[FOM] Definition of "large cardinal axiom"?

Robert M. Solovay solovay at math.berkeley.edu
Wed Apr 14 23:35:11 EDT 2004

On Wed, 14 Apr 2004 JoeShipman at aol.com wrote:

> I have always understood "Large Cardinal Axiom" to mean a sentence Phi in the language of set theory with the following properties:
> 1) Phi is equivalent to a sentence of the form "There exists a cardinal Kappa such that Psi(Kappa)" or "There exist arbitrarily large Kappa Psi(Kappa)", such that
> 2) Psi(Kappa) implies Kappa is strongly inaccessible, or bears some simple relationship to a strongly inaccessible cardinal (for example, Kappa is a singular limit of strong inaccessibles)
> 3) Psi(V) is true, or at least not known to be false (that is, Psi can be straightforwardly modified into a sentence of VNBG or some similar "Class" theory asserting something about the universe of sets that might be true).
> Conditions 2 and 3 are not precise, but they're clear enough in practice.  Can anyone suggest a statement generally regarded as a large cardinal axiom that does not satisfy the above?

	Perhaps "0# exists" would count as a counterexample.

	--Bob Solovay

> -- Joe Shipman
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