MathFoundations16: Why infinite sets don't exist

[F3tusRape]

Nice drumstick.

[andreirocks1992]

how did you manage to say "tomato points" with a straight face? :)

[tothemesosphere]

All of maths is built on axioms. Can we honestly claim to "know" that the axioms of group theory truly describe the symmetries of a given geometrical object? To some extent, as brangelito already said, all that one does in maths is begin with a set of axioms (often motivated by some application, such as symmetry in the case of groups) and develop a theory of all the logical consequences of these axioms. How is modern set theory any different?

[njwildberger]

@tothemesosphere This is not the way Newton, Archimedes, Euler or Gauss approached mathematics. So there must be alternatives to thinking that `mathematics is built on axioms'. I would prefer: mathematics is built on common sense, careful thinking and clear use of language. What these concepts mean is surely more obvious to an educated layperson than what an àxiom' is..

[tothemesosphere]

@njwildberger do axioms not arise from 'common sense', surely this is what we use when working out a system of axioms? 'careful thinking and clear use of language' - what you are describing here is logic (and more precisely, mathematical logic) on which the whole of maths is already based!

Just because an 'educated layperson' may find it difficult to understand certain concepts is no grounds to reject these concepts. Any 'educated layperson' has the potential to become an educated mathematician

[njwildberger]

@tothemesosphere If you look at the Zermelo-Frankel axioms objectively, together with the Axiom of Choice, and other even more bizarre variants that are out there, I think you would be hard pressed to say that they arise from common sense.

[tothemesosphere]

@njwildberger Barring AC, which axioms of ZFC would you say do not arise from common sense? I find all to be very intuitive.

I tend to feel more wary towards AC. In its usual formulation using choice functions I find it to be very intuitive but its equivalent formulations such as the well ordering principle seem less obvious to me.

But we don't need AC to talk about infinite sets anyway, so this is irrelevant to this discussion I suppose.

[njwildberger]

@tothemesosphere The common sense approach would be: define what you mean by a set. Then state clearly the various things you believe sets satisfy. Then prove those things.

The idea that we ought to accept statements without proof, and indeed without proper formulation of the words employed, is wrong--even if we try to get away with it by calling those statements 'axioms'. Here is a little test: state your favourite ZF 'axiom' completely clearly.

[njwildberger]

@njwildberger I ought to have said: without looking it up!



[tothemesosphere]

@njwildberger ok: There exists a set that has no elements.

[njwildberger]

@tothemesosphere I notice you used the word `set'. Do you propose to define this? For example, I happen to believe that all sets are finite. Is this the same concept that you are using?

[tothemesosphere]

@njwildberger Ok thanks I think I'm beginning to understand some of your objections: you're saying that ZF uses statements about sets to define sets, which is in some way circular.

Can we not get around this by more precisely saying:

Define a class to be a collection of mathematical objects.

Then the ZF axioms define which classes are sets.

ie I should have said: "the class consisting of no elements is a set"

[njwildberger]

@tothemesosphere Modern set theory does try to use a term called `class' to try to remove the ambiguities in the term called 'set'. How they can seriously hope to get away with this is a mystery. Whatever word you use: class, collection, set, menagerie, bunch etc the problem comes when you try to clarify what an ''infinite'' one is going to mean. Is anything we can possibly think of a class/collection/set/menagie/b­unch?

[tothemesosphere]

@njwildberger What problem comes when we try to think of infinite classes/sets? Personally, I think the concept of "the collection/set of all natural numbers" is fairly easy to understand.

[tothemesosphere]

@njwildberger With ZF though, aren't these axioms attempting to formalise, in a consistent way, our (very fuzzy) intuitive notion of what a set is, and what sort of things we can do with sets?

For example, with group theory we have looked at the symmetries of a geometric object, studied the way we perceive its symmetries interact, and have then defined a set of axioms hoping to capture the idea of symmetry. How is this different from set theory?

[tothemesosphere]

To what extent does a finite group "exist"? Sure the concept of a finite group might be easier to grasp than that of an infinite set but why should that mean a finite group has more right to "exist" within mathematics? I have done plenty of problems on finite groups in which I cannot claim to have fully understood everything that has been going on - what one does in this situation in mathematics is follow the axioms (and resulting theorems) to put together a logically valid argument.

[rakanuj]

Ok, I'll drop the heavy philosophical names and terms (Wittgenstein, intension, etc.), and see what else is in this series first. And yes, I am aware that many mathematicians do things without critically thinking about them, but that's an ad hominem that's not likely to win you a lot of converts. As for manipulating objects without understanding them: people (in general) do it all the time, even if it /isn't/ honest :)

[xknowledgeisfreex]

Actualy most modern mathematicians do understand axiomatics. Have you ever heard of model theory?

[xknowledgeisfreex]

How do you deal with measure theory without ininite sets? Then you can only define measures of finite sets, which is boring.

[njwildberger]

@xknowledgeisfreex In fact modern measure theory is a highly problematic subject. This will be explained later on in this series.

[GlorifiedTruth]

LOWLY line segment? My grandfather was a line segment and he fought heroicly in the Spanish Civil War!

But seriously, folks, I'm interested to see where this goes. Currently my sympathies lie with the viewpoint that a circle is both a finite object and an infinite set of points... at least it's consistent with the math I've learned since childhood. We'll see.

Keep up the good work, Dr. W.

[brangelito]

I don't think anything "exists". Different axioms leads to different logical systems; some of which may seem more natural than others.

[njwildberger]

@brangelito If no thing ''exists'', then the word seems rather unnecessary. What do you mean by different logical systems?

[rakanuj]

Sorry if I'm a bit incoherent here, as my thoughts aren't entirely settled. I happen to agree

with you on the unreasonable of the "actual" infinite set, but I think the fact that we talk

about these things with such regularity is evidence that something else is going on. That is, we often don't talk about the objects themselves, but rather how we manipulate those objects, and therein how we can play with those manipulations themselves (as in writing a program, rather than editing a file).

[njwildberger]

@rakanuj Yes, people do this routinely in modern mathematics: largely by repeating what they have learnt without much critical analysis. But how can we honestly --in mathematics -- manipulate objects if we don't know precisely what they are?

[rakanuj]

Since we can always substitute one map into another, we can keep nesting intensions inside intensions to make new (finite) specifications of algorithms. These in turn *must* be applied to something before they can really describe anything extended or a legimate abstraction of something physical.

[njwildberger]

@rakanuj Your discussion is starting to sound ominously philosophical. In mathematics we are obliged to use terminology only after we have carefully established its meaning.

[rakanuj]

I think this is the lesson of a lot of the recent talk about category theory, topos theory, and Bob Harper's "computational trinitarianism" (see his blog, youtube won't let me link :().

The idea here is that the algorithm can be listed as a finite object that acts on some other unspecified input, so the "in-" in "infinite" is because we omit the specification of something ahead of time, rather than anything to do with a proceptual (sorry) view of mathematical objects.

[rakanuj]

I have more but youtube is being obnoxious about letting me post.

[rakanuj]

My understanding is that one of Brouwer's critiques, and later Wittgenstein (see is page on the

Stanford Encyclopedia of Philosophy, views on mathematics) was that a "completed infinite set"

confused extension and intension. That is, the finite sets are the only extended sets, and

all the infinite sets can *only* be defined by set comprehension. Equating a listing of elements

in the set with a rule for generating elements in the set leads to all the paradoxes most people

find.(more to come..)

[rakanuj]

Wittgenstein's own example was one of the lines not being a set of points, but rather a rule for determining new points "on the line", or whether another given point was collinear. (I'm not trying to name drop here, I'm just not sure of a better description off the cuff). I've always felt a resolution was to admit that the discussion of most mathematical objects is one step removed in this way - we talk about what we can do with something, as a map of the input, rather than the object itself.

[njwildberger]

@rakanuj There are quite a few lectures in this series that address very concretely what to do in such situations.

[xknowledgeisfreex]

Does it matter?

[bewertow69]

Why the hate for infinite sets? Set theory works very well except for the axiom of choice.

[njwildberger]

@bewertow69 Why hate? Tell me what an infinite set is, and we are all good.

[bewertow69]

@njwildberger take the set of real numbers between 0 and 1. How many elements are in this set?

[njwildberger]

@bewertow69 Take the set of all bewertows in the universe. How many elements are in this set?

[bewertow69]

@njwildberger lol what?

You ignored the point that any interval of the real numbers is an infinite set. Even if you consider only rationals you still have to deal with infinite sets whenever you give an interval. I don't see how rejecting the Axiom of Infinity improves set theory in any way. Maybe you should focus your efforts on fixing the axiom of choice, which is actually a genuine issue with set theory.

[recursion11]

what do you do about continuous spaces where traversing any interval means traversing an infinite set of points? how can any space in geometry be continuous or over the rational number without having an infinite number of points in each interval?

[njwildberger]

@recursion11 When we start doing mathematics correctly, we replace idle chat about infinite numbers of this and that, with concrete, finite expressions. Instead of talking whimsically about the "infinite number of points on the unit circle", we discuss rather the equation x^2+y^2=1; turns out we can compute anything we could before, but more accurately and logically.

Or think about the slice of cheese in your lunch sandwich. Does it really have an infinite number of cheese points?

[MemeMachine1]

Aren't irrational numbers like sqrt(2) infinite sets and therefore don't exist? They consist of an infinite number of decimals which you can only approximate, but for every approximation there is still another infinite number of decimals beyond which you don't know about and can't define. Hmm, this is starting to sound like an infinite set, like the set of natural numbers.

[njwildberger]

@MemeMachine1 Your intuition is good---``irrational numbers'' like sqrt(2) are in fact highly problematic to define properly. I will be discussing this at some length later in this series.

[agentredlum]

I say more power to you. Please continue to challenge accepted norms in mathematics. I find you to be an excellent lecturer, very careful, precise, and clear. I enjoy all of your lectures on youtube and look for you frequently. My own mathematical training goes up to calculus 3 and Linalgebra. I wish at some point you would make videos of mathematical 'tricks' Thank you

[tommyrjensen]

My claim is that you quote the letter out of its context, presumably with aim to misinform the viewers. The best source to substantiate that claim has to be the letter itself, reprinted in Gauss: Werke 8, pp. 215-218. I do not have a copy here, but Bruce Director has nice details on the schillerinstitute website that I am not allowed to post here, but it can be googled. See also The thirteen books of Euclid's Elements by Sir Thomas Little Heath, Dover 1956, p 193.

[njwildberger]

@tommyrjensen In other words, substantiating your claim is too much work.

[tommyrjensen]

@njwildberger Not at all, any interested person can do it easily. Look up William C. Waterhouse "Gauss on Infinity" Historia Mathematica Volume 6, Issue 4, November 1979, Pages 430-436, Abstract: "In opposing the use of completed infinity in mathematics, Gauss was making a valid criticism of one particular kind of argument. His celebrated statement has no connection with the set theory to which it was later applied."

[tommyrjensen]

"Gauss's remark was meant as a rebuke to occasional transgressors who used the notion of infinity - and even the symbol for infinity - as though it was an ordinary number, subject to the same rules of arithmetic as ordinary numbers are." (Eli Maor, To Infinity and Beyond: A Cultural History of the Infinite, Princeton 1991, p. 55)

[njwildberger]

@tommyrjensen Okay, but how about some reference to Gauss's writings themselves to substantiate your claim?

[tommyrjensen]

Gauss's original comment was aimed at usage of infinite sets in a "misunderstood" way. Actually in a way which is popular among cranks even today. It is a despiccable abuse and dishonest to quote him as if the context was an entirely different one.

[njwildberger]

@tommyrjensen Despicable abuse and dishonest to quote someone? Perhaps you are exaggerating. How about being clearer about your claim to know what Gauss's comment really meant. And how about substantiating your claim?

[PhantomHoover]

I like the way you don't actually use any mathematics; rather, you drop names as if they are infallible, make baseless accusations and act as though unintuitive is the same thing as wrong.

[EclecticSceptic]

The tomato thing was hilarious.

[chimamire]

“Critics are men who watch a battle from a high place then come down and shoot the survivors.”—Hemingway

[LaureanoLuna]

I think the words attributed to Weyl are actually from Brouwer.

[Kurtlane]

I was taught elementary set theory (sets, unions, intersections, negation, Venn diagrams) without getting into infinity. I found it extremely useful later when I had to deal with systems of inequalities and quadratic, cubic, etc. inequalities. I grasped them very easily. I don't think I would grasp them at all without knowing some set theory.

As for infinity - that's another matter.

[mateo3470]

And furthermore, what do you make of p-adic numbers?

[mateo3470]

Quit embarrassing yourself with these selective responses. Your dishonesty is quite apparent for all to see.

"No one shall expel us from the Paradise that Cantor has created."

~David Hilbert ("Über das Unendliche" Mathematische Annalen 95, (1926))

I think this is saying that reason will not persuade him. So be it. I guess it's your position as well.

Mathematics, at any given point, can be described with other set theoretical metaphors, but it's redundant. The map is not the territory!

[mateo3470]

@ja524309

You are such a terrible bore! Just like Cantor and his mytho-logic!

People might say otherwise, but their actions reveal that the criterion for Truth is a pragmatic one. Only foolish people reverse this. I think that would make you a fool.

I will not spoon feed you information out of context. Suffice it to say that there is more than one reason to reject Hilbert's "paradise". For one, we are neglecting "bad" indexings or reindexings (odds, evens, primes, etc...).

[mateo3470]

@ja524309

See Zenkin's paper on the "Cantor-Hodges hidden postulate" for more on the indexing issue.

[mateo3470]

@ja524309

"According to W. Hodges (and all modern axiomatic set theories after Cantor himself), only the set N = {1,2,3,4,5,...} is allowed in indexing the real numbers of the countable set X, i.e., in the mapping X —>У N. However, any reindexing of real numbers in (E) is forbidden. Why?—Because only the indexing using all elements of N = {1, 2, 3 , . . . } leads to the desired Cantonan conclusion"

[mateo3470]

@ja524309

What do I care about a poser's argument anyway. I have posted the sources on here numerous times. You are too lazy to go back to a previous post. I can't post links in these comments!

[mateo3470]

@ja524309

You made a statement about countably infinite subsets. WOW! I'm impressed. As a matter of fact, the index argument I talk about uses this notion. Reindexing the reals listing can easily be accomplished at each stage of diagonal construction.

I can't even use subscripts on these comments. This is a YouTube post. Read the paper.

[mateo3470]

@ja524309

I can't use any of the proper notion there either. I will give you the links in a personal message.

[mateo3470]

@ja524309

Tone down the noise level pal.

watch?v=AP92NNe8Tkw

[mateo3470]

Search: Logic of Actual Infinity and G. Cantor's Diagonal Proof of the Uncountability of the Continuum by A. A. Zenkin

Source: Rev. Mod. Log. Volume 9, Number 3-4 (2004), 27-82.

Cantor replaced "infinitum actu non datur" with "omnia seu finita seu infinita definita sunt et excepto Deo ab intellectu determinari possunt". (Page 14 of Michael Hallet's "Cantorian Set Theory and Limitations of Size")

[mateo3470]

@ja524309

Nikolai Nikolaevich Luzin wrote an interesting objection to Cantor's proofs involving "effective enumeration". See: Naming Infinity by Graham and Kantor (p. 206)

Search: Logic of Actual Infinity and G. Cantor's Diagonal Proof of the Uncountability of the Continuum by A. A. Zenkin

Source: Review of Modern Logic Volume 9, Number 3-4 (2004), 27-82. (Project Euclid)

Search: Doron Zeilberger Opinion 68 (math.rutger.edu)

Wildberger has more to say on the subject at his website.

[mateo3470]

@ja524309

Forget axioms. One of the criteria for making mathematical sense or having something sensible to communicate at all is that you don't contradict yourself. Actual or completed infinity is self-contradictory. If you want to understand Cantor's reasoning the read "Cantorian Set Theory and Limitation of Size" (Oxford Logic Guides) by Michael Hallett. Zenkin and Zeilberger are against Bourbakian set theory.

[mateo3470]

@ja524309

Another Poser! You must really hate those "bad" indexings don't you? I refuse to be a victim of left-hemispheric criminality. Read the things I cited. This is a YouTube post so I can't explain everything. But I will give you a very general reason. Read this: "completed infinity". I rest my case.

[mateo3470]

@ja524309

I will quote what I wrote:

"Read the things I cited."

Hard concept?

[mateo3470]

@ja524309

"You're an idiot lol"

What a demonstration of intellect!

"You obviously know nothing about mathematics "

It's hard to come to that conclusion when you refuse to discuss any.

[mateo3470]

@ja524309

You sir are a poser. Doron Zeilberger is a practicing mathematician at Rutgers. Zeilberger was a co-recipient of the 1998 Steele Prize of the American Mathematical Society for his research on hypergeometric summation.

"your ambiguous phrase "completed infinity"

That's not my phrase. If you had the slightest clue about what you were talking about then you would know that completed or actual infinity and potential infinity goes back to the greeks.

[mateo3470]

@ja524309

Search for example "Completed versus Incomplete Infinity in Arithmetic" by Edward Nelson. He's another practicing mathematician at the department of mathematics at Princeton.

[mateo3470]

@ja524309

Here's a funny video:

watch?v=Y0Z0raWIHXk

[mateo3470]

@ja524309

Wrong again. That is what Euclid's mathematics is. Formal axiomatic systems come after the math. We do always makes assumptions, if that's what you mean.

[affablegiraffable]

also how do you feel about the ZFC axioms?

[affablegiraffable]

I don't understand why you bring up the opinions of other mathematicians instead of just being succinct.

[truenorthcorn]

This is just a baby's view

Consult a real textbook instead of being mislead

[TheSpunFire]

Voodoo? I really hope you were being sarcastic when you said that. I'm not trying to find holes in that you said or anything, but voodoo?

[SynergeticsCollab]

I like the criticisms of infinity given by Buckminster Fuller and Doron Zeilberger. Fuller points out "Universe" is the aggregate of all experience and as the sum of finites, must be finite. Zeilberger points out that representations of infinity are finitely presented (e.g., in formulas); so they are, ipso facto, finite.

My synthesis is there are patterns (general principles) that govern all special cases, but the pattern itself is finite.

You seem to say "infinity" is logically flawed?

[Luineancaion]

Ouch. This was one of the most stupid videos I've seen. He's just arguing over semantics it seems, this is made even more obvious by one of the comments here where he suggests the defining of a new mathematical object with the suggested name of 'Even' for the set of even numbers.

Cantor's diagonalisation proof is so easy to understand, and I'm not even university educated.

Also Georg is not George. It's pronounced 'gay-org'. Seriously I expect better from a Prof. of Mathematics.

[Viridian85]

It may be easy to understand but it's based on Cantor's definitions. Those definitions happen to lead to logical flaws. Look into comparing the infinitude of different sets and it should become apparent.

[mateo3470]

@Luineancaion

Nikolai Nikolaevich Luzin wrote an interesting objection to Cantor's proofs involving "effective enumeration". See: Naming Infinity by Graham and Kantor (p. 206)

Search: Logic of Actual Infinity and G. Cantor's Diagonal Proof of the Uncountability of the Continuum by A. A. Zenkin

Source: Review of Modern Logic Volume 9, Number 3-4 (2004), 27-82. (Project Euclid)

Search: Doron Zeilberger Opinion 68 (math.rutger.edu)

Wildberger has more to say on the subject at his website.

[mattmoss]

Is this nomenclature, or are there significant reasons for distinguishing sets and infinite "sets"? If I want to discuss the even numbers, do we not call them a set because they don't meet the (traditional) defn. of a set, or because they can't be considered in a collective manner?

I appreciate this video to see some of the history, but it doesn't really explain your reasons for why infinite sets don't exist (as you suggest at the end of the vid). Is there another vid that continues this topic?

[njwildberger]

Hi mattmoss--- If we want to discuss even numbers, we should define a new type of mathematical object, perhaps called `Even'. This way we avoid talking about `all' the even numbers as a completed whole, which is not allowed.

The reason it is not allowed is that one ends up with logical conundrums if we consider `infinite sets'. I will be talking about this important issue a lot more in this series.

[ja524309]

@mattmoss No, this is not nomenclature, there really is a difference between finite and infinite sets: namely, that infinite sets can be mapped 1-1 onto a proper subset of themselves, whereas with finite sets this is impossible. For example, the positive integers can be mapped 1-1 onto the even integers. For more on this matter, see my other posts.

[karaool]

Looking forward to the next vid. I've long suspected that Cantor's theory of transfinite numbers and infinite sets was a crock., and I've seen many a bright and gifted math student browbeaten into accepting its validity by their profs.

[prunsunk]

I enjoyed your use of the word "voodoo."

[Pebbe496]

Where do I find Gauss' opinion about that? I would be very interested in what he had to say about that topic.

[njwildberger]

I have added some annotations to the video which give the relevant quotes. Gauss's comments were expressed in a letter to Schumacher of July 12, 1831.

[Pebbe496]

Thanks! That's interesting. Especially Weyl's comment.

[wbell539]

I'm definitely 'ordinary'. So I'm looking forward to the next video.

[mkrulic517439]

look forward to your next vid. didnt really sell me on this one. If the counting numbers were finite then what would be the biggest one?

[Viridian85]

He never said they were finite, he never said infinity doesn't exist, he says that infinite sets don't exist. The real arguing point should be "what is a set?" I also have no idea if he's opinion on infinite sets is correct, but it's better than the current framework I've studied.