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

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16 MENACHEM MAGIDORcan be a good guidance for sharpening our concepts and getting a betterand better axiom systems for Set Theory. In what sense the imaginedset is specified? As first approximation let us say that a set is simplyspecified as any satisfying a certain property. So we should rephraseour slogan asIf a set satisfying a given property is possible and thereis no clear obstruction to the existence of such set thensuch set exists.What do we mean by ”possible”? I think that a good approximationis ”can be forced to exists” So let us tryIf one can force the existence of a set satisfying a givenproperty and there is no clear obstruction to its existencethen such a set exists.Still this principle is problematic. One can introduce by forcing a setwhich is an enumeration of all the reals of order type ω 1 but also onecan introduce by forcing a list of ω 2 different reals. Of course it isinconsistent to have sets satisfying both properties. The statement iseven more problematic if in the property one allow parameters say agiven set A. Because suppose that our parameter A is uncountable .By using Levy’s collapse one can make A countable, so introduce a 1-1mapping between A and ω. Obviously such a mapping does not existsin the our universe.I consider forcing axioms as an attempt to try and get a consistentapproximation to the above intuitive principle by restricting theproperties we talk about and the the forcing extensions we use. Therestriction of the forcing notions is usually following the intuition of allowingonly forcing notions that do not make a very dramatic change inthe universe , like making an uncountable set countable.This is somewhatsimilar to restricting in the interpretations of the modalities ”itis possible that...” the set of possible universes to universes which arenot too different from the current universe.So we restrict the forcingwhich we consider to ”mild” forcing extensions.When we extend the universe by using forcing, what we add to theuniverse is the generic object with respect to the forcing notion. Sinceany other set introduced by the forcing notion is defined (over theground model) from the generic filter, our typical forcing axiom is ofthe form:For a given class of ”mild” forcing notions P and forevery forcing notion P ∈ P there is a rich family offilters in P which are generic enough
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.
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