[FOM] Arithmetical soundness of ZFC

Harvey Friedman friedman at math.ohio-state.edu
Sun May 24 13:36:44 EDT 2009

> On May 24, 2009, at 1:47 AM, joeshipman at aol.com wrote:
> If ZFC proved "ZFC is inconsistent" (which is weaker than proving 1=0
> because ZFC could still be consistent but omega-inconsistent) then  
> that
> could be converted into a proof that ZFC is not sound.
> We would expect a sound theory to be omega-consistent with respect to
> arithmetical statements.
> I cannot think of any other possible way in which we could come to
> believe in the non-arithmetical soundness of ZFC than a proof that ZFC
> is omega-inconsistent.
> -- JS

But in order to prove in ZFC that "ZFC is inconsistent", we are going  
to have to find an inconsistency in ZFC + "an inaccessible cardinal".  
This is just a variant of the program to find inconsistencies.

So it remains unclear to me how "the finding arithmetical unsoundness  
adventure" differs from "the finding inconsistencies adventure".

Harvey Friedman

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