Fri May 2, 2008
60 Evans Hall, 4:10–6 PM
M. C. Stanley (Mathematics, San Jose State University)
Outer Model Satisfiability
A remarkable development in Set Theory is the discovery that if large cardinals exist, then, up to a certain level of logical complexity, what is true cannot be changed by set forcing. According to a celebrated theorem of Woodin, if the Continuum Hypothesis holds and there exist unboundedly many measurable Woodin cardinals, then an existential statement in third-order arithmetic having real parameters holds in a set generic extension that preserves the CH if and only if it already holds in V. Just beyond this level of logical complexity, not only is it impossible to complete the universe in this sense, but in ZFC there is not even a first-order characterization of the family of sentences that are false in all outer models. This is peculiar if one is accustomed to thinking of mathematics as governed by logical necessity.
The theorem is that if V satisfies ZFC^+ = ZFC + “every definable closed unbounded class of ordinals includes a Ramsey cardinal” and V is “sufficiently nonminimal”, then this anticharacterization phenomenon disappears.
But first, what are “outer models”? Talk of outer models is inherently second-order. The least restrictive way to understand claims about outermodel existence is to understand “V” as a countable set model in some larger ambient universe. The letter “V” is just intended to suggest a privileged role for this model as proxy for the real universe of sets – “real” at least in the sense that our discussion is formalizable in first-order ZFC. If V is a standard transitive model of ZFC, for the purpose of this abstract, say that W is an outer model of V if V is contained in W, the same ordinals are in W and V, and W is also a standard transitive model of ZFC.
THEOREM. There exists a parameter-free formula good(x) in the language of set theory as follows: Work inside a model V of ZFC^+. Let kappa be a regular uncountable cardinal. Let T in H_kappa be a set of first-order axioms in the language of set theory, extending ZFC and perhaps using parameters from H_kappa.
- If H_kappa satisfies good(T), then there exists an outer model of V that satisfies T.
- If H_kappa does not satisfy good(T) and V is sufficiently nonminimal, then T is not satisfiable in any outer model of V. The formula good(x) can be taken to be parameter-free Pi_2 or, if kappa is greater than omega-1, to be Pi_1 in the parameter omega-1.
If “V” is actually V_delta, where delta is an inaccessible cardinal in some longer model of ZFC, then V is sufficiently nonminimal. Some non-first-order hypothesis, like nonminimality, is necessary.