[FOM] Definition of "large cardinal axiom"?
JoeShipman at aol.com
Thu Apr 15 01:10:11 EDT 2004
Solovay suggests "0# exists" as a counterexample to my conditions for a statement of Set Theory to be a "Large Cardinal Axiom".
But I distinguish between "Large Cardinal Axioms" and statements which imply the consistency of very large sets but not their existence. "0# exists" is consistent with "there are no inaccessible cardinals". (Proof: 0# is absolute for transitive models of ZF containing all countable ordinals; so if kappa is the 1st inaccessible then V(kappa) has no inaccessibles but does satisfy "0# exists" -- see Corollary 14.12 in Kanamori's "The Higher Infinite".)
I see "0# exists" as a statement akin to "Con(Inacc)" or RVM -- independent of the EXISTENCE of very large sets, but entailing their consistency.
Of course, the consequences of Large Cardinal axioms in "ordinary mathematics" USUALLY require only consistency and not existence of large cardinals; but if "0# exists" is to be regarded as a full-fledged Large Cardinal Axiom, shouldn't RVM and Con(Inacc) also be so labeled?
Projective Determinacy is another example of an axiom with strong consistency consequences that does not entail the actual existence of very large sets. Is this also to be characterized as a Large Cardinal Axiom, or just as equiconsistent with one?
In some of these cases (0# exists, Con(Inacc), PD) the proposed axioms are directly implied by axioms asserting the existence of large sets, so even if they are not "Large Cardinal Axioms", their negations negate large cardinal axioms. RVM is different -- it could be false no matter how large sets can get, and could be true even if there are no inaccessibles (as long as MC is consistent). So it is more of a stretch to call RVM a "Large Cardinal Axiom" than for the other axioms discussed above.
-- Joe Shipman
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