Re: Conservativity and CH

On Dec 6, 6:30 am, "Robert E. Beaudoin" <rbeaud...@xxxxxxxxxxx> wrote:
kleptomaniac6...@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

Yeah I forgot about the relationship between metamathematical
statements and arithmetical statements!

But how about the following wild conjecture: For any non-arithmetical
statement S which is independent of some system R where the
arithmetical part of R extends minimal arithmetic or Q, for any system
T which extends R and proves S, there is a system T' which extends R
and disproves S and has the same arithmetical part as T.

All the systems at hand are consistent.

.

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