[FOM] Arithmetical soundness of ZFC

[joeshipman@aol.com]

Sure, just say "As N --> infinity, the fraction of grammatically 
well-formed sentences of length N that are decidable in PA approaches 
0".

However, this would probably depend on the precise formulation of the 
grammar. Almost all sufficiently large sentences will have Con(PA) as a 
conjunct or disjunct, but that is not enough to render the sentence 
undecidable.

In fact, for the most obvious and user-friendly formulations of PA,  
the above is false because a nonzero fraction of sentences of PA begin 
"((0=0) V (" and so are decidably true and a nonzero fraction begin 
"((0=S(0)) & (" and so are decidably false. The actual probability that 
a well-formed sentence will be decidable (in any reasonable notion of 
probability) is likely to be equivalent to Chaitin's number Omega 
(which is also coding-dependent) in a strong sense.

As for the arithmetical unsoundness of ZFC: if there is an Inaccessible 
Cardinal k, then V(k) models ZFC and Th(V(k)) cannot include any false 
arithmetical sentences so ZFC must be arithmetically sound. Therefore 
any evidence for AFC's arithmetical unsoundness is also evidence there 
are no inaccessibles. In fact the same argument works for any Standard 
Model of ZFC because of the absoluteness of arithmetical sentences. 
Therefore you won't be able to argue for the arithmetical unsoundness 
of ZFC unless you start by assuming that there is no Standard Model.

I think that this is actually a reasonable assumption to make, although 
I happen to believe that ZFC *is* arithmetically sound.

-- JS

-----Original Message-----
From: Timothy Y. Chow <tchow at alum.mit.edu>

Harvey Friedman <friedman at math.ohio-state.edu> wrote:
> WHAT WOULD EVIDENCE OF THE NON ARITHMETICAL SOUNDNESS OF ZFC LOOK 
LIKE?

One example would be a proof of "ZFC is inconsistent" in ZFC.

Actually, what I think is more interesting is Nik Weaver's implicit
suggestion that a random sentence in the first-order language of
arithmetic is undecidable in PA.  Is there any way to make this precise?