[FOM] eliminating AC from statements of analysis

[Paul Levy]

Thanks Ali for answering my questions.  I have another one!  Going  
back to your previous point:

> Message: 4
> Date: Sun, 31 Mar 2013 21:55:50 -0400
> From: Ali Enayat <ali.enayat at gmail.com>
> To: fom at cs.nyu.edu
> Subject: Re: [FOM] Eliminating AC [from statements of analysis]
> Message-ID:
> 	<CAPzKPNvVbYcNnaa3sSa7MSAe1URHYbgkhxLiffxvQzsPX2BJ6A at mail.gmail.com>
> Content-Type: text/plain; charset=ISO-8859-1
>
> 3. ZFC + CH is conservative over ZFC for statements of analysis. This
> follows from coupling the completeness theorem for first order logic
> with the well-known fact that every countable model M of ZFC has a
> generic extension N such that:
>
> (a) CH holds in N, and
>
> (b) N has the same reals as M (hence the truth-value of statements of
> analysis does not change in the passage between M and N).
>
> The notion of forcing at work is the set of countable injective
> partial functions from omega_1 into the reals (in the sense of M),
> ordered under (reverse) inclusion; see, e.g, Kunen's text on set
> theory (Theorem 8.3, p.227) for the forcing argument, however, note
> that expositions of forcing--Kunen's included--state their theorems in
> terms of countable *transitive* models, but  the assumption of
> transitivity can be dropped in such arguments by taking a detour
> through  Boolean-valued models.


Does ZFC + GCH have the same second-order consequences as ZFC?

Paul


--
Paul Blain Levy
School of Computer Science, University of Birmingham
+44 121 414 4792
http://www.cs.bham.ac.uk/~pbl