[FOM] eliminating AC from statements of analysis

[Paul Levy]

Thanks Ali.  Your helpful answers prompt more questions:

> 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>
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>
> This is a reply to Paul Levy's posting (March 27), who asked whether
> the conservativity of ZFC over ZF for arithmetical sentences holds
> also for sentences of analysis (aka second order number theory).

> 1. The answer to Levy's question is in the negative; for example, ZFC
> can prove that the so-called full "choice scheme" holds in analysis;
> but it has been known (since the work of Feferman and (Azriel) Levy in
> the 1960's) that there are models of ZF that `believe' that the choice
> schema fails in analysis; see Remark VII.6.3 (p.295) of Simpson's text
> on Subsystems of Second Order Arithmetic.

Do ZFC and ZF + dependent choice have the same second-order  
consequences?

> 2. (Paul) Levy had also asked whether ZF + CH is conservative over ZF
> for statements of analysis (of course in the choice-less setting CH
> stands for "the reals can be injected into every uncountable subset of
> the reals").  I do not know the answer to this question; hopefully
> someone reading this note can enlighten us.

On second thoughts CH doesn't seem very interesting if AC is not  
assumed, so I withdraw that question.

> 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.

Do ZFC + Martin's axiom at aleph_1 and ZFC have the same second-order  
consequences?

regards,
Paul

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