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

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18 MENACHEM MAGIDORforcing notions for which the axiom applies. It is Martin Maximum(MM) introduced by Foreman,Magidor and Shelah and shown to beconsistent relative to the existence of supercompact cardinals.([11])Axiom 6.6 (MM). If P is a forcing notion that does not kill thestationarity of subsets of ω 1 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 )).In [11] it was shown that now the this forcing axiom, besides decidingmany other independent problems, settles the size of the Continuum,namelyMM implies that the 2 ℵ 0= ℵ 2 . In particular in thestatement of the theorem the only κ for which the statement is interestingis κ = ℵ 2 . This result was improved by Todorcevic and Velickovic(see [41]) showing that the same conclusion follows from PFA. In factadditional results of Moore (in [28]) get the same result from weakeningof PFA. In some sense even the tiniest strengthening of MA fixes thecontinuum at ℵ 2 . The fascinating fact is that ℵ 2 keeps appearing as avery special cardinal.Thus I would have loved to suggest MM as natural axiom, decidinga large class of problems , including the size of the Continuum, thathas intuitive appeal and therefore should be considered to be a naturalcandidate for adaption. But MM has a competitor: Woodin’s axiom(*) ([44],see also [21]) has the same intuitive motivation: Namely theuniverse of sets is rich. (At least H ω2 is rich.) Formally (*) is equivalentto the statement that every π 2 statement that can be forced to holdfor H ω2 is already true in H ω2 of the ground model. The remarkablefact ([44]) is that (*) implies many of the consequences of MM forH ω2 .In particular it implies 2 ℵ 0= ℵ 2 . There is a strong evidence that(*) is the right axiom to assume for the structure 〈L[P (ω 1 )], ɛ, NS ω1 〉where NS ω1 is the non stationary ideal on ω 1 in the same sense that theAxiom of Determinacy (AD) seems the right axiom for the structure〈L[P (ω)], ɛ〉. So it also seems a natural axiom to be adapted. 7So we have two competing axioms to , motivated by the same intuition,supported by similar slogans. Are they compatible? can weadapt both of them? Till recently it seems that the combination isproblematic. Larson in [22] showed , assuming that consistency of supercompactlimit of supercompacts , that one can get model in whichMM holds but (*) fails. In fact what one has in this model that in7 (*) does not imply the results of MM for the larger segment of the universeof Set Theory because it is essentially an axiom about L[P (ω 1 ]) . It is equivalentto the statement that L[P (ω 1 ]) is a forcing extension of L[R] by a particular niceforcing notion P max .
SOME SET THEORIES ARE MORE EQUAL 19the structure 〈H ω2 , ɛ, NS ω1 〉 there is a definable well ordering while (*)implies that such a well ordering does not exists. On the other handthe only known way to get a model of (*) was to force over L[R] andsince MM is a global axiom one can not get a model of MM by forcingover a ”small” model like L[R].But there is a conjecture that ,if true will, change the situation dramatically.It involves a natural extension of MM which is denoted byMM ++ .Each forcing axiom has its ++ version. Remember the originalmotivation of the forcing axioms:If one can force the existence of a set satisfying a givenproperty and there is no clear obstruction to its existencethen such a set exists.then the ++ version seems like even a better formulation of the intuitiveconcept than the formulation we adapted. MM ++ is the followingaxiom:Axiom 6.7. Let P be a forcing notion which preserves stationary subsetsof ω 1 . Let α be an ordinal such that P ∈ V α . Then the set ofM ∈ P ω2 (V α ) such that there is a M generic filter G M ⊆ P ∩ M suchthat if S ⊆ ω 1 , S ∈ M[G M ] then M[G M ] S ∈ NS ω1 ⇔ V S ∈ NS ω1is stationary in P ω2 (V α ).Namely the ++ version claims that we not just that we can findenough elementary substructures of V α which has a generic object in Vbut also that we can assume that this generic object is correct aboutsubsets of ω 1 being stationary. So the object we get in the groundmodel is even a better approximation to the object we get by forcing.This is clearly very much in the spirit which let us study the forcingaxioms in the first place.So the conjecture is:Conjecture 6.8. MM ++ implies Woodin’s (*) axiom.If this conjecture is true then it will be strong evidence for adaptingMM ++ . I think that a proof of this conjecture will be a confirmationfor both MM ++ (hence for MM) and for (*) in the same sense that thefact two separate scientific theories with desirable consequences can bemerged into one unified theory can be considered to be confirmationfor both of them.MM or even better MM ++ is a global axiom that has many consequencesthroughout the universe . (See for instance [5] for the impactof MM on variations of the combinatorial principle □ κ for many cardinalsκ. ) But MM impact is mostly on sets of size ≤ ℵ 1 so a naturalresearch program is to try and find reasonable forcing axioms that applyto sets of size ℵ 2 and more. There are some initial steps in this
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