Martin’s Maximum and Definability in H(א2)
نویسنده
چکیده
In [6], we modified a coding device from [14] and the consistency proof of Martin’s Maximum from [3] to show that from a supercompact limit of supercompact cardinals one could force Martin’s Maximum to hold while the Pmax axiom (∗) fails. Here we modify that argument to prove a stronger fact, that Martin’s Maximum is consistent with the existence of a wellordering of the reals definable in H(א2) without parameters, from the same large cardinal hypothesis. In doing so we give a much simpler proof of the original result.
منابع مشابه
Bounded forcing axioms and the continuum
We show that bounded forcing axioms (for instance, the Bounded Proper Forcing Axiom and the Bounded Semiproper Forcing Axiom) are consistent with the existence of (!2; !2)-gaps and thus do not imply the Open Coloring Axiom. They are also consistent with Jensen’s combinatorial principles for L at the level !2, and therefore with the existence of an !2-Suslin tree. We also show that the axiom we ...
متن کاملAlmost-disjoint Coding and Strongly Saturated Ideals
We show that Martin’s Axiom plus c = א2 implies that there is no (א2,א2,א0)-saturated σ-ideal on ω1. Given cardinals λ, κ and γ, a σ-ideal I on a set X is said to be (λ, κ, γ)-saturated if for every set {Aα : α < λ} ⊂ P(X) \ I there exists a set Y ∈ [λ] such that for all Z ∈ [Y ] , ⋂ {Aα : α ∈ Z} ∈ I. Laver [6] was the first to show the consistency of an (א2,א2,א0)-saturated ideal on ω1, using ...
متن کاملA theorem of Todorcevic on universal Baireness
Given a cardinal κ, a set A ⊂ ω is κ-universally Baire if there exist trees S, T which project to A and its complement respectively, and which continue to project to complements after the forcing Coll(ω, κ). Todorcevic [3] has shown that under Martin’s Maximum every set of reals of cardinality א1 is א1-universally Baire but not א2-universally Baire. Here we present alternate proofs of these two...
متن کاملCardinal characteristics, projective wellorders and large continuum
We extend the work of [7] by presenting a method for controlling cardinal characteristics in the presence of a projective wellorder and 2א0 > א2. This also answers a question of Harrington [11] by showing that the existence of a ∆3 wellorder of the reals is consistent with Martin’s axiom and 2א0 = א3.
متن کاملModels of Set Theory
1. First order logic and the axioms of set theory 2 1.1. Syntax 2 1.2. Semantics 2 1.3. Completeness, compactness and consistency 3 1.4. Foundations of mathematics and the incompleteness theorems 3 1.5. The axioms 4 2. Review of basic set theory 5 2.1. Classes 5 2.2. Well-founded relations and recursion 5 2.3. Ordinals, cardinals and arithmetic 6 3. The consistency of the Axiom of Foundation 8 ...
متن کامل