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

[Roger Bishop Jones]

I thought it might help discussion of what a large
cardinal axiom is to say something about my reasons
for being interested in this question.

It seems to me that the intuition behind the acceptance
of large cardinal axioms is that they are essentially
just placing lower bounds on the height of the cumulative
hierarchy.
It is built into the usual conception of the cumulative
hierarchy that it goes as far as it possibly can, and
therefore any axiom which does no more than place a lower
bound on height, and which is consistent, must be also be true.

This rationale only holds good if the axiom does no more
than place a lower bound on height, so one might wish
that large cardinal axioms did no more than that.
We know in fact that large cardinal axioms always do
more than raise the bar, since they give us new theorems
in arithmetic.

Is there any limit to what kind of questions large cardinal
axioms will solve just as a by-product of adding height?
Should we expect large cardinals to settle CH, or should
we suspect that a large cardinal which does so is saying
more than a large cardinal axiom should?

Dehornoy's definition of a large cardinal axiom, which is
part of his description of Woodin's work on CH contains
an extra clause not present in Joe Shipman's.
He requires that psi(kappa) is satisfied in every generic
extension of V associated with a forcing set of cardinality
smaller than kappa (where "psi" is the large cardinal property
and "kappa" is the existentially bound variable).
Is this just a peculiarity of the work with which it is
presented or does this has merit as a criterion of propriety
for large cardinal axioms which makes it more credible that
the axiom is true?

Roger Jones

- rbj01 at rbjones.com
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