was one of the most mind blowing moments of my maths degree
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.
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.
Of course, but what is your point?
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.
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.)
They don't work in some axiomized higher logic because they chose the rules to avoid them.
HoL not having traditional NOT is an example.
Even in FoL, Peano arithmetic uses the SoL induction to be usable.
There is no free lunch.
HoL not having traditional NOT is an example.
Even in FoL, Peano arithmetic uses the SoL induction to be usable.
There is no free lunch.
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.
Note the following section for the first part.
"Non-Absoluteness of Truth in Second-Order Logic"
https://plato.stanford.edu/entries/logic-higher-order/#NonAb...
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.
Note the following section for the first part.
"Non-Absoluteness of Truth in Second-Order Logic"
https://plato.stanford.edu/entries/logic-higher-order/#NonAb...
Well-founded relations is probably a good lens on why the above is important.
Now where did these classical paradoxes originate in? They stem from Cantor's Mengenlehre
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.
Edit: wording
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.
Edit: wording
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.
[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.
> But the axiom doesn't imply the existence of "all" subsets of A.
I feel like that would be a consequence of the axiom of choice.
I feel like that would be a consequence of the axiom of choice.
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.
Admittedly I know nothing about the theorem you’re referring to, but is there a compelling reason to limit ourselves to first-order logic?
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.
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.
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.
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Infinities simplify various things in math. Weird multi-sized infinities though don't appear very useful.
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.
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.
> The number of physical positions you can be in in space is uncountably infinite
You don't know that. Spacetime may be quantized.
Actually, doesn't the notion of Planck length / Planck time already suggest that shorter distances have no physical meaning?
You don't know that. Spacetime may be quantized.
Actually, doesn't the notion of Planck length / Planck time already suggest that shorter distances have no physical meaning?
> 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.
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.
Sure, those infinities. I meant the huge stack of new ones that Cantor started on in his book. What's Aleph 1 actually good for?
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.
I think such an argument would need quite a lot more work, the lack of Solomonoff induction doesn't mean we don't have probability theory.
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.
I wrote a bit about Cantor's quest to get people to take infinity seriously here: https://superbowl.substack.com/p/church-of-reality-cantor-on...
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.
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There is no actual infinity. The Cantorians forgot this, and so have fallen into contradiction. (Henri Poincaré, 1906)
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)
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.
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"
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"
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.
[1] https://en.wikipedia.org/wiki/Parmenides
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.
[1] https://en.wikipedia.org/wiki/Parmenides
Isn't that a completely different question? OP was demonstrating that the existence of natural numbers is trivial to show, unlike actual infinity.
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.
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)
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)
So what you're trying to say is that he has infinite apples?
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.
[1] https://math.dartmouth.edu/~matc/Readers/HowManyAngels/Canto...
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.
[1] https://math.dartmouth.edu/~matc/Readers/HowManyAngels/Canto...
But this is mathematics, not physics. It's about axioms and logical contradictions, not apples.
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.
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.
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.
The spectrum of the hydrogen atom and the volume of the universe both appear to be actually infinite based on our best physical models.
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.
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.
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.
How can you prove that? We literally can't see further than 98 billion light years.
How's that not also a religion? :p
How's that not also a religion? :p
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.
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.
It's weird to you. Who says the universe must be isotropic?
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:
https://arxiv.org/abs/1605.07178
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.
https://arxiv.org/abs/1605.07178
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.
> appear to be actually infinite
Was it shown that the volume in this models are actually infinite or just potentially infinite?
Was it shown that the volume in this models are actually infinite or just potentially infinite?
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.
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.
> being infinite in size in the normal sense of the word.
Well, can you maybe define what this normal sense of this word is?
This is a thread about Cantor, whose whole work is based on subtle differences in the treatment of "infinity".
Well, can you maybe define what this normal sense of this word is?
This is a thread about Cantor, whose whole work is based on subtle differences in the treatment of "infinity".
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.
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.
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?
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 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?
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.
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.
They don't religiously defend their existence, they're defending the second form of exist in which Unicorns are said to exist.
Have you ever seen a negative number of things?
Yes. I have seen debt before.
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.
> 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.
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.
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.
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.
> So in what sense is either one longer?
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
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
> proof of infinity
This combination of words seems strange.
Like: proof of ‘zero’ or proof of ‘left’.
All of them have definitions, not proofs.
(Qualitative distinction of different infinity types has a proof though)
This combination of words seems strange.
Like: proof of ‘zero’ or proof of ‘left’.
All of them have definitions, not proofs.
(Qualitative distinction of different infinity types has a proof though)
> Literally any definition of something infinite can always be reduced to a procedure that recursively transforms or observes some prior state.
Could you come up with or point to such a procedure for R (the reals)?
As I understand the diagonalization argument you can do that for N, but not for R.
Could you come up with or point to such a procedure for R (the reals)?
As I understand the diagonalization argument you can do that for N, but not for R.
Yes? Take any natural or real number. If it has no decimal point add one. Then append any digit. Repeat as necessary.
1, 1.1, 1.01, 1.001, …
This will never reach 2, so it will not generate all real numbers. (Which was what parent was asking for, to recusively generate all R)
This will never reach 2, so it will not generate all real numbers. (Which was what parent was asking for, to recusively generate all R)
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.
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.
Thanks for reply.
When will this reach PI or e or sqrt(2)?
(There are infinitely many numbers that will not be reached by this procedure)
When will this reach PI or e or sqrt(2)?
(There are infinitely many numbers that will not be reached by this procedure)
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.
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.
Fun.
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?
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.
> 1/3 is also a special kind of procedure.
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.
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.
If you could, you could use that procedure to create a mapping
1 => first step in the procedure 2 => second step ...
That in turn would mean that N and R have the same cardinality. This would be news.
1 => first step in the procedure 2 => second step ...
That in turn would mean that N and R have the same cardinality. This would be news.
If infinities don't exist then R and N don't have cardinality so that's not really a problem.
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.
But in all probability we are discussing the wrong thing here. Our difference it's likely at a deeper conceptual level than this.
> Literally any definition of something infinite can always be reduced to a procedure that recursively transforms or observes some prior state.
You can’t generate R this way. This is a consequence of Cantor’s proof.
You can’t generate R this way. This is a consequence of Cantor’s proof.
Sure you can. Generate all 1 digit numbers, then generate all 2 digit numbers and so forth. Which number won't be generated?
Why is “longer” any less confusing than “bigger”?
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.
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.
Can you write down the complete decimal expansion of let's say sqrt(5) ?
No; it’s infinitely long and non-repeating. Are you trying to refute my original comment? I think we’re on the same page.
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.
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 don’t know what you’re talking about. Are you responding to something I’ve said?
I like this. Nicely put!
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?
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 :)
Interval Arguments: Two Refutations of Cantor’s 1874 and 1878 Arguments:
https://www.academia.edu/93528167/Interval_Arguments_Two_Ref...
https://www.academia.edu/93528167/Interval_Arguments_Two_Ref...
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)
the infinite we can do right away. the finite takes longer. -SU
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Does anyone actually understand this?
"several thousand in each generation"
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."
https://www.ams.org/notices/200910/rtx091001268p.pdf
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."
https://www.ams.org/notices/200910/rtx091001268p.pdf
None of the 84 extant comments in this thread address the (putative) substance of the article. Maybe next generation, then?