[FOM] Follow-up to Eliminability of AC

Juliette Kennedy jkennedy at cc.helsinki.fi
Thu Mar 27 14:50:02 EDT 2008

Dear Richard,

Regarding *explicit* references in Goedel to the observation you ask about
I can find (sofar) the following remarks, from the supplement
to the 1964 version of Goedel's
"What is Cantor's Continuum Hypothesis": (Collected Works vol.2, p.267)

"The generalized continuum hypothesis, too, can be shown to be sterile for
number theory and to be true in a model constructible in the original
system, whereas for some other assumption about the power of 2^aleph_alpha
this is perhaps not so."

(Goedel could have added that not-CH is also sterile for number theory,
but he was frying other fish in that passage.)

With Solovay Simon Kochen also credits Kreisel with the observation in
Kochen's 1961 paper "Ultraproducts in the Theory of Models," referring to
Kreisel's 1956 "Some Uses of Metamathematics," a review of Robinson's Theorie
Metamathematique des Ideaux" British Journal for the Philosophy of
Science, vol.7, number 26, 1956.

And indeed on page 165 of that review Kreisel makes that
observation "as a consequence of Goedel's work on the consistency of the
axiom of choice and the continuum hypothesis..."

All best,

Department of Mathematics and Statistics
P.O. Box 68 (Gustaf Hällströmin katu 2b)
FI-00014 University of Helsinki, Finland
tel. (+358-9)-191-51446, fax (+358-9)-191-51400
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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