The collection of branches (maximal linearly ordered sets of nodes) of the tree $^{ (ordered by inclusion) forms an almost disjoint family (of sets of nodes). This family is not maximal--for example, any level of the tree is almost disjoint from all of the branches. How many sets must be added to the family of branches to make it maximal? This question leads to a series of definitions and results: a set of nodes is off-branch if it is almost disjoint (...) from every branch in the tree; an off-branch family is an almost disjoint family of off-branch sets; and o is the minimum cardinality of a maximal off-branch family. Results concerning o include: (in ZFC) a ≤ p, and (consistent with ZFC) o is not equal to any of the standard small cardinal invariants b,a,d, or c = 2 ω . Most of these consistency results use standard forcing notions--for example, $\mathfrak{b = a in the Cohen model. Many interesting open questions remain, though--for example, whether d ≤ o. (shrink)

Given a stationary subset T of $\omega_{1}$ , let $\tilde{T}$ be the set of ordinals in the interval $(\omega_{1}, \omega_{2})$ which are necessarily in the image of T by any embedding derived from the nonstationary ideal. We consider the question of the size of $\tilde{T}$ , givenT, and use Martin's Maximum and $\mathbb{P}_{max}$ to give some answers.

Working in the context of restricted forms of the Axiom of Choice, we consider the problem of splitting the ordinals below λ of cofinality θ into λ many stationary sets, where θ < λ are regular cardinals. This is a continuation of [4].

James Earl Baumgartner (March 23, 1943–December 28, 2011) came of age mathematically during the emergence of forcing as a fundamental technique of set theory, and his seminal research changed the way set theory is done. He made fundamental contributions to the development of forcing, to our understanding of uncountable orders, to the partition calculus, and to large cardinals and their ideals. He promulgated the use of logic such as absoluteness and elementary submodels to solve problems in set theory, he applied (...) his knowledge of set theory to a variety of areas in collaboration with other mathematicians, and he encouraged a community of mathematicians with engaging survey talks, enthusiastic discussions of open problems, and friendly mathematical conversations. (shrink)

We show that the members of a certain class of semi-proper iterations do not add countable sets of ordinals. As a result, starting from suitable large cardinals one can obtain a model in which the Continuum Hypothesis holds and every function from ω1 to ω1 is bounded on a club by a canonical function for an ordinal less than ω2.

We show that none of the following statements is provable in Zermelo-Fraenkel set theory (ZF) answering the corresponding open questions from Brunner in ``The axiom of choice in topology'':(i) For every T2 topological space (X, T) if X is well-ordered, then X has a well-ordered base,(ii) For every T2 topological space (X, T), if X is well-ordered, then there exists a function f : X × W T such that W is a well-ordered set and f ({x} × W) is (...) a neighborhood base at x for each x X,(iii) For every T2 topological space (X, T), if X has a well-ordered dense subset, then there exists a function f : X × W T such that W is a well-ordered set and {x} = f ({x} × W) for each x X. (shrink)

Let κ be a regular cardinal and P a partial ordering preserving the regularity of κ. If P is (κ-Baire and) of density κ, then there is a mad family on κ killed in all generic extensions (if and) only if below each p∈P there exists a κ-sized antichain. In this case a mad family on κ is killed (if and) only if there exists an injection from κ onto a dense subset of Ult(P) mapping the elements of onto nowhere (...) dense sets. If 2< κ =κ, then in each generic extension of V, in which κ is the minimal cardinal obtaining new subsets, some mad family on κ is killed or an independent subset of κ appears. Also, the κ-Suslin Hypothesis holds iff there exists a mad family on κ which is killed in each generic extension containing new subsets of κ and preserving P(λ) for λ<κ. (shrink)

We establish the hierarchy among twelve equivalence relations on the class of relational structures: the equality, the isomorphism, the equimorphism, the full relation, four similarities of structures induced by similarities of their self-embedding monoids and intersections of these equivalence relations. In particular, fixing a language L and a cardinal κ, we consider the interplay between the restrictions of these similarities to the class ModL of all L-structures of size κ. It turns out that, concerning the number of different similarities and (...) the shape of the corresponding Hasse diagram, the class of all structures naturally splits into three parts: finite structures, infinite structures of unary languages, and infinite structures of non-unary languages. (shrink)

We provide an exposition of supercompact Radin forcing and present several methods for iterating Radin forcing.

We strengthen a theorem of Gitik and Shelah [6] by showing that if κ is either weakly inaccessible or the successor of a singular cardinal and S is a stationary subset of κ such that $NS_{\kappa} \upharpoonright S$ is saturated then $\kappa \S$ is fat. Using this theorem we derive some results about the existence of fat stationary sets. We then strengthen some results due to Baumgartner and Taylor [2], showing in particular that if I is a $\lambda^{+++}-saturated$ normal ideal (...) on $P_{\kappa} \lambda$ then the conditions of being $\lambda^{+}-preserving$ , weakly presaturated, and presaturated are equivalent for I. (shrink)

Based on formal arguments from Zermelo–Fraenkel set theory we develop the environment for explaining and resolving certain fundamental problems in physics. By these formal tools we show that any quantum system defined by an infinite dimensional Hilbert space of states interferes with the spacetime structure M. M and the quantum system both gain additional degrees of freedom, given by models of Zermelo–Fraenkel set theory. In particular, M develops the ground state where classical gravity vanishes. Quantum mechanics distinguishes set-theoretic random forcing (...) such that M and gravitational degrees of freedom are parameterized by extended real line. The large scale smooth geometry compatible with the forcing extensions is one of exotic smoothness structures of \. The amoeba forcing makes the old real line to have Lebesgue measure zero in the extended one. We apply the entire procedure to the cosmological constant problem, especially to discard the zero-modes contributions to the gravitational vacuum density. Moreover, there exists certain exotic smooth \ from which one determines the realistic, agreeing with observation, small value of the vacuum energy density. (shrink)

Quantum mechanics (QM) predicts probabilities on the fundamental level which are, via Born probability law, connected to the formal randomness of infinite sequences of QM outcomes. Recently it has been shown that QM is algorithmic 1-random in the sense of Martin–Löf. We extend this result and demonstrate that QM is algorithmic $$\omega$$ -random and generic, precisely as described by the ’miniaturisation’ of the Solovay forcing to arithmetic. This is extended further to the result that QM becomes Zermelo–Fraenkel Solovay random on (...) infinite-dimensional Hilbert spaces. Moreover, it is more likely that there exists a standard transitive ZFC model M, where QM is expressed in reality, than in the universe V of sets. Then every generic quantum measurement adds to M the infinite sequence, i.e. random real $$r\in 2^{\omega }$$, and the model undergoes random forcing extensions M[r]. The entire process of forcing becomes the structural ingredient of QM and parallels similar constructions applied to spacetime in the quantum limit, therefore showing the structural resemblance of both in this limit. We discuss several questions regarding measurability and possible practical applications of the extended Solovay randomness of QM. The method applied is the formalization based on models of ZFC; however, this is particularly well-suited technique to recognising randomness questions of QM. When one works in a constant model of ZFC or in axiomatic ZFC itself, the issues considered here remain hidden to a great extent. (shrink)

We show that in 4-spacetime modified at very short distances due to the weakening of classical logic, the higher dimensions emerge. We analyse the case of some smooth topoi, and the case of some class of pointless topoi. The pointless topoi raise the dimensionality due to the forcing adding “string” objects and thus replacing classical points in spacetime. Turning to strings would be something fundamental and connected with set theoretical forcing. The field theory/strings dualities originate at the set theoretical level (...) of the theories. It is argued that this fundamental level can help solving some difficulties of the physical dualities. (shrink)

, which uses the intuitionistic propositional calculus, with the only connective →. It is very important, because the well known Curry-Howard correspondence between proofs and programs was originally discovered with it, and because it enjoys the normalization property: every typed term is strongly normalizable. It was extended to second order intuitionistic logic, in 1970, by J.-Y. Girard [4], under the name of system F, still with the normalization property.More recently, in 1990, the Curry-Howard correspondence was extended to classical logic, following (...) Felleisen and Griffin [6] who discovered that the law of Peirce corresponds to control instructions in functional programming languages. It is interesting to notice that, as early as 1972, Clint and Hoare [1] had made an analogous remark for the law of excluded middle and controlled jump instructions in imperative languages.There are now many type systems which are based on classical logic; among the best known are the system LC of J.-Y. Girard [5] and the λμ-calculus of M. Parigot [11]. We shall use below a system closely related to the latter, called the λ c -calculus [8, 9]. Both systems use classical second order logic and have the normalization property.In the sequel, we shall extend the λ c -calculus to the Zermelo-Frænkel set theory. The main problem is due to the axiom of extensionality. To overcome this difficulty, we first give the axioms of ZF in a suitable (equivalent) form, which we call ZF ɛ. (shrink)

The concept of indiscernibility in a structure is analysed with the aim of emphasizing that in asserting that two objects are indiscernible, it is useful to consider these objects as members of (the domain of) a structure. A case for this usefulness is presented by examining the consequences of this view to the philosophical discussion on identity and indiscernibility in quantum theory.

We propose developing the theory of consequences of morasses relevant in mathematical applications in the language alternative to the usual one, replacing commonly used structures by families of sets originating with Velleman’s neat simplified morasses called 2-cardinals. The theory of related trees, gaps, colorings of pairs and forcing notions is reformulated and sketched from a unifying point of view with the focus on the applicability to constructions of mathematical structures like Boolean algebras, Banach spaces or compact spaces. The paper is (...) dedicated to the memory of Jim Baumgartner whose seminal joint paper :109–129, 1987) with Saharon Shelah provided a critical mass in the theory in question. A new result which we obtain as a side product is the consistency of the existence of a function \ with the appropriate \-version of property \ for regular \ satisfying \. (shrink)

Informally, structural properties of mathematical objects are usually characterized in one of two ways: either as properties expressible purely in terms of the primitive relations of mathematical theories, or as the properties that hold of all structurally similar mathematical objects. We present two formal explications corresponding to these two informal characterizations of structural properties. Based on this, we discuss the relation between the two explications. As will be shown, the two characterizations do not determine the same class of mathematical properties. (...) From this observation we draw some philosophical conclusions about the possibility of a ‘correct’ analysis of structural properties. (shrink)

This is part two of a two-part paper in which we develop an axiomatic theory of the relation of partial ground. The main novelty of the paper is the of use of a binary ground predicate rather than an operator to formalize ground. In this part of the paper, we extend the base theory of the first part of the paper with hierarchically typed truth-predicates and principles about the interaction of partial ground and truth. We show that our theory is (...) a proof-theoretically conservative extension of the ramified theory of positive truth up to. (shrink)

Recently, Gitik, Kanovei and the first author proved that for a classical Prikry forcing extension the family of the intermediate models can be parametrized by $\mathscr{P}(\omega)/\mathrm{finite}$. By modifying the standard Prikry tree forcing we define a Prikry-type forcing which also singularizes a measurable cardinal but which is minimal, i.e., there are \emph{no} intermediate models properly between the ground model and the generic extension. The proof relies on combining the rigidity of the tree structure with indiscernibility arguments resulting from the normality (...) of the associated measures. (shrink)

We show that strongly compact cardinals and MM are sensitive to $\lambda$-closed forcings for arbitrarily large $\lambda$. This is done by adding ‘regressive' $\lambda$-Kurepa trees in either case. We argue that the destruction of regressive Kurepa trees requires a non-standard application of MM. As a corollary, we find a consistent example of an $\omega_2$-closed poset that is not forcing equivalent to any $\omega_2$-directed-closed poset.

We present Woodin's proof that if there exists a measurable Woodin cardinal δ, then there is a forcing extension satisfying all $\Sigma _{2}^{2}$ sentences ϕ such that CH + ϕ holds in a forcing extension of V by a partial order in V δ . We also use some of the techniques from this proof to show that if there exists a stationary limit of stationary limits of Woodin cardinals, then in a homogeneous forcing extension there is an elementary embedding (...) j: V → M with critical point $\omega _{1}^{V}$ such that M is countably closed in the forcing extension. (shrink)

We construct a model [Formula: see text] of [Formula: see text] which lies between [Formula: see text] and [Formula: see text] for a Cohen real [Formula: see text] and does not have the form [Formula: see text] for any set [Formula: see text]. This is loosely based on the unwritten work done in a Bristol workshop about Woodin’s HOD Conjecture in 2011. The construction given here allows for a finer analysis of the needed assumptions on the ground models, thus taking (...) us one step closer to understanding models of [Formula: see text], and the HOD Conjecture and its relatives. This model also provides a positive answer to a question of Grigorieff about intermediate models of [Formula: see text], and we use it to show the failure of Kinna–Wagner Principles in [Formula: see text]. (shrink)

Set theory is an autonomous and sophisticated field of mathematics, enormously successful not only at its continuing development of its historical heritage but also at analyzing mathematical propositions cast in set-theoretic terms and gauging their consistency strength. But set theory is also distinguished by having begun intertwined with pronounced metaphysical attitudes, and these have even been regarded as crucial by some of its great developers. This has encouraged the exaggeration of crises in foundations and of metaphysical doctrines in general. However, (...) set theory has proceeded in the opposite direction, from a web of intensions to a theory of extensionpar excellence, and like other fields of mathematics its vitality and progress have depended on a steadily growing core of mathematical structures and methods, problems and results. There is also the stronger contention that from the beginning set theory actually developed through a progression ofmathematicalmoves, whatever and sometimes in spite of what has been claimed on its behalf.What follows is an account of the development of set theory from its beginnings through the creation of forcing based on these contentions, with an avowedly Whiggish emphasis on the heritage that has been retained and developed by current set theory. The whole transfinite landscape can be viewed as the result of Cantor's attempt to articulate and solve the Continuum Problem. (shrink)

A generic extension L[x]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf{L}[x]}$$\end{document} by a real x is defined, in which the E0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathsf{E}_0}$$\end{document}-class of x is a lightface Π21\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\it \Pi}^1_2}$$\end{document} set containing no ordinal-definable reals.

We show that either of the following hypotheses imply that there is an inner model with a proper class of strong cardinals and a proper class of Woodin cardinals. 1) There is a countably closed cardinal k ≥ N₃ such that □k and □(k) fail. 2) There is a cardinal k such that k is weakly compact in the generic extension by Col(k, k⁺). Of special interest is 1) with k = N₃ since it follows from PFA by theorems of (...) Todorcevic and Velickovic. Our main new technical result, which is due to the first author, is a weak covering theorem for the model obtained by stacking mice over $K^c ||k.$. (shrink)

A stationary subset S of a regular uncountable cardinal κ reflects fully at regular cardinals if for every stationary set $T \subseteq \kappa$ of higher order consisting of regular cardinals there exists an α ∈ T such that S ∩ α is a stationary subset of α. Full Reflection states that every stationary set reflects fully at regular cardinals. We will prove that under a slightly weaker assumption than κ having the Mitchell order κ++ it is consistent that Full Reflection (...) holds at every λ ≤ κ and κ is measurable. (shrink)

We study an extensive connection between quotient forcings of Borel subsets of Polish spaces modulo a σ-ideal and quotient forcings of subsets of countable sets modulo an ideal.

In Zermelo-Fraenkel set theory with the axiom of choice every set has the same cardinal number as some ordinal. Von Rimscha has weakened this condition to “Every set has the same cardinal number as some transitive set”. In set theory without the axiom of choice, we study the deductive strength of this and similar statements introduced by von Rimscha.

Two Fraenkel-Mostowski models are constructed in which the Boolean Prime Ideal Theorem is true. In both models, AC for countable sets is true, but AC for sets of cardinality 2math image and the 2m = m principle are both false. The Principle of Dependent Choices is true in the first model, but false in the second.

We analyze the representation of binary relations in general, and in particular of functions and of total antisymmetric relations, in monadic third order logic, that is, the simple typed theory of sets with three types. We show that there is no general representation of functions or of total antisymmetric relations in this theory. We present partial representations of functions and of total antisymmetric relations which work for large classes of these relations, and show that there is an adequate representation of (...) cardinality in this theory. The relation of our work to similar work by Henrard and Allen Hazen is discussed. This work can be understood as part of a program of assessing the capabilities of weak logical frameworks: our results are applicable for example, to the framework in David Lewis’s Parts of Classes. (shrink)

We formulate a restricted version of the Tukey-Teichmüller Theorem that we denote by (rTT). We then prove that (rTT) and (BPI) are equivalent in ZF and that (rTT) applies rather naturally to several equivalent forms of (BPI): Alexander Subbase Theorem, Stone Representation Theorem, Model Existence and Compactness Theorems for propositional and first-order logic. We also give two variations of (rTT) that we denote by (rTT)+ and (rTT)++; each is equivalent to (rTT) in ZF. The variation (rTT)++ applies rather naturally to (...) various Selection Lemmas due to Cowen, Engeler, and Rado. (shrink)

Assuming every real has a sharp, we prove that for any $\mathop \prod \limits_\~ _1^1 $ equivalence relation either Borel reduces E₀ or in a $\mathop \Delta \limits_\~ _3^1 $ manner allows the assignment of bounded subsets of ω₁ as complete invariants.

Category theory and topos theory have been seen as providing a structuralist framework for mathematics autonomous vis-a-vis set theory. It is argued here that these theories require a background logic of relations and substantive assumptions addressing mathematical existence of categories themselves. We propose a synthesis of Bell's many-topoi view and modal-structuralism. Surprisingly, a combination of mereology and plural quantification suffices to describe hypothetical large domains, recovering the Grothendieck method of universes. Both topos theory and set theory can be carried out (...) relative to such domains; puzzles about ‘large categories’ and ‘proper classes’ are handled in a uniform way, by relativization, sustaining insights of Zermelo. (shrink)

We investigate uncountable maximal antichains of perfect trees and of splitting trees. We show that in the case of perfect trees they must have size of at least the dominating number, whereas for splitting trees they are of size at least \\), i.e. the covering coefficient of the meager ideal. Finally, we show that uncountable maximal antichains of superperfect trees are at least of size the bounding number; moreover we show that this is best possible.

A pointwise definable model is one in which every object is \loos definable without parameters. In a model of set theory, this property strengthens $V=\HOD$, but is not first-order expressible. Nevertheless, if \ZFC\ is consistent, then there are continuum many pointwise definable models of \ZFC. If there is a transitive model of \ZFC, then there are continuum many pointwise definable transitive models of \ZFC. What is more, every countable model of \ZFC\ has a class forcing extension that is pointwise definable. (...) Indeed, for the main contribution of this article, every countable model of Gödel-Bernays set theory has a pointwise definable extension, in which every set and class is first-order definable without parameters. (shrink)

We present several generalizations of the well-known Kunen inconsistency that there is no nontrivial elementary embedding from the set-theoretic universe V to itself. For example, there is no elementary embedding from the universe V to a set-forcing extension V[G], or conversely from V[G] to V, or more generally from one set-forcing ground model of the universe to another, or between any two models that are eventually stationary correct, or from V to HOD, or conversely from HOD to V, or indeed (...) from any definable class to V, among many other possibilities we consider, including generic embeddings, definable embeddings and results not requiring the axiom of choice. We have aimed in this article for a unified presentation that weaves together some previously known unpublished or folklore results, several due to Woodin and others, along with our new contributions. (shrink)

If we assume the axiom of choice, then every two cardinal numbers are comparable, In the absence of the axiom of choice, this is no longer so. For a few cardinalities related to an arbitrary infinite set, we will give all the possible relationships between them, where possible means that the relationship is consistent with the axioms of set theory. Further we investigate the relationships between some other cardinal numbers in specific permutation models and give some results provable without using (...) the axiom of choice. (shrink)

We prove versions of the Dual Ramsey Theorem and the Dual Ellentuck Theorem for families of partitions which are defined in terms of games.

We show, assuming weak large cardinals, that in the context of games of length \ with moves coming from a proper class, clopen determinacy is strictly weaker than open determinacy. The proof amounts to an analysis of a certain level of L that exists under large cardinal assumptions weaker than an inaccessible. Our argument is sufficiently general to give a family of determinacy separation results applying in any setting where the universal class is sufficiently closed; e.g., in third, seventh, or (...) \\)th order arithmetic. We also prove bounds on the strength of Borel determinacy for proper class games. These results answer questions of Gitman and Hamkins. (shrink)

The new result of this paper is that for θ-stable we have S1[θ] = D[θ, L, ∞]. S1 is Hrushovski's rank. This is an improvement of a result of Kim and Pillay, who for simple theories under the assumption that either of the ranks be finite obtained the same identity. Only the first equality is new, the second equality is a result of Shelah from the seventies. We derive it by studying localizations of several rank functions, we get the followingMain (...) Theorem. Suppose that μ is regular satisfying μ ≥ |T|+, p is a finite type, and Δ is a set of formulas closed under Boolean operations. If either R[p, Δ, μ+] < ∞ or p is Δ-stable and μ satisfies “for every sequence {μi : i < |Δ| + ℵ0} of cardinals μi < μ we have that equation image holds”, then S[p, Δ, μ+] = D[p, Δ, μ+] = R[p, Δ, μ+].The S rank above is a localized version of Hrushovski's S1 rank. This rank, as well as our systematic use of local stability, allows us to get a more conceptual proof of the equality of D and R, which is an old result of Shelah. A particular case of the theorem offers a new sufficient condition for the equality of S1 and D[·, L, ∞]. We also manage, due to a more general approach, to avoid some combinatorial difficulties present in Shelah's original exposition. (shrink)

We show that if there is a nonconstructible real, then every perfect set has a nonconstructible element, answering a question of K. Prikry. This is a specific instance of a more general theorem giving a sufficient condition on a pair $M\subset N$ of models of set theory implying that every perfect set in N has an element in N which is not in M.

It was questions about points on the real line that initiated the study of set theory. Points paved the way to point sets and these to ever more abstract sets. And there was more: Reflection on structural properties of point sets not only initiated the study of ordinary sets; it also supplied blueprints for defining extra-ordinary, “large” sets, transcending those provided by standard set theory. In return, the existence of such large sets turned out critical to settling open conjectures about (...) point sets.How to explain such action at a distance between the very large and the rather small? Rather than having an air of magic, could these results rest on deep structural similarities between the two superficially distant species of sets?In this essay I dissect one group of such two-way results. Their linchpin is the notion of measure.§1. Vitali's impossibility result. Our starting point is a problem in measure theory regarding the notion of “Lebesgue measure.” Before presenting the problem, I would like to review the notion of Lebesgue measure. Rather than listing its main properties, I would like to show how Lebesgue measure is born out of an attempt to generalize the notion of the length of an interval to arbitrary sets of reals. One tries to approximate arbitrary sets of reals by intervals, in the hope that the lengths of the intervals will induce a measure on these sets. (shrink)

A generic extension of the constructible universe by reals is defined, in which the union of ‐classes of x and y is a lightface set, but neither of these two ‐classes is separately ordinal‐definable.

We show that the theory, consisting of the usual axioms of but with the power set axiom removed—specifically axiomatized by extensionality, foundation, pairing, union, infinity, separation, replacement and the assertion that every set can be well‐ordered—is weaker than commonly supposed and is inadequate to establish several basic facts often desired in its context. For example, there are models of in which ω1 is singular, in which every set of reals is countable, yet ω1 exists, in which there are sets of (...) reals of every size, but none of size, and therefore, in which the collection axiom fails; there are models of for which the Łoś theorem fails, even when the ultrapower is well‐founded and the measure exists inside the model; there are models of for which the Gaifman theorem fails, in that there is an embedding of models that is Σ1‐elementary and cofinal, but not elementary; there are elementary embeddings of models whose cofinal restriction is not elementary. Moreover, the collection of formulas that are provably equivalent in to a Σ1‐formula or a Π1‐formula is not closed under bounded quantification. Nevertheless, these deficits of are completely repaired by strengthening it to the theory, obtained by using collection rather than replacement in the axiomatization above. These results extend prior work of Zarach. (shrink)

We deal with some natural properties of ultrafilters which trivially fail for normal ultrafilters.