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

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22 MENACHEM MAGIDORIn the presence of large cardinals the picture is much simpler asfar as the first two versions: If there is a Woodin cardinal then everyUniversally Baire set of reals is either countable or contains a perfectsubset. The situation is even more dramatic with respect to the wellordering version: Under the existence of Woodin cardinal then everyUniversally Baire well ordering of a set of reals in countable. So ifwe assume the existence of a Woodin cardinal then the well orderingversion has no bearing on the Continuum problem. Also as Martin andKoellner argue in [27] and [21] the interpolation version does not shedlight on the continuum problem.The third version:the surjection version is more interesting. Herethere is a chance even assuming the existence of large cardinals that if2 ℵ 0= κ then we can have a Universally Baire evidence for that. Wefeel that there is some intuitive appeal to following principle:If 2 ℵ 0≥ κ then there is a Universally Baire Surjectionof R onto κ.This intuition motivated what Koellner in [21] called the Foreman-Magidor program. This was an attempt in [12] to prove that if thereis a Universally Baire surjection of R onto an ordinal α then α < ω 2 .If this program was successful then we believe that it can be construedas evidence for CH. The program as stated was shot down by WoodinTheorem 6.14 (Woodin [44]). If the NS ω1 is ω 2 saturated and there isa measurable cardinal then there is a ∆ 1 3 surjection of R onto ω 2 . Sincethis surjection is in L[R] then if we assume the existence of a properclass of Woodin cardinals then it is a Universally Baire surjection of Ronto ω 2 .If we combine this theorem with a result of Schimmerling ([35]) aboutPFA and AD and the fact from [11] that MM implies that NS ω1 isω 2 saturated. we getTheorem 6.15. MM implies that there is a Universally Baire surjectionof R onto ω 2So our favored axiom MM implies that really there is a ”nice” evidencefor the size of the continuum is ℵ 2 . (in the surjection sense.)What is being called The Foreman-Magidor program is not completelydead because we do not know any model of ZFC in whichthere is a Universally Baire surjecton of R onto ω 3 . 99 We are talking here about model with choice. In a universe that satisfies ADthe first cardinal such there is no surjection of R onto it is rather large and definitelylarger than ω 3 .This cardinal is usually denoted by Θ.
SOME SET THEORIES ARE MORE EQUAL 23So one can still conjecture which makes sense whether one assumeslarge cardinals or not, even though of proof of it even under the assumptionof large cardinals will be very interesting:Conjecture 6.16. There is no Universally Baire surjection of R ontoω 3 .If this conjecture is true then it could be considered as evidenceassuming that 2 ℵ 0≤ ℵ 2 is a very natural assumption .7. codaI hope that the previous sections give some good arguments why itis a meaningful question to ask whether one strengthening of ZFC isbetter than another and it is relevant question even if we consider thatthe main goal of Set Theory is extrinsic, namely to give a foundationalsupport for Mathematics and through it to all of Science. We describedsome process by which we evaluate different set theories and decide onour favorite axioms.Admittedly this decision can change over time whenmore mathematical facts become available. This is a process that is notdissimilar to the process by which any branch of science decides on thecurrent theory, where at any given time there may be several candidatesbut we are never in a situation in which we allow wild pluralism. Theprocess, which is an on going process is based both on the consequencesof the theory, or on its coherence with some intuitive feeling of whatthe right theory should be. This intuition is also an ongoing processand the intuition is refined by additional evidence.I believe that thisdescription of the process is close to what Maddy ([25] was refereing toas a ”proper set theoretic method”.As far as the most important open problem: CH, we believe thatthe process we described above leads in directions that will eventuallywill refine our theory to the extent that we shall have a definite answerfor the value of the Continuum as well as answers to many other independentproblems. Interesting fact is that the three directions chartedin the previous section leads us to only two possible values for thecontinuum: either ℵ 1 or ℵ 2 . We of course have to remember that theapproaches in the previous section are based on unproved conjectures.Another interesting fact is the prominence of the cardinal ℵ 2 whichkeeps appearing over and over again in many seemingly different contexts.Itis a historical curiosity that Gödel in his last years believedthat the right value of the continuum is ℵ 2 and tried to find argumentssupporting it though his attempted proof of 2 ℵ 0= ℵ 2 from the axiomsthat he considered natural was wrong. See the introduction by Solovayto the Gödel unpublished paper in [16], vol 3 pages 405-425.
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