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Topic: Matheology § 214
Replies: 19   Last Post: Feb 11, 2013 4:56 PM

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Posts: 18,076
Registered: 1/29/05
Matheology § 214
Posted: Feb 10, 2013 4:38 AM
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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 self-evident
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 non-empty 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? non-empty sets of real
numbers, at least as long as you remain within the world of standard
Zermelo-Fraenkel 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 Zermelo-Fraenkel 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

Regards, WM

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