[FOM] Eliminating AC

[Mitchell Spector]

Gitik constructed a model of ZF in which all infinite cardinals are cofinal with omega. Such a model has no class forcing extension which is a model of ZFC.

Mitchell Spector
spector at alum.mit.edu


On Mar 29, 2013, at 4:59 AM, "Robert Lubarsky" <Lubarsky.Robert at comcast.net> wrote:

> 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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