[FOM] Weak independent statements of arithmetic

[Harvey Friedman]

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