SOME SET THEORIES ARE MORE EQUAL ... - Logic at Harvard

SOME SET THEORIES ARE MORE EQUAL 17A more rigorous statement will beFor the given class of ”mild” forcings P , for every P ∈ Pand for every ordinal α such that P ∈ V α there is richfamily of elementary substructures of 〈V α , ɛ, P 〉 M suchthat there is a M generic object for M ∩ P .The different forcing axioms differ only in the choice of the class ofthe forcing notions and the notion of ”rich” collection of elementarysubstructures. Since for a countable model M we can always find aM generic object for every forcing notion in M, the schemata above isinteresting only if we assume that there is a rich family of uncountableM for which we can find a generic. Typically the notion of ”rich”means that there the set of M which are elementary substructure ofV α and for which there is a generic is stationary subset of P κ (V α ) forsome set of cardinals κ. 6For instance the first forcing axiom was Martin’s axiom which isAxiom 6.4 (MA). If P is a forcing notion satisfying the countablechain condition (c.c.c) then for every α such that P ∈ V α and forevery κ ≤ 2 ℵ 0the set M ∈ P κ (V α ) for which there is a M generic filterfor P ∩ M is stationary in P κ (V α )).The way we presented the axiom is trivially true under CH so ifCH holds we do not get any new interesting mathematical statementand the same is true for other forcing axioms. So when we state anyforcing axiom we shall implicitly assume that 2 ℵ 0> ℵ 1 . MA whichwas proved consistent by Martin and Solovay in [26] decides manyindependent statements but it still leaves a lot of freedom as far as thesize of the continuum.The next step is enlarging the class of forcing notions for which theaxiom applies so the next step was the Proper Forcing Axiom:Axiom 6.5 (PFA). If P is a proper forcing notion (see [1] for definition.)then for every α such that P ∈ V α and for every κ ≤ 2 ℵ 0theset M ∈ P κ (V α ) for which there is a M generic filter for P ∩ M isstationary in P κ (V α )).This axiom was shown to be consistent relative to the consistencyof supercompact cardinal by Shelah ([37]). The next step was themaximal possible strengthening of this axioms as far as the class of6 P κ (V α ) is the set of all subsets of V α of cardinality less than κ. A subset ofP κ (V α ), S is stationary if for every enrichment of the structure 〈V α , ɛ〉 by countablymany new relations and functions there is an elementary substructure of thisstructure which is in S and whose intersection with κ is an ordinal.