[FOM] Weak independent statements of arithmetic
Harvey Friedman
friedman at math.ohio-state.edu
Tue Feb 23 14:35:05 EST 2010
On Feb 23, 2010, at 8:34 AM, Colin McLarty wrote:
> This may be very familiar but I do not know it: what is known about
> new axioms which can be added to ZFC without increasing the
> consistency strength (as for example CH, or V\neq L, or Martin's
> axiom) but which do imply first order statements of arithmetic which
> are not implied by ZFC alone?
>
> Well, one example would be any statement of first order arithmetic
> which is independent of ZFC but provably equiconsistent with it. I do
> not care if the equiconsistency proof uses all of ZFC. That is not an
> issue to me.
>
> Are such statements known? Is there some easy way to find them?
ZFC + "a Rosser sentence for ZFC" is equiconsistent with ZFC.
A Rosser sentence for ZFC asserts
"to every proof of me in ZFC there exists a proof in ZFC of my
negation, with smaller Goedel number".
Harvey Friedman
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