[FOM] A different type of cardinal "collapse"
joeshipman@aol.com
joeshipman at aol.com
Tue May 15 22:52:30 EDT 2007
I am interested in a different kind of cardinal collapse, which does
not involve forcing but something much simpler, where you take models
of the form V_kappa for kappa inaccessible, and vary kappa. Cardinals
remain cardinals, but they don't necessarily have the same large
cardinal properties anymore. "Intrinsic" large cardinal properties like
being a measurable cardinal will still be the same, but properties
defined with reference to much larger sets like supercompactness may
change (a cardinal which is supercompact with respect to V_kappa may no
longer be supercompact with respect to larger ranks).
This is motivated by the criticism that large cardinal axioms are about
sets so large that there is an unavoidable vagueness about them. I can
distinguish 3 levels of concreteness for a large cardinal axiom:
1) an axiom which is equiconsistent with an axiom speaking only about
"small sets". In my previous post discussed weak inaccessibles, weak
Mahlos, and atomlessly-measurable cardinals, which can be defined in a
way that only speaks about sets in a finitely iterated powerset of
omega but which are equiconsistent with "strong" versions speaking
about very large sets. These can go quite high up in consistency
strength, for example if you count "projective determinacy" as a large
cardinal axiom you can use it to get the consistency of any finite
number of Woodin cardinals, I think.
2) an axiom which describes an "intrinsic" property of the cardinal
(not necessarily one which refers only to sets of lower rank, but which
certainly cannot refer to sets of arbitrary rank). These can go
extremely high up, for example the axiom that there is an elementary
embedding from a rank into a rank.
3) an axiom which refers to sets of arbitrary rank. I am willing to
grant that these axioms do involve an essential vageness, precisely
because the existence of such a cardinal may depend on how big the
whole universe is (if it gets bigger the large-cardinal property may
"break").
-- JS
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