[FOM] Strong Hypotheses and the Theory of N

[Richard Heck]

On 03/25/2010 12:36 AM, Robert Solovay wrote:
> In a recent posting (March 20th) Joe Shipman asks the following question:
>
> Would the problem be easier if I asked for a (presumably consistent)
> “natural” set of arithmetical axioms which implied all the arithmetical
> consequences of ZF but was not necessarily limited to them?
>
> This relates to his prior question (in a posting on March 15th):
>
> A related question: is there a natural way to represent the
> "arithmetical content" of ZF by arithmetical axioms; in other words, a
> natural decidable set of arithmetical statements which have the same
> arithmetical consequences as ZF?
>
> [snip]
>
> Then the answer to the first of Shipman's questions that I propose is:
>
> 1) The axioms of Peano Arithmetic;
>
> 2) For each positive integer $n$, the arithmetical formulation of "ZFC
> is $n$-consistent".
>
> That all these arithmetical formulas are true can be proved, e. g., in
> ZFC + "There is an inaccessible cardinal".
>
>
And that EACH of these is true can be proved in ZFC itself, of course, 
since ZFC is reflexive.

rh