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

[Roger Bishop Jones]

On Wednesday 21 April 2004  7:16 pm, JoeShipman at aol.com wrote:
> 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.

Do you think the claim that the orderings are compatible
can be proven using your definition of "large cardinal axiom"?

On the basis of your proposed definition I would have thought
that any large cardinal axiom, no matter what size the
cardinal it asserts, could be modified to have arbitrarily
high consistency strength without increasing the size of
the cardinal asserted to exist, by adding a statement of
arithmetic (e.g. con(ZFC + stronger large cardinal axiom)).
If you agree this is the case, can you think of any way
of fixing the definition so as to make your claim above
that the two orderings are compatible provable?

Roger Jones

- rbj01 at rbjones.com
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