
Matheology § 214
Posted:
Feb 10, 2013 4:38 AM


Matheology § 214
What?s wrong with the axiom of choice? Part of our aversion to using the axiom of choice stems from our view that it is probably not ?true?. {{In fact it is true for existing sets  but there it is not required as an axiom but is a selfevident truth.}} A theorem of Cohen shows that the axiom of choice is independent of the other axioms of ZF, which means that neither it nor its negation can be proved from the other axioms, providing that these axioms are consistent. Thus as far as the rest of the standard axioms are concerned, there is no way to decide whether the axiom of choice is true or false. This leads us to think that we had better reject the axiom of choice on account of Murphy?s Law that ?if anything can go wrong, it will?. This is really no more than a personal hunch about the world of sets. We simply don?t believe that there is a function that assigns to each nonempty set of real numbers one of its elements. While you can describe a selection function that will work for ?nite sets, closed sets, open sets, analytic sets, and so on, Cohen?s result implies that there is no hope of describing a de?nite choice function that will work for ?all? nonempty sets of real numbers, at least as long as you remain within the world of standard ZermeloFraenkel set theory. And if you can?t describe such a function, or even prove that it exists without using some relative of the axiom of choice, what makes you so sure there is such a thing? Not that we believe there really are any such things as in?nite sets, or that the ZermeloFraenkel axioms for set theory are necessarily even consistent. Indeed, we?re somewhat doubtful whether large natural numbers (like 80^5000, or even 2^200) exist in any very real sense, and we?re secretly hoping that Nelson will succeed in his program for proving that the usual axioms of arithmetic?and hence also of set theory?are inconsistent. (See E. Nelson. Predicative Arithmetic. Princeton University Press, Princeton, 1986.) All the more reason, then, for us to stick with methods which, because of their concrete, combinatorial nature, are likely to survive the possible collapse of set theory as we know it today. [Peter G. Doyle, John Horton Conway: "Division by Three" 1994, ARXIV math/0605779v1] http://arxiv.org/abs/math/0605779v1
Regards, WM

