# Are real numbers “a joke”? [closed]

Recently I stumbled upon the website of Prof. N. J. Wildberger, who has an written a thought-provoking article about set theory and the real numbers. In his opinion, real numbers are actually "a joke". An excerpt, where he talks about the definition of the real numbers as equvalence classes of Cauchy secquences:

(...) But here is a very important point: we are not obliged, in modern mathematics, to actually have a rule or algorithm that specifies the sequence. In other words, 'arbitrary' sequences are allowed, as long as they have the Cauchy convergence property. This removes the obligation to specify concretely the objects which you are talking about. Sequences generated by algorithms can be specified by those algorithms, but what possibly could it mean to discuss a 'sequence' which is not generated by such a finite rule? Such an object would contain an 'infinite amount' of information, and there are no concrete examples of such things in the known universe. This is metaphysics masquerading as mathematics. (...)

For me this actually makes some sense. Why are opinions like this so rarely seen? Shouldn't we take Wildberger's arguments seriously? Are there really flaws in the foundations of mathematics?

EDIT #1: I think this question is not a duplicate (see my statement in the comments section).

EDIT #2: To clarify my question a bit: If I understand Wildberger correctly, he sees a problem in handling objects that contain an infinite amount of information (because no human being can fully understand such objects), like equivalence classes of Cauchy sequences. (He does not have a problem with infinite sets in general as I remember a video lecture of him where he says that he believes in the rational numbers). My question is: Can this (talking about objects that contain an infinite amount of information) be problematic for the science of mathematics? Or is it just a "believe-it-or-not-thing" like the Axiom Of Choice? I'm not looking for an answer like "Wildberger is right/wrong", but rather for a simple explaination of the key points, as I have the feeling of lacking a bit of advanced knowledge that is required to fully understand this topic, or to fully understand Wildbergers arguments. I've read the answers to the question this one is a possible duplicate of, but as far as I've understood them (I did not understand everything), they do not adress my question.

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This is a legitimate question, but I fear you won't get any good answers unless you choose a less provocative title. –  MJD Jun 9 at 14:20
I vote against closing this question. –  MJD Jun 9 at 14:34
What does mathematics have to do with "concrete examples in the known universe" –  Hagen von Eitzen Jun 9 at 14:52
In my opinion, philosophical questions like this are not fact-based, and will lead to debate and extended discussion instead of any definitive answer. And as for "Why are opinions like this so rarely seen?", how could anyone have something other than speculation on why most people don't have this opinion, or why they don't feel like sharing it at any rate? I have voted to close as "not constructive". –  Zev Chonoles Jun 9 at 16:27
Wildberger's essay has been significantly misrepresented in the "answers" at the other question. The high voted answer authors did not correct their postings when basic errors were pointed out, so there are now high scoring answers there that badly mischaracterize the essay and are effectively trolling for votes on mockery (arguably W asked for it, but it's still inaccurate). It is not a good idea to take that thread as an indication of anything about Wildberger's essay. It got bad enough that a prominent Australian user was prodded out of lurking and came forward to defend his countryman. –  zyx Jun 9 at 23:20

## closed as not constructive by Grigory M, Myself, rschwieb, Asaf Karagila, Carl MummertJun 15 at 21:11

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This is an old issue about constructivism.

The real numbers are often represented by the example $\sqrt2$, which is easily shown not to be a rational number. This is somewhat misleading since this type of example is depending on numbers which can be defined. These (definable) numbers are countable and therefore they have a one-to-one correspondence with the rational numbers. However, the real numbers – as they are understood to exist by an analogy to the geometrical concept of a number line – are uncountable. We can also say, more provokingly: if they existed they would be uncountable.

The question is: if something exists, should you also be able to point to it, define it, and construct it in a final number of steps?

We have therefore agreement in all quarters that a countable set of real numbers exist. However to quote Prof Podnieks: Any Goedel-style enumeration can cover only those real numbers that are definable by formulas (in some fixed language). Thus, if in one's set theory, there are uncountably many real numbers, then some of these numbers must be undefinable (by formulas).

These (undefinable) real numbers are creating a division between people (mathematicians). “If they exist why can’t you point to them” say the constructivists, which are said to have have many subdivisions. I have also heard the following pun: mathematics is superior to physics: The rules of physics apply to everything that exists. But the rules of mathematics apply even to things that don't exist.

I believe you may be OK either way, we can go to any church we like, and we don’t necessarily even have to choose between them.

Edit: I believe the more commonly used term for what Podnieks mention is "indefinable" rather than "undefinable". In any case it refers to a real number that can't be defined in a finite number of steps. Observe that a limit of an infinite series is definable in this context.

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What do you mean by "definable number"? –  Michael Greinecker Jun 15 at 21:12
What are definable numbers and why are them countable? –  MyUserIsThis Jun 15 at 21:55
Wikipedia: “In mathematical logic, a Gödel numbering is a function that assigns to each symbol and well-formed formula of some formal language a unique natural number, called its Gödel number”. Gödel number translates a definition into a number. Any uncountable set such as the real numbers must contain elements that cannot be given such a number. They cannot be defined in a finite number of steps. –  Mikael Jensen Jun 16 at 6:23
@MikaelJensen: Good answer, thanks. So the constructivity issue here (numbers that are "undefinable") is basically the same as with the Axiom Of Choice (a choice function that is "undefinable"), is this correct? –  zero-divisor Jun 16 at 11:54
Joel David Hamkins gives an extensive discussion of some notion of definability based on enumerating formulas here and seems to argue that it is ultimately untenable. I am confused. How does this relate to what you are discussing here? –  Martin Jun 16 at 14:01