[FOM] Eliminability of AC

[Juliette Kennedy]

Dear Richard,

It must have been Goedel that made this observation,
based on the straightforwardness of the argument. (But I will try to
find a reference for you once I get home.)

The argument goes like this: Suppose an arithmetic sentence Phi follows
from ZFC. Let M be a model of ZF and consider L(M). M thinks L(M) is
transitive, therefore M and L(M) have the same natural numbers. Phi holds
in L(M), since choice holds in L(M). By absoluteness of arithmetic
statements Phi must hold in M.

More or less the same argument works in the CH case, so althought Solovay
credits Kreisel, again I would include Goedel there. Once you have L you
have these 2 observations.

All best,
Juliette
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Department of Mathematics and Statistics
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e-mail: juliette.kennedy at helsinki.fi
mobile: +358-50-371-4576
On Tue, 25 Mar 2008, Richard Zach wrote:

> An incidental question: Who first made the observation Joe has been
> posting about (viz., that if an arithmetical (or even Sigma^1_2)
> sentence follows from ZFC, it already follows from ZF)?  Solovay credits
> a similar observation, namely that if an arithmetical sentence follows
> from ZFC + GCH, it follws from ZFC alone, to Kreisel (Gödel's Collected
> Works, vol II, p. 19).
>
> Best,
> -RZ
>
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