[FOM] Re: Why the definition of "large cardinal axiom" matters
JoeShipman at aol.com
Wed Apr 21 14:16:48 EDT 2004
The interesting thing about large cardinal axioms, empirically, is that they fall into a hierarchy. They are all compatible with each other, and are naturally ordered in 2 compatible but distinct ways: consistency strength and cardinality.
What would be truly new is an axiom which did not fall into this hierarchy; for example, a statement of the form "Therexists kappa Phi(kappa)" such that
1) Phi(V) is plausible [abusing notation in the usual way]
2) Phi(kappa) implies kappa is an inaccessible cardinal
3) "Therexists kappa Phi(kappa)" refutes some other large cardinal axiom
As stated, this is easy: Phi could be "kappa is the unique inaccessible cardinal", but this ought to be regarded as cheating. A more careful definition of "large cardinal axiom" might require Phi to be "intrinsic", that is, to refer to a first-order property of V(kappa) and not say anything about what happens for larger sets. But wouldn't such a restriction rule "An extendible cardinal exists", which is generally considered a legitimate example of "large cardinal axiom"?
-- Joe Shipman
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