[FOM] Cohen's Minimal Model and others

joeshipman@aol.com joeshipman at aol.com
Sat May 9 00:07:21 EDT 2009

Cohen's minimal model M consists of all the "strongly constructible 
sets", the sets which must appear in any standard model of ZF (that is, 
models which can be obtained by taking the actual membership relation 
for some real set).

If there are no standard models of ZF, then M is not a set but a proper 
class which can be obtained by the same kind of construction that L is, 
except that the operations allowed to build new sets are restricted to 
the ones made necessary by the ZF axioms.

The conjunction of "V=L" and "there is no standard model" can be viewed 
as a strengthening of V=L, and I will denote this statement as V=M. V=M 
is certainly consistent since if there is a standard model then the 
minimal model M is a set which satisfies V=M.

What I am interested in is what restrictions statements like V=L and 
V=M place on large cardinals. V=M implies no inaccessibles, and V=L 
implies no measurables (and in fact rules out some smaller cardinals 
too, the smallest of which is 0#).

What other results of this kind are there? Hamkins has shown that a 
certain very large cardinal axiom (larger than n-huge for all n), if 
consistent, is consistent with V=HOD, but I am looking for results in 
the other direction, that show an inconsistency of a large cardinal 
axiom with "V=X" for various inductively defined classes X.

-- JS 

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