[FOM] A different type of cardinal "collapse"

[joeshipman@aol.com]

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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