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

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14 MENACHEM MAGIDORconstructed thus far, where at each stage you throw in some rathercanonical objects like the minimal (partial) extender missing on thesequence constructed thus far seems to fit the intuitive picture of beingcanonical.The obvious example for such a L like universe is L it self. It has theadvantage of deciding most of the interesting independent problems.The disadvantage of L is that the direction it decides the independentproblems is usually in the less elegant and coherent direction. In somesense ”L is the paradise of counter examples”. In view of our decisionof adapting any of the reasonable large cardinals axioms , L should bedropped as an option because it rules out rather mild axioms of stronginfinity.The alternative is to adapt some inner models for large cardinals,where the slogan for justifying it is that if while we may want largerand larger cardinals to exist in our universe, we want only sets that arenecessary in some sense. (Given all the ordinals.) .The most appealinginner model is the ”Ultimate L” suggested by Woodin. ([45]) which issupposed to be a the canonical inner model for supercompact cardinal,but it has the pleasant feature that it catches stronger cardinals , ifthey happen to exist. The construction of the ultimate L is still a workin progress and if successful then the axiom ”V=Ultimate L” maybe avery strong competitor for the set theory that decides CH (It decidesCH in the positive direction) but I have my doubts for several reasons.It is very likely that the Ultimate L, like the old L, will satisfy manyof the combinatorial principles like ♦ ω1 . These principles are usuallythe reason that ”L is the paradise of counter examples”. They allowone to construct counter examples to many elegant conjectures . (TheSouslin Hypothesis is a famous case). In the next subsection we shalltry to suggest as intuitive principle that the universe of sets shouldbe as rich as you can reasonably expect. that this principle obviousconflicts with the axiom ”V=Ultimate L”.I am going to take a greaterrisk by stating a conjecture that, if true, will violate the possibility ofbuilding the ultimate L, at least along the lines considered now.The conjecture has to do with the absoluteness of different set theoreticaltheories under forcing extensions. As we mentioned above,one of the most remarkable features the work done in the late 80’s onthe connection of AD with large cardinals is that in the presence of aproper class of Woodin cardinals , the theory of L[R] is invariant underforcing extension. In particular there is no well ordering of the realswhich is definable in L[R]. A stronger result is the following:
SOME SET THEORIES ARE MORE EQUAL 15Theorem 6.2 (Woodin). Assume that there is a proper class of measurableWoodin cardinals and that CH holds. Let Φ be a Σ 2 1 sentence.(Namely Φ has the form ”There is a set of reals A such that Ψ(A)holds where all the quantifies in Ψ are over reals.) Then Φ is true inthe ground model.In particular under our assumptions there is no Σ 2 1well ordering of the reals.So the theorem claims that if CH is true (and we have the appropriatelarge cardinals) then the Σ 2 1 sentences which are are true is maximalamong all forcing extensions of our universe.Results of Abraham andShelah ([2],[3]) show that this result is the best possible in the sensethat for any model of set theory there is a forcing extension not addingany reals (hence preserving CH) and introducing a Σ 2 2 well ordering ofthe reals. Also CH is necessary because over any model of set theoryone can introduce by forcing a Σ 1 1 well ordering of the reals. Naturallyone may ask about possible generalizations of the last theorem to Σ 2 2sentences. In view of the Abraham-Shelah results one should replaceCH by a stronger statement. We make the following conjecture:Conjecture 6.3. (Under the assumption of the appropriate large cardinals)Assume that the combinatorial ♦ ω1 holds and let Φ be a Σ 2 2sentence, then if Φ can be made true in a forcing extension that satisfyCH then Φ is true in ground model.In particular in the presence ofappropriate large cardinals and ♦ ω1 there is no Σ 2 2 well ordering of thereals.Namely ♦ ω1 implies in the presence of strong enough large cardinalsthat the set of true Σ 2 2 sentences is maximal for forcing extensions satisfyingCH. A partial results giving some support for this conjectureswere obtained in [6]. The problem is that the present attempts forconstructing the Ultimate L , if successful, will very likely satisfy ♦ ω1and will have a Σ 2 2 well ordering of the reals.So the last conjecture , iftrue, will kill the possibility of constructing the Ultimate L along thesuggested lines.6.2. Forcing Axioms. Forcing axioms like Martin’s Axiom (MA),the Proper Forcing Axiom (PFA), Martin’s Maximum (MM) andother variations were very successful in settling many independentproblems. The intuitive motivation for all of them is that the universeof sets is as rich as possible, or at the slogan levelA set that its existence is possible and there is no clearobstruction to its existence does existsThis ”slogan”is obviously very vague and each of the terms used isproblematic , but the spirit of this talk is that such vague principles
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