[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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