[FOM] A different type of cardinal "collapse"

[joeshipman@aol.com]

Responding to my own post pre-emptively:

1) Of course all large cardinal axioms are equiconsistent with 
arithmetical statements, but that's not what I mean when I ask for 
alternative axioms about "small sets"; I hope everyone understands the 
distinction I am making, because it's a difficult one to formalize.

2) A large-cardinal property of a cardinal kappa whose definition 
refers to arbitrarily large sets can be replaced by an axiom about the 
"intrinsic" property of being a universe large enough to have a 
cardinal with the original property. Thus, "kappa is supercompact" can 
be replaced by "there exists kappa < lambda such that V_lambda |- 
'kappa is supercompact' "-- as far as lambda is concerned, this is an 
"intrinsic" property which doesn't change as the universe expands. How 
much weaker is this alternative (in general, and in the specific case 
of supercompactness)?

-- JS

-----Original Message-----
From: joeshipman at aol.com
...
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"...- 
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)... 
3) an axiom which refers to sets of arbitrary rank... 
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