[FOM] Arithmetical soundness of ZFC

joeshipman@aol.com joeshipman at aol.com
Mon May 25 14:53:38 EDT 2009

I wrote

....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. 

I'd like to clarify my attitude here. In my opinion, the mathematics of 
hypothetical alien civilizations may differ from ours in many ways, but 
I do not expect it to differ in the sense that there will exist an 
arithmetical sentence S such that we think S is provably true and they 
think S is provably false. In support of this, I observe that all of 
the mutually incompatible extensions of ZFC have been proposed (large 
cardinals, negations of large cardinals, V=L, nonexistence of a 
standard model, projective determinacy, real-valued measurable 
cardinal, Martin's Axiom, etc.) do not appear to have incompatible 
arithmetical consequences (if you replace each such extension by its 
arithmetical consequences you can include them all without 

I state this as a conjecture, and challenge anybody to identify two 
extensions to ZFC which have been seriously proposed (and not 
subsequently shown inconsistent) such that there exists an arithmetical 
sentence decided oppositely by the two systems.

-- JS

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