Re: Conservativity and CH

From: "Robert E. Beaudoin" <rbeaudoin@xxxxxxxxxxx>
Date: Thu, 06 Dec 2007 01:30:13 -0500

kleptomaniac666_@xxxxxxxxxxx wrote:

I actually have one more question I was wondering about: Is it the
case that if for some axiom T, ZF+T is conservative over ZF for
arithmetical statements, then ZF+notT is also conservative over ZF for
arithmetical statements?

Let's take it for granted that ZF is consistent. (Else ZF entails every
arithmetical statement, so any extension of ZF would be conservative for
arithmetical statements.) Let A be the axiom "there is no measurable
cardinal". Then ZF + A is satisfied by the constructible universe, and
so is conservative over ZF for arithmetical statements. But ZF + ~A
entails Con(ZF), which is arithmetical and not derivable from ZF alone.

Robert E. Beaudoin
.

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