Set-theoretic Definition of Natural Numbers - Problem

Problem

A consequence of Kurt Gödel's work on incompleteness is that in any effectively generated axiomatization of number theory (i.e. one containing minimal arithmetic), there will be true statements of number theory which cannot be proven in that system. So trivially it follows that ZFC or any other effectively generated formal system cannot capture entirely what a number is.

Whether this is a problem or not depends on whether you were seeking a formal definition of the concept of number. For people such as Bertrand Russell (who thought number theory, and hence mathematics, was a branch of logic and number was something to be defined in terms of formal logic) it was an insurmountable problem. But if you take the concept of number as an absolutely fundamental and irreducible one, it is to be expected. After all, if any concept is to be left formally undefined in mathematics, it might as well be one which everyone understands.

Poincaré, amongst others (Bernays, Wittgenstein), held that any attempt to define natural number as it is endeavoured to do so above is doomed to failure by circularity. Informally, Gödel's theorem shows that a formal axiomatic definition is impossible (incompleteness), Poincaré claims that no definition, formal or informal, is possible (circularity). As such, they give two separate reasons why purported definitions of number must fail to define number. A quote from Poincaré: "The definitions of number are very numerous and of great variety, and I will not attempt to enumerate their names and their authors. We must not be surprised that there are so many. If any of them were satisfactory we should not get any new ones." A quote from Wittgenstein: "This is not a definition. This is nothing but the arithmetical calculus with frills tacked on." A quote from Bernays: "Thus in spite of the possibility of incorporating arithmetic into logistic, arithmetic constitutes the more abstract ('purer') schema; and this appears paradoxical only because of a traditional, but on closer examination unjustified view according to which logical generality is in every respect the highest generality."

Specifically, there are at least four points:

1. Zero is defined to be the number of things satisfying a condition which is satisfied in no case. It is not clear that a great deal of progress has been made.
2. It would be quite a challenge to enumerate the instances where Russell (or anyone else reading the definition out loud) refers to "an object" or "the class", phrases which are incomprehensible if one does not know that the speaker is speaking of one thing and one thing only.
3. The use of the concept of a relation, of any sort, presupposes the concept of two. For the idea of a relation is incomprehensible without the idea of two terms; that they must be two and only two.
4. Wittgenstein's "frills-tacked on comment". It is not at all clear how one would interpret the definitions at hand if one could not count.

These problems with defining number disappear if one takes, as Poincaré did, the concept of number as basic i.e. preliminary to and implicit in any logical thought whatsoever. Note that from such a viewpoint, set theory does not precede number theory.