SOME SET THEORIES ARE MORE EQUAL ... - Logic at Harvard
SOME SET THEORIES ARE MORE EQUAL ... - Logic at Harvard
SOME SET THEORIES ARE MORE EQUAL ... - Logic at Harvard
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14 MENACHEM MAGIDORconstructed thus far, where <strong>at</strong> each stage you throw in some r<strong>at</strong>hercanonical 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 th<strong>at</strong> 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 r<strong>at</strong>her mild axioms of stronginfinity.The altern<strong>at</strong>ive is to adapt some inner models for large cardinals,where the slogan for justifying it is th<strong>at</strong> if while we may want largerand larger cardinals to exist in our universe, we want only sets th<strong>at</strong> arenecessary in some sense. (Given all the ordinals.) .The most appealinginner model is the ”Ultim<strong>at</strong>e L” suggested by Woodin. ([45]) which issupposed to be a the canonical inner model for supercompact cardinal,but it has the pleasant fe<strong>at</strong>ure th<strong>at</strong> it c<strong>at</strong>ches stronger cardinals , ifthey happen to exist. The construction of the ultim<strong>at</strong>e L is still a workin progress and if successful then the axiom ”V=Ultim<strong>at</strong>e L” maybe avery strong competitor for the set theory th<strong>at</strong> decides CH (It decidesCH in the positive direction) but I have my doubts for several reasons.It is very likely th<strong>at</strong> the Ultim<strong>at</strong>e L, like the old L, will s<strong>at</strong>isfy manyof the combin<strong>at</strong>orial principles like ♦ ω1 . These principles are usuallythe reason th<strong>at</strong> ”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 th<strong>at</strong> the universe of sets shouldbe as rich as you can reasonably expect. th<strong>at</strong> this principle obviousconflicts with the axiom ”V=Ultim<strong>at</strong>e L”.I am going to take a gre<strong>at</strong>errisk by st<strong>at</strong>ing a conjecture th<strong>at</strong>, if true, will viol<strong>at</strong>e the possibility ofbuilding the ultim<strong>at</strong>e L, <strong>at</strong> 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 fe<strong>at</strong>ures the work done in the l<strong>at</strong>e 80’s onthe connection of AD with large cardinals is th<strong>at</strong> 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: