[FOM] Re: Why the definition of "large cardinal axiom" matters

[JoeShipman@aol.com]

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