Generic Absoluteness and the Continuum

Stevo Todorcevic
2002 Mathematical Research Letters  
Let H ω2 denote the collection of all sets whose transitive closure has size at most ℵ 1 . Thus, (H ω2 , ∈) is a natural model of ZFC minus the power-set axiom which correctly estimates many of the problems left open by the smaller and better understood structure (H ω1 , ∈) of hereditarily countable sets. One of such problems is, for example, the Continuum Hypothesis. It is largely for this reason that the structure (H ω2 , ∈) has recently received a considerable amount of study (see e.g. [15]
more » ... udy (see e.g. [15] and [16]). Recall the well-known Levy-Schoenfield absoluteness theorem ([10, §2]) which states that for every Σ 0 −sentence ϕ(x, a) with one free variable x and parameter a from H ω2 , if there is an x such that ϕ(x, a) holds then there is such an x in H ω2 , or in other words, Strictly speaking, what is usually called the Levy-Schoenfield absoluteness theorem is a bit stronger result than this, but this is the form of their absoluteness theorem that allows a variation of interest to us here. The generic absoluteness considered in this paper is a natural strengthening of (1) where the universe V is replaced by one of its boolean-valued extensions V B , i.e. the statement of the form for a suitably chosen boolean-valued extension V B . This sort of generic absoluteness has apparently been first considered by J. Stavi (see [11] ) and then by J. Bagaria [4] who has also observed that any of the 'Bounded Forcing Axioms' introduced by M.Goldstern and S.Shelah [9] is equivalent to the corresponding generic absoluteness statement for the structure (H ω2 , ∈) . The most prominent such statement (besides of course Martin's axiom; see [6] ) is the Bounded Martin's Maximum which asserts (2) for any boolean-valued extension V B which preserves stationary subsets of ω 1 (see [1]-[4], [6]-[9], [11]-[12], [14]-[16] ). The purpose of this note is to answer the natural question (appearing explicitely or implicitly in some of the listed papers) which asks whether any of the standard forms of the generic absoluteness (2) discussed above decides the size of the continuum. Theorem 1. Assume generic absoluteness (2) for boolean-valued extensions which preserve stationary subsets of ω 1 . Then there is a well ordering of the continuum of length ω 2 which is definable in the structure (H ω2 , ∈).
doi:10.4310/mrl.2002.v9.n4.a6 fatcat:zbcp5hkutjgz7nvqq2pkod3ibe