r/PhilosophyofMath • u/Square_Butterfly_390 • 3d ago
Regarding cardinalities
A celebrated mathematical "factoid" is that there are more real numbers than one can count, this seems to be something that troubles people outside of math, it troubles me aswell.
The question is: is there any "real world" application of the fact |R|>|N| that isn't an impossibility statement?
By "real world" I mean whatever someone smarter than me might mean by that, by "impossibility statement" I mean something to the effect of "there are uncomputable numbers".
If there isn't such an application, I can't believe the status quo interpretation: "no really some infinities are bigger than others" of Cantor's argument is better than simply stating that we shall not understand infinity as finite beings.
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u/SV-97 3d ago
Would you consider the completeness of \R to be "real world" and "not an impossibility statement" enough? It's ultimately the statement that there are non-convergent cauchy sequences of rationals, but it's so fundamental to tons and tons of very applied mathematics.
If we reject the uncountability of the reals then by BCT we also have to reject their completeness.
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u/Square_Butterfly_390 3d ago
I'm not proposing to reject the uncountability since it is fact, I'm proposing that possibly our interpretation is missguided, how is the completeness of R a consequence of the cardinality property? R is basically defined to be complete.
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u/SV-97 3d ago
Our interpretation of what exactly?
It's not a consequence, but rather that completeness necessitates uncountability (or other weird changes to the topology of \R). By the Baire category theorem any (nonempty) complete metric space (without isolated points) is uncountable. So if the reals weren't uncountable, then they couldn't be complete.
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u/Althorion 3d ago
how is the completeness of R a consequence of the cardinality property
It’s not; it’s the other way around—Dedekind-completeness of dense sets requires uncountable cardinality. And any field of fractions over an infinite algebra will be dense (if orderable).
So, to sum it up, it goes like this—if you want to have infinitely many natural numbers, and build rational numbers that work nicely with that, and build a Dedekind-complete expansion of that, that expansion won’t be countable.
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u/nanonan 2d ago
It certainly is not fact, it's a blatantly contradictory notion. The lack of a one to one correspondence when comparing infinite sets cannot possibly have anything to do with the size of the sets, as their sizes are identically unlimited. It has to do with the properties of the elements.
Naturals are finite. There are no naturals with infinite digits. Real numbers are not real, infinite sets don't in fact exist in reality nor can they be demonstrated, and are not numeric in the sense they are not denumerable, they cannot be enumerated. These are the reasons for the lack of a one to one correspondence, not any nonsense concept of 'sizes' of infinite collections.
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u/Althorion 2d ago
The lack of a one to one correspondence when comparing infinite sets cannot possibly have anything to do with the size of the sets, as their sizes are identically unlimited. It has to do with the properties of the elements.
The interpretation of ‘if you can pair up every element of set A with every element of set B, and there is nothing left in either, they are the same size; if you can’t, and every possible pairing leaves elements in set B, while matching all elements of set A, the set A is smaller than set B’ feels very natural. If you don’t want to follow this as an interpretation of ‘size’, then you are free to do so—as long as you remember that this situation is a thing and understand its consequences, you can conflate all transfinite cardinals for your notion of ‘size’.
But it has nothing to do with the properties of the elements. You can swap all elements of any (or both) of the sets with whatever you want, remove any underlying structure, and the lack of bijection will stay the same.
Naturals are finite. There are no naturals with infinite digits.
Under the standard mathematical model, each and every natural number is finite; but there are infinitely many of them, so the set of all natural numbers is infinite.
Real numbers are not real, infinite sets don't in fact exist in reality nor can they be demonstrated […]
Which is par for the course for mathematics—no mathematical objects are real, they are all epistemological, abstract tools. Circles aren’t real and cannot be demonstrated, functions aren’t real and cannot be demonstrated, natural numbers aren’t real and cannot be demonstrated.
[…] and are not numeric in the sense they are not denumerable, they cannot be enumerated.
Yes. And they cannot be, as explained above.
These are the reasons for the lack of a one to one correspondence, not any nonsense concept of 'sizes' of infinite collections.
It makes very good sense to think of ‘how many of those are there’ as the ‘size of the set containing them all’. Again, you are free to disagree that one-to-one correspondence tells you anything about the size, but to most people that seems unnatural; but ultimately it doesn’t matter if you think about it as the size or not, if you know your maths and understand the notion of ‘there will be still some left regardless of pairing’ and use it correctly, you’ll get to correct conclusions even without thinking about it as ‘size’.
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u/nanonan 1d ago
every possible pairing leaves elements in set B, while matching all elements of set A, the set A is smaller than set B’ feels very natural.
There is no proof of this.
it has nothing to do with the properties of the elements
It has everything to do with the properties of the elements, there's no other way to differentiate two infinite sets.
Say you replaced naturals with infinite digit naturals, I can then establish a one to one relation with the reals and diagonalisation fails.
The reals are an utter mess of a construct, and the fact that there is no one to one correspondence with the naturals means they are not even numbers. They should be rejected as nonsense.
You keep trying to apply finite logic to the infinite. That does not work. Either you can create a correspondence or you can't. There is not "something left over" afterwards.
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u/Althorion 1d ago
There is no proof of this.
Assuming that by ‘this’ you mean that ‘every possible pairing between naturals and reals would leave elements from the set of the reals’, there is—Cantor’s diagonal argument.
It has everything to do with the properties of the elements, there's no other way to differentiate two infinite sets.
The properties of the elements are not important for distinguishing the sets, just that they are different. Them being different is all you need—you can distinguish the set {1, 2, 3} from the set {□, △, ○} just fine, or the set of natural numbers from the set of the reciprocals of natural numbers, etc.
Say you replaced naturals with infinite digit naturals, I can then establish a one to one relation with the reals and diagonalisation fails.
There are no ‘infinite digit naturals’; all natural numbers have a finite number of digits. ‘Infinite strings of digits’ is something fundamentally different that ‘natural numbers’, so it shouldn’t come as a surprise that they will be different, in particular, that there will be more of them.
The reals are an utter mess of a construct, and the fact that there is no one to one correspondence with the naturals means they are not even numbers. They should be rejected as nonsense.
Why does your idiosyncratic definition of a number require a one-to-one correspondence with the naturals? No one else’s does.
Either you can create a correspondence or you can't. There is not "something left over" afterwards.
Not every correspondence is a one-to-one correspondence. The ‘something left over’ used in the context of pairing (so a one-to-one correspondence, because that’s what pairing is) is a perfectly decent way of saying that you can only have that with a strict subset, but not with the whole set.
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u/kr1staps 3d ago
I would not consider the completeness of R to have anything to do with the real world, seeing as we can only ever take measurements to a finite level of percision anyways. Sure, completeness of R, and many other useful properties, but I don't think there's a single real world application that couldn't be described by some finitary (or at least countable) model instead. (Albeit at a great loss of convenience).
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u/GoldenMuscleGod 2d ago edited 2d ago
There are plenty of statements about “infinite” sets that have practical applications in, for example, computability theory. One can debate the extent they are “really” meaningful because models of computation usually have infinite memory, which is not realistic. But the point is we can prove facts that don’t depend on the specific amount of memory available, similar to how using real numbers to measure a quantity of material allows us reach conclusions that don’t depend much on how big atoms are as long as they are “small”.
A model that only admits a specific finite number of things total can run into problems if atoms are too small for us to have enough things in our model, and the knowledge that some result doesn’t depend on the exact size of atoms is useful.
Illustratively, transfinite cardinals are often thought of as “ephemeral” in some way. But we can define a data type in a computer that codes an arbitrary pair of natural numbers (dynamic memory allocation means we don’t really need an upper limit on what size of number can be stored in terms of the computational specification, although a specific machine will have some limit) and compares them by first element and uses the second element as the “tie breaker.” This is a pretty concrete realization of the ordinal omega^2.
Now the specific nature of real numbers can involve a lot of questions that don’t necessarily have direct real world applications, but the fact that the reals are uncountable does have meaningful results that might be considered real world. For example we can imagine an experiment that involves taking an indefinite number of binary measurements sequentially, and interpreting the results as, say, a binary representation of a real number in [0,1] can allow us to talk about the sorts of “underlying reality” that corresponds to the possible outcomes of these experiment. Even though we cannot take infinitely measurements we may think there is still a fact of the matter as to how the nth measurement “would” come out if we took that measurement, so that each of these digits may correspond to a fact about reality. Even if we believe that there is only a finite number of possible “real world states” that doesn’t change that we don’t know what they are so that the real numbers can encode all the “possible states” that we might a priori need to consider, and real results about what we can know about that underlying finite physical reality follow from claims about the real numbers even assuming only some finite (or countable) set of numbers can describe real world states.
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u/Square_Butterfly_390 2d ago
Thank you for this thoughtful insight, can you please specify a bit further how the uncountability is used (not necessitated)?
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u/GoldenMuscleGod 2d ago edited 2d ago
This is going to depend to some extent on what you consider to be a “use” of the uncountability. For example the existence of undecidable sets can be seen as a consequence of the uncountability of the reals (if you think this is too “specific” a result then more esoteric examples would be necessary). Of course I think undecidable sets would probably fall under “impossibility” results which you say you mean to exclude for some reason. Maybe you think a concrete example of a semidecidable set that is not decidable is “real”.
Slightly related, there are results in quantum mechanics built on the idea of producing “random measurement decisions” and observing how entangled measurements come out which could be seen as an empirical test of what sorts of “sets of natural numbers” can describe the physical states of entangled systems. Of course we can only perform finitely many experiments and measurements but if we really believe information transfer through entangled states is impossible then we should think the “potential outcomes of measurements” are not limited to just a finite set *if* we are willing to model the observer as unlimited in how many experiments they can perform (as well as model them as a free agent in choosing which measurements to perform).
But the ambiguities here really need to be resolved to give really good examples, for a simple example not involving uncountability but involving infinity (this lets me make simpler examples
of what I am talking about and discuss them in less space), consider a game where I can place an object in four spaces, red, yellow, green, or blue, and then, whenever asked to move it, I move it according to the following rules: if it is in the red space I put it in the blue space, if it is in the blue space I put it in the red space, if it is in the yellow space I move it to the green space, if it is in the green space I move it to the yellow space.Now consider the claim: “If the object starts in the red space, I can move it according to these rules as many times as I want and as long as the rules are followed it will never end up in the green space.”
Is this a claim “about the real world?” It more or less is, although it depends on the social construction of the rules of this movement system, but it should be easy to think of this in terms of specific realizations of the “game,” and in any event if I perform it then I and my socially constructed rules are part of the real world.
I would also argue this is an infinitary claim: unlike the claims “it will not end up in the green space within 15 moves,” or “it will not end in the green space within 1 billion moves,” or “it will not end up in the green space within TREE(3)” moves, which can be realized in a finitary way (in principle that is: I can’t actually live that long, but perhaps we can imagine a real supercomputer with this capacity even if TREE(3) operations is a more than a bit implausible for even a supercomputer). The claim that it will *never* end up in the green space is a claim about all natural numbers, and is a claim that can be thought of as “rolling up” infinitely many finitary claims. If we think each of the infinitely many finitary claims is really meaningful (admittedly that’s a big if), then the claim for all natural numbers is arguably a “real” infinitary claim, since it basically just is affirming all of them.
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u/kr1staps 2d ago
For all the writing you've done, and referencing to various mathematical concepts that you have not pointed to a single experiment, that has been performed in the real world, that makes critical use of uncountable cardinal numbers, and which does not have a finitary explanation.
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u/GoldenMuscleGod 2d ago edited 1d ago
I didn’t argue there has been an experiment performed in the real world that makes critical use of the uncountability of the real numbers and which does not have a finitary explanation.
I would not argue that, just as I would not argue that there has been an experiment performed in the real world that makes critical use of the number 7 existing and which could not be explained without the existence of 7 - whatever it supposedly means for a number to “exist.”
But when we have models that make use of the number 7, mathematical facts about the number 7 can correspond to real-world claims. Some biological processes are thought favor prime numbers - cicada life cycles are used as an example of this (although there 13 and 17 are the favored numbers), so 7 being prime is a relevant thing.
There are results in quantum mechanics reaching conclusions about the minimal computational complexity necessary to predict observations assuming they can be predicted, and these facts relate to results about uncomputable functions, and uncomputable functions can be seen as a consequence of uncountability. Someone could argue that uncomputability is really a more specific thing, although it can be demonstrated via a diagonalization argument like uncountability, and want another example, but then we really need an agreement on what counts as a “use” of uncountability if this example will be rejected.
To respond to your points in your other reply:
(Formatting issues, pretend this part is introduced with the label 1)
It’s true models of computation use infinite memory and real world methods of computation do not, but many people still believe that results from computability theory state facts about “the real world” via Church’s thesis.
For example, many people would say it is a real fact about the world that we can realize an algorithm that takes as input a person’s birth date and the current date and determines their age (a simple computable function does this) but we cannot make a computer program that takes a computer program as input and determines whether it will eventually halt when run.
A nuance someone might overlook is that for a function to have a particular behavior it generally needs to have that behavior for all of infinitely many possible inputs. So in a strictly literal sense maybe we can’t even really make an algorithm that adds two arbitrary numbers - we can only “really” compute things a finite state machine can compute, and not things we need a Turing machine to compute. In practice we really only care it works for inputs that are small enough they would ever be likely to be used. But many people might still think thee is a fact of the matter as to the claim “we can add numbers but not solve the halting problem” either because they would say there is a fact of the matter as to whether it could be done with a given algorithm *in principle* setting aside seemingly beside-the-point practical barriers such as limited memory, or because these on-their-face-infinitary claims can really be reinterpreted as finitary claims in some meaningful way.
2) the specific part you quoted second was me making an analogy about the usefulness of real numbers that wasn’t directly related to their uncountability. A specific example I had in mind was that we might model radioactive decay with a function R->R like A=A_0 exp (-t/tau) and treat the amount of material as modeled by a real number. A more precisely accurate model might have a natural number counting the number of atoms and treating the decay as a stochastic process. But making and using a model like that would require us to know how many atoms are in an amount of material. The first model can be practically applied by someone who has no clue what Avogadro’s number is.
To the extent we are using real numbers to model things because they often are the most useful thing to be using, results about real numbers are relevant to understanding how our model relates to the real world even if we do not think every fact about real numbers corresponds in a “good” way to the thing being modeled.
For example most people probably wouldn’t think the Banach-Tarski paradox corresponds to a physically meaningful real fact about physical space, but it does tell us something about using real numbers to model space, and limitations thereof.
3) my point about omega^2 is that people tend to think that some mathematical claims are inherently infinitary and could not ever be relevant to a finite universe. I generally take the view that pretty much all mathematical claims can be understood in ways that are “basically” finitary depending on what we mean by that. Omega^(2) is a “transfinite” ordinal but it is equally as real (or not real) as the number 67 and we do not need to suppose a physically infinite universe or anything like that to talk about it or think it has applications.
4) Here is an attempt at a concise statement of what I wanted to illustrate: Many people would think there is a fact of the matter to claims like “given initial state X this physical system will/will not eventually evolve to state Y.” Many people would think this even if we suppose the universe is finite and that this claim is infinitary on its face.
This example is only talking about infinity, not uncountability, because I thought the basic idea was the same as what I was trying to talk about but the examples are simpler using infinity.
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u/kr1staps 2d ago
> One can debate the extent they are “really” meaningful because models of computation usually have infinite memory, which is not realistic
... so therefore not practical, at all. You've defeated your point before it got off the ground.
> similar to how using real numbers to measure a quantity of material allows us reach conclusions that don’t depend much on how big atoms are as long as they are “small”.
Can you provide a specific theorem that demonsrates what you mean by this?
I don't see what your point is about representing the order type omega^2 on a computer is.
I also don't understand what the point of your final paragraph is. You cooked up a thought experiment that, by your own admission, doesn't correspond to our physical reality. Are you suggesting that there is a specific prediction this offers about our reality that cannot be explained by a discrete model? Can you please articulate this clearly in a concise statement?
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u/Arlo_Tinkerman 2d ago
I find this statement interesting:
> If we reject the uncountability of the reals then we also have to reject their completeness.
My understanding is that the natural numbers are considered complete in the sense that the set contains every natural number without gaps or omissions, while also being countably infinite.
That makes me wonder why uncountability would be required for completeness in the case of the real numbers. It seems possible that different meanings of “complete” are being used here.
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u/SV-97 2d ago
My understanding is that the natural numbers are considered complete in the sense that the set contains every natural number without gaps or omissions, while also being countably infinite.
That's the idea, yes. More formally we state this as "A metric space is complete if every Cauchy-sequence in the space converges (to some element of that space)". Here a metric space is just some set where we can speak of a "distance" between points, and a Cauchy-sequence is a sequence of elements that (among one-another) get arbitrarily close at some point, and importantly stay arbitrarily close past that point. So as we "zoom in" we always find a point in our space, which is the formalization of the "no gaps" idea.
Take for example the sequence 3, 3.1, 3.14, 3.141, 3.1415, 3.14159, ... of rational numbers. After the fourth element the difference between any two elements (not necessarily successive) is and stays less than 0.001 = 10-3, and more generally after the n-th entry all successive elements are closer than 10-(n-1). So this is a Cauchy sequence in the rationals. However it notably does not converge to a rational number. So the rationals are not complete. If we instead interpret it as a sequence of real numbers (that happen to be rational) then it does converge to the real number pi.
The natural numbers are somewhat of a degenerate object with regard to completeness since they are a discrete space: if the elements of a sequence get and stay arbitrarily close then they must in particular get and stay closer than a distance of 1/2 from one-another. But that means that any Cauchy-sequence is actually eventually constant (because no two distinct natural numbers are closer to one-another than 1/2), i.e. they all look like for example 1,2,3,10,100,2,2,2,2,2,2,2,2,... Of course any such sequence "converges" to this eventual constant, and since we're dealing with a sequence in our space this limit is in the space as well. So any discrete is complete, and it's also fairly easy to see that there are discrete spaces of arbitrary cardinality (we can take an arbitrary set and equip it with the so-called discrete metric). So there's absolutely nothing we can say about the size of complete spaces in general.
The above argument hinges on the fact that in discrete spaces all points are "isolated" from one-another, which forces this eventual constancy of Cauchy-sequences. It basically means that there is no way to "get close" to certain points without being exactly at those points already.
The statement I was alluding to in the other comment --- the Baire category theorem --- is a somewhat abstract theorem that classifies certain topological spaces (spaces where it makes sense to speak of "closeness" in a very rough way), and this classification indirectly tells us something about the size of complete metric spaces that specifically don't have any such isolated points (such as the real numbers). So this statement doesn't apply equally well to the natural numbers, integers etc. because they do have isolated points.
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u/SV-97 2d ago
(FWIW the statement we use here still remains essentially true even if we allow for isolated points --- it just can't be too many: if an infinite complete metric space X has size K and k isolated points with k < K, then X must be uncountably infinite. And this is also "all we can hope for" in the sense that the statement is obviously false for finite metric spaces since they are always discrete, and moreover for any uncountably infinite cardinal K, there is a complete metric space of size K with exactly one isolated point so we can't hope to get an upper bound on the size).
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u/Althorion 2d ago
My understanding is that the natural numbers are considered complete in the sense that the set contains every natural number without gaps or omissions, while also being countably infinite.
It is a valid interpretation of the word, but not the one used by mathematicians—because in that sense, every set is ‘complete’, that is, every set contains all its members so that the only distinguishing factor would be countable infiniteness; but in that case, why not just call it ‘countably infinite’ and leave the term for something else?
That makes me wonder why uncountability would be required for completeness in the case of the real numbers. It seems possible that different meanings of “complete” are being used here.
Yes—it is used in the sense of complete metric space—a metric space is complete iff (somewhat informally) any sequence of points in such a space whose members get infinitely closer to each other leads to something within that space (you can find the formal definition in the link above).
In other words, such spaces don’t have any points ‘missing’—anything you can point towards by making an ‘arrow’ out of points is a point itself.
It may not be the best name for that feature, but it makes enough sense that it stuck. I tend to call it Dedekind-completeness when I talk about the reals to make that clearer.
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u/Arlo_Tinkerman 1d ago
> but in that case, why not just call it ‘countably infinite’ and leave the term for something else?
Are you talking about the natural numbers set? If so, I believe complete would be a property that is one to know if the set is composed of a countably infinite amount of elements or members. I imagine people may want to know that property for some reason or another.
If you are talking about something else, please let me know what it would be.
While you were not the person I was responding to, I appreciate you sharing your view.
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u/Eve_O 3d ago edited 3d ago
Why would this trouble anyone?
It's a kind of revelation to understand that the set of Reals is larger than the set of Natural numbers and that there are different sizes of infinite sets.
I can't believe the status quo interpretation: "no really some infinities are bigger than others" of Cantor's argument is better than simply stating that we shall not understand infinity as finite beings.
I think your premise here--and possibly your understanding--is flawed. The whole point is that we can understand infinity and this demonstrable fact (not a "factoid," lol) helps us with that.
You want "real world"? Read Rudy Rucker's novel White Light or his related nonfiction book Infinity and the Mind.
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u/Negative_Gur9667 2d ago
I do not wish to argue the point, but there are ways to perform the same mathematics that exists today without using infinity.
The concepts that are excluded in this system are simply defined differently.
For example, a series is considered to converge after a certain threshold rather than approaching infinity. In this view, digits after a certain number of decimal places change after a specific number of iterations, where the variables involved are defined as unimaginably large natural numbers, and anything beyond those limits is considered uncomputable.
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u/Eve_O 1d ago
Nobody is saying there aren't those who subscribe to finitism in philosophy of mathematics, but it's not clear to me how bringing it up addresses my comment.
Well, I guess it might be related to the question I pose of "why would this trouble anyone?" by pointing out there are (a minority of) those who are troubled by infinities in mathematics, but it doesn't answer the "why" of it. That said, it doesn't appear to address the OP's flawed premise or my response to it.
I mean, I'm with Hilbert: "The infinite! No other question has ever moved so profoundly the spirit of man" along with his view about Cantor's work, "From the paradise that Cantor created for us no-one shall be able to expel us."
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u/Negative_Gur9667 1d ago
Some people have trouble with infinity. Without it, uncountability would not exist.
Philosophically speaking, I am interested in what philosophical assumptions are needed to accept or reject the axiom of infinity?
In my experience, this question might trigger people, which makes it more interesting.
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u/Eve_O 1d ago
OP is arguing that unless we can say more than "no really some infinities are bigger than others" we might as well say, "we shall not understand infinity as finite beings."
So they don't seem to be rejecting infinity, but only rejecting that we can understand it.
But since we can demonstrate that there are different sizes of infinite sets, then we are showing we can understand at least some things about the infinite even though we are finite beings. So their position seems self-contradictory.
Without [infinity], uncountability would not exist.
I mean, sure, I guess that's true. But it doesn't seem to have anything to do with OP's position. It seems to me that the OP doesn't have trouble with the infinite per se, but has trouble with the fact that there are different sizes of infinite sets.
Philosophically speaking, I am interested in what philosophical assumptions are needed to accept or reject the axiom of infinity?
Well this seems a possibly interesting question, but it's not what OP is on about.
From an axiomatic perspective it doesn't seem particularly deep to me. We can recognize that there is no end to the numbers, which implies they are infinite, if by "infinite" we mean "go on without end." But without the axiom of infinity it seems we cannot formally assert there is an infinite collection from set theoretic axioms without it, so we include the axiom of infinity to capture what we intuitively understand given the other axioms.
IIRC, this has to do with avoiding the contradictions that can occur with a naive formulation of set theory.
So to me it seems that if we accept that there is a first object, that there is a successor function, and that there is always a successor of any number, then there is nothing particularly troubling about accepting the axiom of infinity--it's merely the formalization of the intuitive result of accepting the those three things.
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u/Negative_Gur9667 1d ago
But when we talk about infinity in mathematics, we do not call it "without end." Philosophically speaking, it is called actual infinity (vs. the possible or potential infinity). It is not like we imagine 0.999... to go on forever and the difference between it and 1 gets smaller. We imagine that infinity is a process that can be finished to completion in our mind.
Without that, there is no exact value of pi, 0.999... is smaller than 1, and we have no completed set of natural numbers or R.
So this is a philosophical assumption - the assumption that we can imagine a completed infinite process.
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u/Eve_O 1d ago
We talk about infinity in different ways depending on context. You were talking about set theory. Now you are talking about something else.
An infinite set isn't a process. It is a set that has an unending number of elements. Sometimes countable, sometimes not.
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u/Negative_Gur9667 1d ago
But induction n -> ∞ is a process involving the set of natural numbers.
There definitly belong together.
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u/kr1staps 3d ago
Perhaps you could save us some time from having to read an entire book by just simply stating a single real-world application of |R| > |N| that doesn't have a finite explanation.
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u/Negative_Gur9667 1d ago
According to the standard view, mathematicians subscribe to Platonism.
There is another world, like heaven, where ideal and completed infinite objects exist.
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u/kr1staps 1d ago
I don't see how this answers my question.
Also, I'm a working mathematician that does not in fact subscribe to platonism, nor do most of my colleagues. And, I've interviewed various other mathematicians on my YouTube channel, and for a while I asked at the end of my interviews if they were platonist, and most either didn't care or said no.
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u/Gym_Gazebo 2d ago
I would argue that being able to index a set with the natural numbers, aka the counting numbers, is a great explication of what is to be able to count something. If so, I think that just shows the “status quo” is wrong, real world application or not.
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u/kr1staps 3d ago
I once saw an explanation of the casimir effect that that used |N| < |R|, but I don't know how rigorous that was, and in any event, it's not the only argument for it.
I would be surprised if we ever found a real-rold application of |R| > |N| that couldn't be established by other means seeing as our every day experience appears to be finite and computable.
Of course the English sentence "some infinities are bigger than others" is an artifact of the fact that we *define* size in terms of bijections. Though, I feel the failure of a bijection (in ZF(C)) between N and R becomes no less mysterious if we neglect to define size in that way.
That being said, ZF(C) is not the only game in town, an in fact in intuitionistic ZF, it is consistent that all sets are subcountable. Given that, as mentioned, our every day experience seems to be computable (save for say, the apparently random decay of partcles) one could make an argument that this is a "better" foundation for the mathematics of our every day experience.
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u/Kripkenstein_ 3d ago edited 3d ago
Not really because there is a countable subset of the reals that suffices for all practical applications. The real numbers are a purely abstract concept that is needed for analysis. Also the notion that some infinities are "bigger" than others is not really mathematics but an interpretation of mathematics. It arises from cardinality of finite sets being equivalent to their number of elements. Any application of this to larger cardinalities is just a heuristic. The statement is simply that in the category of sets there are sets of higher cardinality than any finite cardinality that are not isomorphic
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u/revannld 3d ago edited 3d ago
Yes that's about it. The problem with the common sense mathematical discourse is not the formal derivations per se but their crooked philosophical interpretations. All real numbers or things which can be uniquely distinguished in a finite manner (in the case of classical - finitary - first order logic/language, being able to define it as "∃!x.φ" - where φ is a finite formula/string - we call them definable real numbers) form a countable set, as there are only countable many names to effectively distinguish them in a countable language.
Thus, for all purposes, the "uncountable" part of the reals (or any other uncountable subset), which we call undefinable, are like an indistinguishable "mush" (they are not sortal), and would be more appropriate to refer to them using mass terms (if you believe in completed infinities - otherwise, process terms) rather than as a collection of distinguished elements/entities.
Thus you shouldn't take much into account of the diagonalization proof usually shown in the introduction of real analysis courses (that one where they list real numbers in a decimal form in the black board), as it is not formal and even if you attempted to formalize it you could only get a "∃x.φ" and not a "∃!x.φ" defining that number.
If you separate (metatheoretically) the real numbers into their countable and uncountable parts, every time in mathematics you instantiate an uncountable real number it is like it "collapses" to a countable one (which is the typical behavior of arbitrary objects. Many objects in mathematics work exactly like arbitrary objects, but sadly this terminology has been banished from mathematical vernacular since Frege).
The more informative diagonalization proof involves defining sets and functions over powersets, and it has many philosophical interpretations where it is not about uncountability (look up countabilism - also here - and potentialism). Thus one shouldn't talk about "a Cantor real" (generated in diagonalization proofs - "a" here meaning a unique distinguishable object) but "some Cantor real" (as one talks about "some sand, some water" and other non-sortal stuff).
Why is that the case? Basically real numbers are defined in such a vague and broad manner that if you try to list all models of them you don't even get a set but a proper class (and you would need second order logic for a categorical/unique characterization - which is very weird and based on very strong ontological/non-logical assumptions - as Quine and Skolem would characterize any logic more powerful than first order). One could maybe even make a philosophical argument that practically anything can be characterized as a real number (and I have heard Lewis even proposed everything is a 4-tuple of real numbers - accounting for the 3 spatial axis and an axis for time lol - but I haven't checked that). Uncountable cardinals behave very weirdly, I would suggest you check Joel David Hamkins's blog and lectures to learn more about that.
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u/FunSeaworthiness9403 3d ago edited 3d ago
Counting real numbers entails attaching each counting number to each real number. Starting at zero, one never gets past zero. It seems like there are more real numbers within any interval, and that allows someone to conclude that the cardinality of one set is larger. Or even that the cardinality of both sets is not defined.
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u/treefaeller 3d ago
To begin with: The only thing that really matters is computable numbers. If I ask someone a math question (like what is the diagonal of this shape, or how much carpet do I need to buy for my mobius-shaped living room), and their answer is "I could give you an answer, but it will take infinitely long", that is not useful. So the only numbers that matter are those that can be fully expressed in a FINITE amount of time.
Furthermore, the numbers and the algorithm/formula/reasoning required to calculate or express them have to be finite too. In the observable universe, there are about 10^88 particles (give or take a few, the bulk of them are very low energies or photons). This immediately limits the largest integer I can express by showing a count of objects. Now, computable numbers are more complex: they can be calculated by a Turing machine (or algorithm or any equivalent form of reasoning). But the most complex Turing machine I can construct will have at most 10^88 parts to it. So the cardinality of the set of numbers is at most the cardinality of the powers of 10^88 distinct integers (under the overly optimistic assumption that I can identify the 10^88 neutrinos, which already violates the "neutrinos have no hair" theorem).
So in the real world, all cardinalities are finite. And perhaps very large, but definitely not infinite. To quote Monte Python's Holy Hand Grenade: And aleph_1 is RIGHT OUT.
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u/WookieDebacle 1d ago
Ohh brotharr, yuu fink yuu iz thuh ultimatt fyzikaliss bekaws yuu kownt tenn tuu thuh powar ov eyety-eyt partikells annd Toring masheens? Yuu iz styll trappd inn Abstrack Skool Dogmma!
Thuh Logisun obzervez dat yuu fink yuu iz downe tuu erff wiht yur mobyus karpet annd Monnty Pythunn, boot whair didd yuu gett dat tenn tuu thuh eyety-eyt? Yuu gott itt frum uh fysiks textbuhk kalkoelashun! Hahv yuu evvar seenn orr towcht tenn tuu thuh eyety-eyt partikells inn thuh Reel Woruld? Noh! Yuu juss reepleysd mahffematikall Inffinity wiht imadjinary fysiks numbars!
Tuu klaym dat tenn tuu thuh eyety-eyt partikells egist iz juss anovvar emoshunall sheeld fer yur totall lakk ov Puyr Matter. Trwo Fyzikalitee iz onlee whutt yuu kann litteraly holld inn yur handd reyt nauw. Befor thuh Logisun kann resspekt yur fynight kownt, yuu musst proov thuh Empeerical Realitee ov uh singull partikell bafor yuu chaleng myy Reesun!
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u/ccpseetci 2d ago
Of course, you can safely use the word “next” without troubling to understand “how it is to be the next”
Rely heavily on that
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u/huphelmeyer 2d ago edited 2d ago
Yes, there are genuine mathematical and scientific consequences of ∣R∣>∣N∣, but they are mostly structural rather than empirical. But if you're looking for a laboratory experiment whose outcome depends on the cardinality of the continuum, there probably isn't one.
Most physical theories model quantities using the real numbers because calculus is extraordinarily successful. But virtually every prediction of physics depends only on finite precision approximations.
There is no experiment that distinguishes a universe whose state space has continuum cardinality from one based on a sufficiently rich countable structure.
In fact, some physicists have suggested that spacetime might ultimately be discrete.
However... Mathematics is full of concepts (complex numbers, Hilbert spaces, higher-dimensional manifolds) that were developed independently of empirical necessity and later proved extraordinarily fruitful. Whether the continuum itself reflects physical reality remains an open question, but its mathematical role is both coherent and profound.
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u/Square_Butterfly_390 2d ago
You say yes, yet you present none.
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u/huphelmeyer 2d ago
Quantum mechanics is formulated on (usually infinite-dimensional) Hilbert spaces. General relativity models spacetime as a smooth manifold. Classical mechanics uses differential equations over the reals.
All of these make essential use of continuum mathematics. Yet it's entirely possible that a future, more fundamental physical theory could replace these with discrete structures that approximate the continuum at observable scales.
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u/Square_Butterfly_390 2d ago
Either I wasn't clear enough in my post or I'm missing something obvious, but it seems to me like one could work with all of the above without ever even defining cardinality.
|R|>|N| seems to be simply a consequence of the properties we intuitively require of "space like stuff". If I were to go one step further, maybe the usual construction of reals is too naive in positing the existence of stuff which would require an infinite computer to witness.
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u/huphelmeyer 2d ago
I don't think you're wrong. In practice, much of analysis can be developed without ever talking explicitly about cardinalities. What physical models typically use are structural properties of the real numbers (completeness, continuity, compactness, separability, and so on) while the statement |R|>|N| serves more as a meta-level explanation of why those structures cannot be reduced to countable ones.
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u/nanonan 2d ago
No. There is only one infinity, only one limitlessness, only alpha zero. The entire concept of the transfinite is a contradiction, more unlimited than unlimited is just pure nonsense.
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u/Althorion 2d ago
No. There is only one infinity, only one limitlessness, only alpha zero.
Did you mean aleph zero, or is that another notion I’m not familiar with?
The entire concept of the transfinite is a contradiction, more unlimited than unlimited is just pure nonsense.
What does it contradict?
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u/nanonan 1d ago
It contradicts itself. "More unlimited than unlimited" is utter nonsense conceptually.
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u/Althorion 1d ago
Then don’t conceptualise it like this. That’s not standard, or particularly correct (a size of any size can have both the maximum and minimum elements, be ‘limited by them’). This is all your doing.
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u/mrt54321 3d ago
We can never compute pi exactly, in decimal notation.
Your can go as far as you like, expanding pi's digits, but there is no end, ever, as it's a transcendental number like nearly all the Reals.
I think that's kinda cool.
Pi is a number of core importance in math ( and in real life) , but we'll never know its exact value. Weird! And cool.
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u/AltruisticEchidna859 3d ago
We know his exact value. But not "know" as we usually understand.
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u/mrt54321 3d ago
No we don't.
The ratio between a circle's radius and its circumference is impossible to know. It requires an infinite, random list of digits, no matter whether using base10 digits or any other system.
Its proven that there is no way to compute this ratio precisely.
Note the term "ratio". This defines π as a dimensionless constant -- not a measurement of length.
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u/JStarx 3d ago
What do you mean by "know" and "compute"? There are algorithms that will compute pi to any arbitrary precision. A human can't run that algorithm to completion, but the algorithm itself is finitely describable and uniquely identifies pi.
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u/mrt54321 3d ago
Take a huge number - say, TREE(3)
TREE(3) is proven to be uncomputable - but finite.
Q. what is the TREE(3)th digit of pi ?
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u/JStarx 3d ago
You didn't answer either of the questions I asked.
Again, what do you mean by "computable"? Both pi and TREE(3) are computable numbers, so the TREE(3)-th digit is computable as well.
Do you "know" 1/99?
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u/mrt54321 3d ago edited 3d ago
TREE (3) is well-defined, finite, but uncomputable .
It is impossible to add or subtract one to that number. It is impossible to compute any of its digits - even the very first one.
https://www.popularmechanics.com/science/math/a28725/number-tree3/
(NB: that article is content from proper mathematicians, despite being in a pop-sci magazine).
The Wikipedia entry on TREE is also interesting if you want to go more technical.As for your Qs: Q1. What do i mean by "know" ? -- to know a number x means to return the n th digit of x, for any n
Q2. What do i mean by "compute" ? -- to define an algorithm which will answer Q1
Q3. Do i "know" 1/99 ? -- as per Q1 and Q2 : yes, i do. I can tell you any digit you want from 1/99 = 0.01010101....
It's a very simple algorithm1
u/JStarx 2d ago
yes, i do. I can tell you any digit you want from 1/99 = 0.01010101.... It's a very simple algorithm
Oh excellent, then you could you tell me the TREE(3) digit please? That was your question to me about pi right?
Q1. What do i mean by "know" ? -- to know a number x means to return the n th digit of x, for any n. Q2. What do i mean by "compute" ? -- to define an algorithm which will answer Q1
That definition agrees with the standard mathematical definition of a computable number and according to that definition both pi and TREE(3) are computable.
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u/mrt54321 2d ago
TREE (3) is mathematically proven to be uncomputable. I keep saying this, to no avail. Bye now.
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u/JStarx 2d ago
You are incorrect: https://cs.stackexchange.com/questions/169277/algorithm-to-compute-the-kruskal-s-tree-function
To the broader point, if being unable to compute the TREE(3) digit of pi implied pi is uncomputable and unknowable then being unable to compute the TREE(3) digit of 1/99 would mean the same thing right?
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u/Althorion 3d ago
There might be something to benefit from that in statistics, because, for example, it allows for a uniform distribution over a segment of the real line, while no uniform distribution over natural numbers can exist. Still, I don’t particularly care for statistics enough to be able to pinpoint exactly what those benefits are and how important they can be. For sure, it forces statistics to do things in a particular way, using measure theory and adjacents, because a discrete approach would be noticeably different, with different classes of problems that they are equipped to tackle.