[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
> _______________________________________________
> FOM mailing list
> FOM at cs.nyu.edu
> http://www.cs.nyu.edu/mailman/listinfo/fom
>
More information about the FOM
mailing list