Cohen Forcing and Inner Models

Archive for Mathematical Logic, 2009

There is a partial order P preserving stationary subsets of ! 1 and forcing that every partial order in the ground model V that collapses a su ciently large ordinal to ! 1 over V also collapses ! 1 over V P. The proof of this uses a coding of reals into ordinals by proper forcing discovered by Justin Moore and a symmetric extension of the universe in which the Axiom of Choice fails. Also, using one feature of the proof of the above result together with an argument involving the stationary tower it is shown that sometimes, after adding one Cohen real c, there are, for every real a in V [c], sets A and B such that c is Cohen generic over both L[A] and L[B] but a is constructible from A together with B.

Fundamenta Mathematicae, 2001

We define a new principle, SEP, which is true in all Cohen extensions of models of CH, and explore the relationship between SEP and other such principles. SEP is implied by each of CH * , the weak Freeze-Nation property of P(ω), and the (ℵ 1 , ℵ 0)-ideal property. SEP implies the principle C s 2 (ω 2), but does not follow from C s 2 (ω 2), or even C s (ω 2). 1. Introduction. There are many consequences of CH which are independent of ZFC, but are still true in Cohen models-that is, models of the form V [G], where V GCH and V [G] is a forcing extension of V obtained by adding some number (possibly 0) of Cohen reals; see [1, 2, 5, 7, 8]. Roughly, these consequences fall into two classes. One type are elementary submodel axioms, saying that for all suitably large regular λ, there are many elementary submodels N ≺ H(λ) such that |N | = ℵ 1 and N ∩ P(ω) "captures" in some way all of P(ω); these are trivial under CH, where we could take N ∩ P(ω) = P(ω). The other are homogeneity axioms, saying that given a sequence of reals, r α : α < ω 2 , there are ω 2 of them which "look alike"; again, this is trivial under CH. In this paper, we define a new axiom, SEP, of the elementary submodel type, and explore its connection with known axioms of both types. A large number of applications of such axioms may be found in [2, 4, 7, 8]. 2. Some principles true in Cohen models. We begin with a remark on elementary submodels. Under CH, one can easily find N ≺ H(λ) such that |N | = ω 1 and N is countably closed ; that is, [N ] ω ⊆ N. Without CH,

Annals of Mathematical Logic, 1980

The Journal of Symbolic Logic, 1994

The paper is a continuation of [The SCH revisited], In § 1 we define a forcing with countably many nice systems. It is used, for example, to construct a model “GCH below κ, c f κ = ℵ0, and 2κ &gt; κ+ω” from 0(κ) = κ+ω. In §2 we define a triangle iteration and use it to construct a model satisfying “{μ ≤ λ∣c f μ = ℵ0 and pp(μ) &gt; λ} is countable for some λ”. The question of whether this is possible was asked by S. Shelah. In §3 a forcing for blowing the power of a singular cardinal without collapsing cardinals or adding new bounded subsets is presented. Answering a question of H. Woodin, we show that it is consistent to have “c f κ = ℵ0. GCH below κ, 2κ…

Archive for Mathematical Logic, 2012

We construct a variety of inner models exhibiting large cardinal features usually obtained by forcing. For example, if there is a supercompact cardinal, then there is an inner model with a Laver indestructible supercompact cardinal. If there is a supercompact cardinal, then there is an inner model with a supercompact cardinal κ for which 2 κ = κ + , another for which 2 κ = κ ++ and another in which the least strongly compact cardinal is supercompact. If there is a strongly compact cardinal, then there is an inner model with a strongly compact cardinal, for which the measurable cardinals are bounded below it and another inner model W with a strongly compact cardinal, such that H V κ + ⊆ HOD W . Similar facts hold for supercompact, measurable and strongly Ramsey cardinals. If a cardinal is supercompact up to a weakly iterable cardinal, then there is an inner model of the Proper Forcing Axiom and another inner model with a supercompact cardinal in which GCH + V = HOD holds. Under the same hypothesis, there is an inner model with level by level equivalence between strong compactness and supercompactness, and indeed, another in which there is level by level inequivalence between strong compactness and supercompactness. If a cardinal is strongly compact up to a weakly iterable cardinal, then there is an inner model in which the least measurable cardinal is strongly compact. If there is a weakly iterable limit δ of <δ-supercompact cardinals, then there is an inner model with a proper class of Laver-indestructible supercompact cardinals. We describe three general proof methods, which can be used to prove many similar results.

Archive for Mathematical Logic, 2016

We introduce and study the first-order Generic Vopěnka's Principle, which states that for every definable proper class of structures C of the same type, there exist B = A in C such that B elementarily embeds into A in some set-forcing extension. We show that, for n ≥ 1, the Generic Vopěnka's Principle fragment for Πn-definable classes is equiconsistent with a proper class of n-remarkable cardinals. The n-remarkable cardinals hierarchy for n ∈ ω, which we introduce here, is a natural generic analogue for the C (n)-extendible cardinals that Bagaria used to calibrate the strength of the first-order Vopěnka's Principle in [1]. Expanding on the theme of studying set theoretic properties which assert the existence of elementary embeddings in some set-forcing extension, we introduce and study the weak Proper Forcing Axiom, wPFA, which states that for every transitive model M in the language of set theory with some ω 1-many additional relations, if it is forced by a proper forcing P that M satisfies some Σ 1-property, then V has a transitive modelM, which satisfies the same Σ 1property, and in some set-forcing extension there is an elementary embedding fromM into M. This is a weakening of a formulation of PFA due to Schindler and Claverie [2], which asserts that the embedding fromM to M exists in V. We show that wPFA is equiconsistent with a remarkable cardinal and that wPFA implies PFA ℵ 2 , the proper forcing axiom for antichains of size at most ω 2 , but it is consistent with κ for all κ ≥ ω 2 , and therefore does not imply PFA ℵ 3 .

Archive for Mathematical Logic, 2004

We study the preservation of the property of L(R) being a Solovay model under proper projective forcing extensions. We show that every Σ ∼ 1 3 strongly-proper forcing notion preserves this property. This yields that the consistency strength of the absoluteness of L(R) under Σ ∼ 1 3 strongly-proper forcing notions is that of the existence of an inaccessible cardinal. Further, the absoluteness of L(R) under projective strongly-proper forcing notions is consistent relative to the existence of a Σ ∼ ω -Mahlo cardinal. We also show that the consistency strength of the absoluteness of L(R) under forcing extensions with σ-linked forcing notions is exactly that of the existence of a Mahlo cardinal, in contrast with the general ccc case, which requires a weakly-compact cardinal.

The Journal of Symbolic Logic, 1995

We prove that a number of axioms, each a consequence of PFA (the Proper Forcing Axiom) are equivalent. In particular we show that TOP (the Thinning-out Principle as introduced by Baumgartner in the Handbook of set-theoretic topology), is equivalent to the following statement: If I is an ideal on co, with co, generators, then there exists an uncountable X C co,, such that either [X]w n I = 0 or [X]w C I. ?1. Introduction. In this paper we study relations between some consequences of the Proper Forcing Axiom (PFA). Among them we consider the Thinning-out Principle (TOP) introduced by Baumgartner in [B], and the partition calculus axiom co, ) (co, (co,; fin cw1))2 proposed by Todorcevic in [T]. We show that each of these two axioms can be restated in a simpler way, and then we easily deduce that Todorcevic's axiom (which we call Axiom S in this paper) is a consequence of TOP. We will then show how our versions of these axioms give simplified proofs of the applications of these axioms in [B] and [T]. We will show that the following axiom is equivalent to TOP: AXIOM 0. Let S = {S,: a < co } be a collection of (countable) subsets of coi, such that for every uncountable X C co,, there exists a countable set Q C X which cannot be covered by a finite subfamily of S. Then there exists an uncountable subset of co1 which meets each S, in a finite set. We will show that the following (weaker) version of Axiom 0 is equivalent to Axiom S: AXIOM 1. Let S = {S,: a < co, } be a collection of (countable) subsets of W1, such that for every uncountable X C co,, there exists a countable set Q C X which cannot be covered by a finite subfamily of S. Then there exists uncountable set X C co,, such that for each a C X, S, n X is a finite set.

Notre Dame Journal of Formal Logic - NDJFL, 2005

An outline is given of the proof that the consistency of a κ⁺-Mahlo cardinal implies that of the statement that I[ω₂] does not include any stationary subsets of Cof(ω₁). An additional discussion of the techniques of this proof includes their use to obtain a model with no ω₂-Aronszajn tree and to add an ω₂-Souslin tree with finite conditions.

arXiv (Cornell University), 2019

Methods of Higher Forcing Axioms was a small workshop in Norwich, taking place between 10-12 of September, 2019. The goal was to encourage future collaborations, and create more focused threads of research on the topic of higher forcing axioms. This is an improved version of the notes taken during the meeting by Asaf Karagila.