Recently the second author introduced combinatorial principles that characterize supercompactness for inaccessible cardinals but can also hold true for small cardinals. We prove that the proper forcing axiom PFA implies these principles hold for ω 2 . Using this, we argue to show that any of the known methods for forcing models of PFA from a large cardinal assumption requires a strongly compact cardinal. If one forces PFA using a proper forcing, then we get the optimal result that a supercompact cardinal is necessary. * Parts of the results of this paper are from the second author's doctoral dissertation [22] written under the supervision of Dieter Donder, to whom the second author wishes to express his gratitude.
We analyze the notion of guessing model, a way to assign combinatorial properties to arbitrary regular cardinals. Guessing models can be used, in combination with inaccessibility, to characterize various large cardinals axioms, ranging from supercompactness to rank-to-rank embeddings. The majority of these large cardinals properties can be defined in terms of suitable elementary embeddings j : Vγ → V λ . One key observation is that such embeddings are uniquely determined by the image structures j[Vγ ] ≺ V λ . These structures will be the prototypes guessing models. We shall show, using guessing models M , how to prove for the ordinal κM = jM (crit(jM )) (where πM is the transitive collapse of M and jM is its inverse) many of the combinatorial properties that we can prove for the cardinal j(crit(j)) using the structure j[Vγ ] ≺ V j(γ) . κM will always be a regular cardinal, but consistently can be a successor and guessing models M with κM = ℵ2 exist assuming the proper forcing axiom. By means of these models we shall introduce a new structural property of models of PFA: the existence of a "Laver function" f : ℵ2 → H ℵ 2 sharing the same features of the usual Laver functions f : κ → Hκ provided by a supercompact cardinal κ. Further applications of our analysis will be proofs of the singular cardinal hypothesis and of the failure of the square principle assuming the existence of guessing models. In particular the failure of square shows that the existence of guessing models is a very strong assumption in terms of large cardinal strength.
Abstract. We analyze certain subfamilies of the category of complete boolean algebras with complete homomorphisms, families which are of particular interest in set theory. In particular we study the category whose objects are stationary set preserving, atomless complete boolean algebras and whose arrows are complete homomorphisms with a stationary set preserving quotient. We introduce a maximal forcing axiom MM +++ as a combinatorial property of this category. This forcing axiom strengthens Martin's maximum and can be seen at the same time as a strenghtening of Baire's category theorem and of the axiom of choice. Our main results show that MM +++ is consistent relative to large cardinal axioms and that MM +++ makes the theory of the Chang model L( [Ord] ≤ℵ 1 ) with parameters in P(ω 1 ) generically invariant for stationary set preserving forcings that preserve this axiom. We also show that our results give a close to optimal extension to the Chang model L( [Ord] The main objective of this paper is to show that there is a natural recursive extension T of ZFC+ large cardinals which gives a complete theory of the Chang model L( [Ord] ≤ℵ 1 ) with respect to the unique efficient method to produce consistency results for this structure, i.e stationary set preserving forcings. In particular we will show that a closed formula φ relativized to this Chang model is first order derivable in T if and only if it is provable in T that T + φ is forceable by a stationary set preserving forcing. In our eyes the results of this paper give a solid a posteriori explanation of the success forcing axioms have met in providing at least one consistent solution to many ZFC-provably undecidable problems. The paper can be divided in six sections: . We shall also try to argue that the results of this paper give an a posteriori explanation of the success that forcing axioms have met in solving a variety of problems showing up in set theory as well as in many other fields of pure mathematics. • Section 2 presents some background material on stationary sets (subsection 2.1), large cardinals (subsection 2.2), posets and boolean completions (subsection 2.3), stationary set preserving forcings (subsection 2.4), forcing axioms (subsection 2.5), iterated forcing (subsection 2.6) which will be needed in the remainder of the paper.• Section 3 introduces the notion of category forcings. We shall look at subcategories of the category of complete boolean algebra with complete homomorphisms. Given a category (Γ, Θ) (where Γ is the class of objects and Θ the class of arrows) we associate to it the partial order (U Γ,Θ , ≤ Θ ) whose elements are the objects in Γ ordered by B ≥ Θ C iff there is an i : B → C in Θ. We shall also feel free to confuse a set sized partial order with its uniquely defined boolean completion. In this paper we shall focus on the analysis of the category (SSP, SSP) whose objects are the stationary set preserving (SSP) complete boolean algebras and whose arrows (still denoted by SSP) are the complete homomorphisms with a station...
The resurrection axioms are forcing axioms introduced recently by Hamkins and Johnstone, developing on ideas of Chalons and Veličković. We introduce a stronger form of resurrection axioms (the iterated resurrection axioms RAα(Γ) for a class of forcings Γ and a given ordinal α), and show that RAω(Γ) implies generic absoluteness for the first-order theory of H γ + with respect to forcings in Γ preserving the axiom, where γ = γΓ is a cardinal which depends on Γ (γΓ = ω1 if Γ is any among the classes of countably closed, proper, semiproper, stationary set preserving forcings).We also prove that the consistency strength of these axioms is below that of a Mahlo cardinal for most forcing classes, and below that of a stationary limit of supercompact cardinals for the class of stationary set preserving posets. Moreover we outline that simultaneous generic absoluteness for H γ + 0 with respect to Γ0 and for H γ + 1 with respect to Γ1 with γ0 = γΓ 0 = γΓ 1 = γ1 is in principle possible, and we present several natural models of the Morse Kelley set theory where this phenomenon occurs (even for all Hγ simultaneously). Finally, we compare the iterated resurrection axioms (and the generic absoluteness results we can draw from them) with a variety of other forcing axioms, and also with the generic absoluteness results by Woodin and the second author.
We show that the Proper Forcing Axiom implies the Singular Cardinal Hypothesis. The proof uses the reflection principle MRP introduced by Moore in [10].
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