From: hrubin@odds.stat.purdue.edu (Herman Rubin)
Subject: Re: Axiom of choice question
Date: 25 Oct 1999 19:58:13 -0500
Newsgroups: sci.math
Keywords: Axiom of Determinateness, Banach-Mazur paradox

In article <38149E1A.5C69@kodak.com>, Jonathan Hoyle  <jhoyle@kodak.com> wrote:
>>> To each person who rejects the axiom of choice: do you accept that
>>> a countable union of countable sets is countable?  If not, OK, and
>>> I will not ask you to help me prove anything... but otherwise where
>>> do you reject the minimal conditions of choice in between... and
>>> why there and not stronger conditions?

>Yes, I too find it odd that someone argues against an axiom.  Why not
>argue against the Power Set Axiom, or better still the Replacement
>Axiom?

Without the Power Set Axiom, how would you even get the real
numbers?  You could get every constructible real number IF the
Replacement Axiom was true, but not without it.  The ordinal
number \omega + \omega requires it, as does Cantor's argument.

There is just too much which cannot be done without Power Set and
Replacement (the F part of ZF) to give these up.

>My un-asked for opinion...

>The Formalist side of me thinks this whole question is moot anyway. 
>After all, you can prove theorems assuming AC, and you can prove
>different theorems not assuming AC (or even assuming ~AC).  It's like
>arguing if Euclid's Parallel Postulate is true.  If you assume it's
>true, you get Euclidean Geometry; if you assume it's false, you get
>non-Euclidean Geometry.  Is one geometry "truer" than another?  Not to
>me.  Both derive useful results.

>The Platonist in me (whatever's left of him) has never been bothered by
>the Axiom of Choice.  Given an arbitrary collection of non-empty
>disjoint sets, why *shouldn't* I be able to take one element out of each
>of them?  I always felt AC was as intuitively obvious as any of the
>other axioms.  And what do we get by denying Choice anyway?  All subsets
>of the reals are now measurable (thus no Banach-Tarski-isms).

This is much weaker than choice.  It is easy to get from the
Prime Ideal Theorem, which is already weaker, and it has been
obtained from the even weaker Hahn-Banach Theorem.

However, there is another paradox.  The Banach-Mazur paradox
proceeds as follows:  Consider a game, in which the players
alternately put down bits in the binary expansion of a number is
[0,1].  Player A is trying to get the number in a set S, and B
in its complement.  With Choice, there are such games without
values, although the game is one of perfect information.  The
Axiom of Determinateness states that all such games have values;
this is believed, but cannot be proved, to be consistent.  This
one gives enough of choice for analysis to handle all but the
oddest situations, and does have all sets measurable.
-- 
This address is for information only.  I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Dept. of Statistics, Purdue Univ., West Lafayette IN47907-1399
hrubin@stat.purdue.edu         Phone: (765)494-6054   FAX: (765)494-0558