Adding Closed Unbounded Subsets of ω₂ with Finite Forcing

Finite set theory, here denoted ZFfin, is the theory obtained by replacing the axiom of infinity by its negation in the usual axiomatization of ZF (Zermelo-Fraenkel set theory). An ω-model of ZFfin is a model in which every set has at most finitely many elements (as viewed externally). Mancini and Zambella (2001) employed the Bernays-Rieger method of permutations to construct a recursive ω-model of ZFfin that is nonstandard (i.e., not isomorphic to the hereditarily finite sets Vω). In this paper we initiate the metamathematical investigation of ω-models of ZFfin. In particular, we present a new method for building ω-models of ZFfin that leads to a perspicuous construction of recursive nonstandard ω-model of ZFfin without the use of permutations. Furthermore, we show that every recursive model of ZFfin is an ω-model.

Handbook of the History of Logic, 2012

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 > 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) > 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…

Transactions of the American Mathematical Society, 2016

We describe a framework for proving consistency results about singular cardinals of arbitrary cofinality and their successors. This framework allows the construction of models in which the Singular Cardinals Hypothesis fails at a singular cardinal κ \kappa of uncountable cofinality, while κ + \kappa ^+ enjoys various combinatorial properties. As a sample application, we prove the consistency (relative to that of ZFC plus a supercompact cardinal) of there being a strong limit singular cardinal κ \kappa of uncountable cofinality where SCH fails and such that there is a collection of size less than 2 κ + 2^{\kappa ^+} of graphs on κ + \kappa ^+ such that any graph on κ + \kappa ^+ embeds into one of the graphs in the collection.

Strengthening a result of Amir Leshem [7], we prove that the consistency strength of holding GCH together with definable tree property for all successors of regular cardinals is precisely equal to the consistency strength of existence of proper class many Π 1 1-reflecting cardinals. Moreover it is proved that if κ is a supercompact cardinal and λ > κ is measurable, then there is a generic extension of the universe in which κ is a strong limit singular cardinal of cofinality ω, λ = κ + , and the definable tree property holds at κ +. Additionally we can have 2 κ > κ + , so that SCH fails at κ.

Acta Mathematica, 2013

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 .

We show that the analogues of the Hamkins embedding theorems [Ham13], proved for the countable models of set theory, do not hold when extended to the uncountable realm of ω 1 -like models of set theory. Specifically, under the ♦ hypothesis and suitable consistency assumptions, we show that there is a family of 2 ω 1 many ω 1 -like models of ZFC, all with the same ordinals, that are pairwise incomparable under embeddability; there can be a transitive ω 1 -like model of ZFC that does not embed into its own constructible universe; and there can be an ω 1 -like model of PA whose structure of hereditarily finite sets is not universal for the ω 1 -like models of set theory.

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.

Lecture Notes Series, Institute for Mathematical Sciences, National University of Singapore, 2008

This paper is mainly a survey of recent results concerning the possibility of building forcing extensions in which there is a simple definition, over the structure H(ω 2), ∈ and without parameters, of a prescribed member of H(ω 2) or of a well-order of H(ω 2). Some of these results are in conjunction with strong forcing axioms like P F A ++ or M M , some are not. I also observe (Corollary 4.4) that the existence of certain objects of size ℵ 1 follows outright from the existence of large cardinals. This observation is motivated by an attempt to extend the P F A ++ result to a result mentioning M M ++. 1 Main starting questions and some pieces of notation The work presented here deals mostly with the problem of finding optimal definitions of well-orders of the reals and other objects. More precisely, it addresses the following two questions. 1 Question 1: Suppose A is a subset of ω 1. Suppose we are given the task of going over to a nice set-forcing extension 2 in which A admits a simple definition Φ(x), without parameters, over the structure H(ω 2), ∈ or over some natural (definable) extension of this structure, like H(ω 2), ∈, N S ω 1 .