[FOM] On the arithmetical conservativity of ZFC + GCH over ZF

[Joe Shipman]

This almost completely answers my earlier question. What remains of it is the following: 

What is the lowest possible quantifier-complexity for a sentence of analysis which is a theorem of ZF+DC but not of ZF?

-- JS

Sent from my iPhone

> On May 23, 2015, at 3:43 PM, Ali Enayat <ali.enayat at gmail.com> wrote:
> 
> This is a response to a query of Roman Murawski (May 22, 2015) who has asked:
> 
> "I was told about Kreisel's result stating that ZF + AC + GCH is a
> conservative extension of ZF with respect to sentences about natural
> numbers. Is it true? Where one can find it?"
> 
> I will give a three-pronged answer:
> 
> 1. The result is indeed true.  By modern standards it is a routine
> consequence of the fact that, provably in ZF, (a) AC and GCH hold in L
> (Goedel's constructible universe); and (b) arithmetical statements are
> absolute between V (the universe) and L.
> 
> 2. This topic was discussed on FOM a couple of years ago. You will
> find bibliometric information about Kreisel's observation in the
> following FOM posting of mine:
> 
> http://www.cs.nyu.edu/pipermail/fom/2013-March/017169.html
> 
> 3. A similar conservativity result is true even in the realm of
> analysis (second order number theory), namely:  if a theorem of
> analysis is provable in  ZFC+GCH, then it is already provable in ZF+DC
> (where DC is the usual axiom of dependent choice).   For more detail
> see my FOM posting below (this conservativity results seems not to
> have been explicitly noted earlier).
> 
> http://www.cs.nyu.edu/pipermail/fom/2013-April/017209.html
> 
> Best regards,
> 
> Ali Enayat
> 
> 
> 
> 
> 
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