The Continuum Hypothesis - Logic at Harvard

2.1 Three VersionsThere are three different formulations of the continuum hypothesis—theinterpolant version, the well-ordering version, and the surjection version.These versions are all equivalent to one another in ZFC but we shall beimposing a definability constraint and in this case there can be interestingdifferences. 5 There is really a hierarchy of notions of definability—rangingup through the Borel hierarchy, the projective hierarchy, the hierarchy inL(R), and, more generally, the hierarchy of universally Baire sets—and soeach of these three general versions is really a hierarchy of versions, eachcorresponding to a given level of the hierarchy of definability. 62.1.1 Interpolant VersionThe first formulation of CH is that there is no interpolant, that is, there is noinfiniteset Aofrealnumbers suchthatthecardinalityofAisstrictlybetweenthat of the natural numbers and the real numbers. To obtain definableversions one simply asserts that there is no “definable” interpolant and thisleads to a hierarchy of definable interpolant versions, depending on whichnotion of definability one employs. More precisely, for a given pointclassΓ in the hierarchy of definable sets of reals, the corresponding definableinterpolant version of CH asserts that there is no interpolant in Γ.The Cantor-Bendixson theorem shows that there is no interpolant in Γ inthe case where Γ is the pointclass of closed sets, thus verifying this version ofCH. This was improved by Suslin who showed that this version of CH holds1for Γ where Γ is the class of Σ ∼1 sets. One cannot go much further withinZFC—to prove stronger versions one must bring in stronger assumptions. Itturns out that axioms of definable determinacy and large cardinal axioms1achieve this. For example, results of Kechris and Martin show that if ∆ ∼n -1determinacy holds then this version of CH holds for the pointclass of Σ ∼ n+1sets. Going further, if one assumes AD L(R) then this version of CH holds forall sets of real numbers appearing in L(R). Since these hypotheses followfrom large cardinal axioms one also has that stronger and stronger largecardinal assumptions secure stronger and stronger versions of this version ofthe effective continuum hypothesis. Indeed large cardinal axioms imply that5 Our discussion follows Martin (1976).6 For a discussion of the hierarchy of definability see §2.2.1 and §4.6 of the entry “LargeCardinals and Determinacy”.8