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[Prismatic]

The term countable in mathematics is a term of art for the cardinality of a set in 1-1 correspondence with the set of natural numbers. You are apparently taking it in a different sense entirely.

What you’ve given here is what the term countable is, in set theory. Skolem was of the opinion that set theory ‘can't serve as a “foundation for mathematics”’ http://plato.stanford.edu/entries/paradox-skolem/, and as such what counting means in set theory is not the same as what it means in mathematics, or in ordinary English for that matter. There are two entirely different senses here.

As I’ve said above, Wittgenstein was of the same opinion – for him, counting doesn’t involve mapping onto an infinite set, counting is a procedure which outputs a finite set of numbers, the procedure itself is infinite because it doesn’t come to a built-in stop; wherever you get to, you can still keep going; the output isn't - infinity does not mean something that’s very, very huge and even bigger than that, that's just nonsense. There are no infinite sets of numbers, of whatever type (irrationals etc). Victor Rodych’s essay “Wittgenstein on Irrationals and Algorithmic Decidability” gives a thorough account of this.

Yes, counting as a process that determines the cardinality of a finite set and countability as an extension of the notion of cardinality to certain infinite sets are different. If you disallow the set theory notion of countability, it becomes impossible to do certain proofs in set theory and consequently in the rest of mathematics. Notions of countability crop up in topology and analysis all the time and are quite useful.

[Half-Six]

Wayne92587, to whom you responded, was closer to describing counting in mathematics, at least when he says "Numbers do not exist as countable objects", not what is called countability in set theory. Wittgenstein's, and perhaps Skolem's, point was not that the set theory notion of countability should be disallowed, Wittgenstein certainly didn't argue that there was something wrong with it; it's rather that it didn't achieve what Russell in particular felt it would achieve, to 'serve as a “foundation for mathematics”', where one would need to provide a foundation for counting, and countability doesn't fulfil this, as you say they are quite different, and for the numbers, and set theory doesn't achieve this, the numbers aren't contained in infinite sets, this is particularly pertinent for irrationals.

Countability, which is perhaps better named mappability, may well have practical uses, I'm sure mappability would be immensely useful in topology; but it's not counting.

[Prismatic]

Wayne92587, to whom you responded, was closer to describing counting in mathematics, at least when he says "Numbers do not exist as countable objects", not what is called countability in set theory. Wittgenstein's, and perhaps Skolem's, point was not that the set theory notion of countability should be disallowed, Wittgenstein certainly didn't argue that there was something wrong with it; it's rather that it didn't achieve what Russell in particular felt it would achieve, to 'serve as a “foundation for mathematics”', where one would need to provide a foundation for counting, and countability doesn't fulfil this, as you say they are quite different, and for the numbers, and set theory doesn't achieve this, the numbers aren't contained in infinite sets, this is particularly pertinent for irrationals.

Countability, which is perhaps better named mappability, may well have practical uses, I'm sure mappability would be immensely useful in topology; but it's not counting.

The notion of countable set was never a problem and was never intended to provide a foundation for counting—which is not itself a mathematical notion.

The project to found mathematics on set theory has its roots in the rigorization of analysis in the nineteenth century. Newton's calculus was very much ad hoc and lacking in rigor, but much had been built on it and there was a wide recognition among mathematicians of a need to put it on a firm footing. Dedekind, Bolzano, and Weierstrass made contributions to this and it was enormously successful. New problems arose however in the description of the sets of real numbers on which Fourier series converged and this was the spur for Cantor's set theory.

Russell's hope and that of many others was that set theory would furnish a foundation for mathematics by satisfying several requirements. The first object was to formalize set theory, that is to say, axiomatize it. The second was to define the concepts of mathematics within this axiomatic framework and prove the standard theorems. The third was to show the axiomatic system consistent.

This program was necessitated by the various paradoxes of set theory—Russell's paradox, Richard's paradox, Burali-Forti, etc. The two volume work of Russell and Whitehead aimed at carrying out such a program with the theory of types, but proved enormously cumbersome. Gödel's theorems made the project less attractive and Paul Cohen's treatment of the continuum hypothesis put an end to the idea.

[Half-Six]

The notion of countable set was never a problem and was never intended to provide a foundation for counting—which is not itself a mathematical notion.

I think effectively we’re in agreement here, countable sets isn’t a mathematical problem, which is why Wittgenstein didn't argue that there was something wrong with it. It’s a philosophical problem in that set theory has been used, by Russell etc, to attempt a foundation of mathematics (not a foundation of counting, apologies, I shouldn’t have used phrased it that way – counting might be taken as part of that foundation, so can’t by definition be a mathematical notion). For whatever reason, the word countable was chosen as a term in set theory, as well as infinite and transfinite, and it’s natural for people to jump to a conclusion that countable involves what we ordinarily by mean counting, as has been done in this thread, and that it relates to our usual understanding of infinity. If we take that conclusion, then Skolem’s paradox is a paradox, it’s a problem. Without it, it starts to dissipate.

[Keen]

I think there are two notions of countability. One which is relative to the model that is used and one which is used in meta-mathematics. I am certainly not an expert on logic, but it seems to me, that it actually often uses mathematical arguments (countability, sets induction and so on), which is sometimes a bit confusing to me, because as a mathematician, I view logic as a sound mean to prove that mathematics are correct. As for Skolem's paradox I can't say much on it, because I never had such an in depth knowledge in logic to be able to give an opinion on that matter.