[FOM] Eliminating AC

[Robert Lubarsky]

One can use class forcing to get AC to be true. The integers do not change, so the truth of arithmetical statements remain unchanged.

Bob Lubarsky

-----Original Message-----
From: fom-bounces at cs.nyu.edu [mailto:fom-bounces at cs.nyu.edu] On Behalf Of François Dorais
Sent: Thursday, March 28, 2013 10:20 AM
To: Foundations of Mathematics
Subject: Re: [FOM] Eliminating AC

I had a naive idea regarding elimination of AC which turns out to be flawed but I wonder if it could be fixed or permanently shot down.

The naive idea was to replace the uses of AC by uses of finitely many Skolem functions and then syntactically eliminate them to get a proof without AC nor Skolem functions. Of course, that's not how Skolem functions work. We can add a bunch of Skolem functions to get a conservative extension of ZF but the comprehension axioms are not allowed to use these Skolem functions. For example, we do get a new function that picks a representative from each Vitali class, but we cannot comprehend the range of that function to form a Vitali set.
Thus, there is no reason to believe the original ZFC proof of the arithmetical statement has a translation in this expansion of ZF.

Is there a refined version of this naive idea that works? Is there a way to see that such an argument couldn't possibly work?

Best wishes,

François G. Dorais
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