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

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12 MENACHEM MAGIDOR. . . We can conceive of mathematical situations wherenatural concept of probability emerges which is not capturedby the usual axioms of probability theory.What Ihave in mind is not a radical extension of probability. . . but rather a conservative extension. . .We can envision a development in physical theories which will encompassa larger class of functions than accepted today as legitimate functions.So it will not be a complete absurdity to ask whether a functionsatisfying the Pitowsky condition exists. So Physics could be potentiallypartial to the set theoretic context in which its mathematicalmodeling is takeing place.While being a wild speculation it is not impossible that ScientificTheories will prefer one set theory over others because it makes thescientific theory simpler and more elegant. It may even be possiblethat in order to derive certain experimentally testable results one wouldhave to prefer one set theory over others. I am not claiming that it islikely to have an experimental test that will decide between differentSet Theories but that we will be able to compare between differentSet Theories according to what type of mathematical hinterland theyprovide for theoretical Physics. I believe that it is possible.6. Some paths to followIn previous section we described several criteria for evaluating potentialextensions of ZFC , hopefully leading to the preferable set theorythat will decide many of the outstanding independent problems. Awell known success story that meets our criteria is the series of strongaxioms of infinity.They definitely fit an appealing mental image of theuniverse of set theory going into larger and larger levels. Many ofthese axioms can be justified as reflection principles. (We mean reflectionprinciples in somewhat different sense than Koellner in [20] , sohis negative result there does not apply) .Definition 6.1. Given a property of structures in a fixed signaturesuch that the property is invariant under isomorphism of structures. Areflection cardinal for the given property is a cardinal κ such that everystructure in the given signature has a substructure of cardinality lessthan κ having the property.For instance if the property is first order then ℵ 1 is a reflection cardinalfor the property.(This is of course the Löwenheim-Skolem theorem.)The existence of a supercompact cardinal is equivalent to the existenceof a reflection cardinal for every property which is expressible in second
SOME SET THEORIES ARE MORE EQUAL 13order logic. The ultimate reflection principle is the assumption thatevery property of structures has a reflection cardinal and it is equivalentto well known Vopenka’s principles which implies the existence ofunboundedly many extendible cardinals. 5 My favorite slogan for thisprinciple is that the universe of sets is so large that every proper classof structures of the same signature must contain repetitions in somesense.The large cardinals axioms are success story because they decided asubstantial class of undecided statements, the most noteworthy classis the theory of L[R] which contains the theory of projective sets andit provides a very elegant and coherent theory . In the presence of thelarge cardinals this theory is resilient under forcing extensions. As faras verifiable consequences I consider the fact that these axioms providesnew Π 0 1 sentences which so far were not refuted. In some sense we canconsider these Π 0 1 sentences as physical facts about the world that sofar are confirmed by experience .Unfortunately they do not decide somecentral independent problems like the Continuum Hypothesis. In thissection we shall study three potential paths of extending ZFC andshedding light on the Continuum Problem. In view of the success ofstrong axioms of infinity we take for granted that any extensions ofZFC that is considered, is compatible with the standard axioms ofstrong infinity.We shall consider three directions in which ZFC can be extendedand which shed light on the possible values of the continuum. Thesedirections involve unproved conjectures so the decision about their efficacywill have to wait further development. This is not unusual inany scientific discipline that we have some promising theories, someof them competing and the decision about them is delayed till furtherevidence is available.6.1. The Ultimate L. The mental image for any L like model (”canonicalinner model”) is that the the universe of sets is constructed by asequence of stages where each stage is obtained from the previous stageby an operation which is definable in a very canonical way. ”canonical” here is rather vague but for instance if each stage is the powerset of the previous stage , like in the definition of the V α ’s , then itis not ”canonical” enough . We can too easily change the meaningof this operation , for instance by forcing. On the other hand takingthe next stage as the definable subsets of the previous stage , like inthe definition of the the L α ’s is very canonical.The definition that isbeing used in the definition of the inner models for the large cardinals5 This is essentially an unpublished result of Stavi. See also [40]
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