[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
contradiction).
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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