The Continuum Hypothesis - Logic at Harvard

So, of the various theories T A involved in Theorem 5.1, there is one thatstands out: The theory T (∗) given by (∗). This theory maximizes the Π 2 -theory of the structure 〈H(ω 2 ),∈,I NS ,A | A ∈ P(R) ∩ L(R)〉. The fundamentalresult is that in this maximal theory2 ℵ 0= ℵ 2 .5.2 The Parallel Case for CHTheparallelcaseforCHalsohastwosteps, thefirstinvolving Ω-completenessand the second involving maximality.The first result in the first step is the following:Theorem 5.4 (Woodin, 1985). Assume ZFC and that there is a proper classof measurable Woodin cardinals. Then ZFC+CH is Ω-complete for Σ 2 1 .Moreover, up to Ω-equivalence, CH is the unique Σ 2 1-statement that is Ω-complete for Σ 2 1 ; that is, letting T A be the Ω-complete theory given by ZFC+A where A is Σ 2 1 , all such T A are Ω-equivalent to T CH and hence (trivially)all such T A contain CH. In other words, there is a “good” theory and all“good” theories imply CH.To complete the first step we have to determine whether this result isrobust. For it could be the case that when one considers the next level,Σ 2 2 (or further levels, like third-order arithmetic) CH is no longer part ofthe picture, that is, perhaps large cardinals imply that there is an axiom Asuch that ZFC+A is Ω-complete for Σ 2 2 (or, going further, all of third orderarithmetic) and yet not all such A have an associated T A which contains CH.We must rule this out if we are to secure the first step.The most optimistic scenario along these lines is this: The scenario isthat there is a large cardinal axiom L and axioms A ⃗ such that ZFC+L+⃗A is Ω-complete for all of third-order arithmetic and all such theories areΩ-equivalent and imply CH. Going further, perhaps for each specifiablefragment V λ of the universe of sets there is a large cardinal axiom L andaxioms A ⃗ such that ZFC+L+ A ⃗ is Ω-complete for the entire theory of V λand, moreover, that such theories are Ω-equivalent and imply CH. Werethis to be the case it would mean that for each such λ there is a unique Ω-completepictureofV λ andwewouldhaveauniqueΩ-completeunderstandingof arbitrarily large fragments of the universe of sets. This would make for a29