The consistency of a second-order version of Morley’s Theorem on the number of countable models was proved in [EHMT23] with the aid of large cardinals. We here dispense with them.

We give modest upper bounds for consistency strengths for two well-studied combinatorial principles. These bounds range at the level of subcompact cardinals, which is significantly below a κ+-supercompact cardinal. All previously known upper bounds on these principles ranged at the level of some degree of supercompactness. We show that by using any of the standard modified Prikry forcings it is possible to turn a measurable subcompact cardinal into ℵω and make the principle □ℵω,<ω fail in the generic extension. We also (...) show that by using Lévy collapse followed by standard iterated club shooting it is possible to turn a subcompact cardinal into ℵ2 and arrange in the generic extension that simultaneous reflection holds at ℵ2, and at the same time, every stationary subset of ℵ3 concentrating on points of cofinality ω has a reflection point of cofinality ω1. (shrink)

Certain set theoretical notions cannot be split into finer subnotions.

In mathematical practice certain formulas φ are believed to essentially decide all other natural properties of the object x. The purpose of this paper is to exactly quantify such a belief for four formulas φ, namely “x is a Ramsey ultrafilter”, “x is a free Souslin tree”, “x is an extendible strong Lusin set” and “x is a good diamond sequence”.

Part I of this paper introduces a range of mechanistic theories of causality, including process theories and the complex-systems theories, and some of the problems they face. Part II argues that while there is a decisive case against a purely mechanistic analysis, a viable theory of causality must incorporate mechanisms as an ingredient, and describes one way of providing an analysis of causality which reaps the rewards of the mechanistic approach without succumbing to its pitfalls.

We investigate club guessing sequences and filters. We prove that assuming V=L, there exists a strong club guessing sequence on μ if and only if μ is not ineffable for every uncountable regular cardinal μ. We also prove that for every uncountable regular cardinal μ, relative to the existence of a Woodin cardinal above μ, it is consistent that every tail club guessing ideal on μ is precipitous.

We describe the inner models with representatives in all equivalence classes of thin equivalence relations in a given projective pointclass of even level assuming projective determinacy. The main result shows that these models are characterized by their correctness and the property that they correctly compute the tree from the appropriate scale. The main step towards this characterization shows that the tree from a scale can be reconstructed in a generic extension of an iterate of a mouse. We then construct models (...) with this property as generic extensions of iterates of mice under the assumption that the corresponding projective ordinal is below ω2ω2. On the way, we consider several related problems, including the question when forcing does not add equivalence classes to thin projective equivalence relations. For instance, we show that if every set has a sharp, then reasonable forcing does not add equivalence classes to thin provably View the MathML source Δ1/3 equivalence relations, and generalize this to all projective levels. (shrink)

The present paper investigates the power of proper forcings to change the shape of the universe, in a certain well-defined respect. It turns out that the ranking among large cardinals can be used as a measure for that power. However, in order to establish the final result I had to isolate a new large cardinal concept, which I dubbed “remarkability.” Let us approach the exact formulation of the problem—and of its solution—at a slow pace.Breathtaking developments in the mid 1980s found (...) one of its culminations in the theorem, due to Martin, Steel, and Woodin, that the existence of infinitely many Woodin cardinals with a measurable cardinal above them all implies that AD, the axiom of determinacy, holds in the least inner model containing all the reals, L. One of the nice things about AD is that the theory ZF + AD + V = L appears as a choiceless “completion” of ZF in that any interesting question seems to find an at least attractive answer in that theory. Beyond that, AD is very canonical as may be illustrated as follows.Let us say that L is absolute for set-sized forcings if for all posets P ∈ V, for all formulae ϕ, and for all ∈ ℝ do we have thatwhere is a name for the set of reals in the extension. (shrink)

We present two ways in which the model L(R) is canonical assuming the existence of large cardinals. We show that the theory of this model, with ordinal parameters, cannot be changed by small forcing; we show further that a set of ordinals in V cannot be added to L(R) by small forcing. The large cardinal needed corresponds to the consistency strength of AD L (R); roughly ω Woodin cardinals.

We prove that a combinatorial consequence of the negation of the PCF conjecture for intervals, involving free subsets relative to set mappings, is not implied by even the strongest known large cardinal axiom.

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.

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 prove that, in any fine structural extender model with Jensen’s λ-indexing, there is a \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\square(\kappa^{+})}$$\end{document} -sequence if and only if there is a pair of stationary subsets of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\kappa^{+} \cap {\rm {cof}}( < \kappa)}$$\end{document} without common reflection point of cofinality \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${ < \kappa}$$\end{document} which, in turn, is equivalent to the existence of a (...) family of size \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${ < \kappa}$$\end{document} of stationary subsets of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\kappa^{+} \cap {\rm {cof}}( < \kappa)}$$\end{document} without common reflection point of cofinality \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${ < \kappa}$$\end{document}. By a result of Burke/jensen, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\square_\kappa}$$\end{document} fails whenever \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\kappa}$$\end{document} is a subcompact cardinal. Our result shows that in extender models, it is still possible to construct a canonical \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\square(\kappa^{+})}$$\end{document} -sequence where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\kappa}$$\end{document} is the first subcompact. (shrink)

We give some applications of mixed support forcing iterations to the topics of disjoint stationary sequences and internally approachable sets. In the first half of the paper we study the combinatorial content of the idea of a disjoint stationary sequence, including its relation to adding clubs by forcing, the approachability ideal, canonical structure, the proper forcing axiom, and properties related to internal approachability. In the second half of the paper we present some consistency results related to these ideas. We construct (...) a model in which a disjoint stationary sequence exists at the successor of an arbitrary regular uncountable cardinal. We also construct models in which the properties of being internally stationary, internally club, and internally approachable are distinct. (shrink)

We work out the details of a schema for a mixed support forcing iteration, which generalizes the Mitchell model [7] with no Aronszajn trees on ω2.

If we replace first-order logic by second-order logic in the original definition of Gödel’s inner model L, we obtain the inner model of hereditarily ordinal definable sets [33]. In this paper...

After a decade of intense debate about mechanisms, there is still no consensus characterization. In this paper we argue for a characterization that applies widely to mechanisms across the sciences. We examine and defend our disagreements with the major current contenders for characterizations of mechanisms. Ultimately, we indicate that the major contenders can all sign up to our characterization.

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.

We introduce a natural principleStrong Chang Reflectionstrengthening the classical Chang Conjectures. This principle is between a huge and a two huge cardinal in consistency strength. In this note we prove that it implies the existence of an inner model with a huge cardinal. The technique we explore for building inner models with huge cardinals adapts to show thatdecisiveideals imply the existence of inner models with supercompact cardinals. Proofs for all of these claims can be found in [10].1,2.

We argue against Foreman’s proposal to settle the continuum hypothesis and other classical independent questions via the adoption of generic large cardinal axioms.

We characterize the (κ, Λ, < μ)-distributive law in Boolean algebras in terms of cut and choose games $\scr{G}_{<\mu}^{\kappa}(\lambda)$ , when μ ≤ κ ≤ Λ and κ<κ = κ. This builds on previous work to yield game-theoretic characterizations of distributive laws for almost all triples of cardinals κ, Λ, μ with μ ≤ Λ, under GCH. In the case when μ ≤ κ ≤ Λ and κ<κ = κ, we show that it is necessary to consider whether the κ-stationarity (...) of Pκ+Λ in the ground model is preserved by B. In this vein, we develop the theory of κ-club and κ-stationary subsets of Pκ+Λ. We also construct Boolean algebras in which Player I wins $\scr{G}_{\kappa}^{\kappa}(\kappa ^{+})$ but the (κ, ∞, κ)-d.1, holds, and, assuming GCH, construct Boolean algebras in which many games are undetermined. (shrink)

Since the work of Gödel and Cohen, which showed that Hilbert's First Problem was independent of the usual assumptions of mathematics, there have been a myriad of independence results in many areas of mathematics. These results have led to the systematic study of several combinatorial principles that have proven effective at settling many of the important independent statements. Among the most prominent of these are the principles diamond and square discovered by Jensen. Simultaneously, attempts have been made to find suitable (...) natural strengthenings of ZFC, primarily by Large Cardinal or Reflection Axioms. These two directions have tension between them in that Jensen's principles, which tend to suggest a rather rigid mathematical universe, are at odds with reflection properties. A third development was the discovery by Shelah of "PCF Theory", a generalization of cardinal arithmetic that is largely determined inside ZFC. In this paper we consider interactions between these three theories in the context of singular cardinals, focusing on the various implications between square and scales, and on consistency results between relatively strong forms of square and stationary set reflection. (shrink)

Three central combinatorial properties in set theory are the tree property, the approachability property and stationary reflection. We prove the mutual independence of these properties by showing that any of their eight Boolean combinations can be forced to hold at${\kappa ^{ + + }}$, assuming that$\kappa = {\kappa ^{ < \kappa }}$and there is a weakly compact cardinal aboveκ.If in additionκis supercompact then we can forceκto be${\aleph _\omega }$in the extension. The proofs combine the techniques of adding and then destroying (...) a nonreflecting stationary set or a${\kappa ^{ + + }}$-Souslin tree, variants of Mitchell’s forcing to obtain the tree property, together with the Prikry-collapse poset for turning a large cardinal into${\aleph _\omega }$. (shrink)

We prove a number of consistency results complementary to the ZFC results from our paper [J. Cummings, M. Foreman, M. Magidor, Canonical structure in the universe of set theory: part one, Annals of Pure and Applied Logic 129 211–243]. We produce examples of non-tightly stationary mutually stationary sequences, sequences of cardinals on which every sequence of sets is mutually stationary, and mutually stationary sequences not concentrating on a fixed cofinality. We also give an alternative proof for the consistency of the (...) existence of stationarily many non-good points, show that diagonal Prikry forcing preserves certain stationary reflection properties, and study the relationship between some simultaneous reflection principles. Finally we show that the least cardinal where square fails can be the least inaccessible, and show that weak square is incompatible in a strong sense with generic supercompactness. (shrink)

We start by studying the relationship between two invariants isolated by Shelah, the sets of good and approachable points. As part of our study of these invariants, we prove a form of “singular cardinal compactness” for Jensen's square principle. We then study the relationship between internally approachable and tight structures, which parallels to a certain extent the relationship between good and approachable points. In particular we characterise the tight structures in terms of PCF theory and use our characterisation to prove (...) some covering results for tight structures, along with some results on tightness and stationary reflection. Finally, we prove some absoluteness theorems in PCF theory, deduce a covering theorem, and apply that theorem to the study of precipitous ideals. (shrink)

Burke proved that the generalized nonstationary ideal, denoted by NS, is universal in the following sense: every normal ideal, and every tower of normal ideals of inaccessible height, is a canonical Rudin‐Keisler projection of the restriction of NS to some stationary set. We investigate how far Burke's theorem can be pushed, by analyzing the universality properties of NS with respect to the wider class of ‐systems of filters introduced by Audrito and Steila. First we answer a question of Audrito and (...) Steila, by proving that ‐systems of filters do not capture all kinds of set‐generic embeddings. We provide a characterization of supercompactness in terms of short extenders and canonical projections of NS, without any reference to the strength of the extenders; as a corollary, NS can consistently fail to canonically project to arbitrarily strong short extenders. We prove that ω‐cofinal towers of normal ultrafilters, e.g., the kind used to characterize I2 and I3 embeddings, are well‐founded if and only if they are canonical projections of NS. Finally, we provide a characterization of “ is Jónsson” in terms of canonical projections of NS. (shrink)

Given an ideal $I$ , let $\mathbb{P}_{I}$ denote the forcing with $I$ -positive sets. We consider models of forcing axioms $MA(\Gamma)$ which also have a normal ideal $I$ with completeness $\omega_{2}$ such that $\mathbb{P}_{I}\in \Gamma$ . Using a bit more than a superhuge cardinal, we produce a model of PFA (proper forcing axiom) which has many ideals on $\omega_{2}$ whose associated forcings are proper; a similar phenomenon is also observed in the standard model of $MA^{+\omega_{1}}(\sigma\mbox{-closed})$ obtained from a supercompact cardinal. (...) Our model of PFA also exhibits weaker versions of ideal properties, which were shown by Foreman and Magidor to be inconsistent with PFA. Along the way, we also show (1) the diagonal reflection principle for internally club sets ( $\mathit{DRP}(IC_{\omega_{1}})$ ) introduced by the author in earlier work is equivalent to a natural weakening of “there is an ideal $I$ such that $\mathbb{P}_{I}$ is proper”; and (2) for many natural classes $\Gamma$ of posets, $MA^{+\omega_{1}}(\Gamma)$ is equivalent to an apparently stronger version which we call $MA^{+\operatorname{Diag}}(\Gamma)$. (shrink)

It is well known that saturation of ideals is closely related to the “antichain-catching” phenomenon from Foreman–Magidor–Shelah [10]. We consider several antichain-catching properties that are weaker than saturation, and prove:If${\cal I}$is a normal ideal on$\omega _2 $which satisfiesstationary antichain catching, then there is an inner model with a Woodin cardinal;For any$n \in \omega $, it is consistent relative to large cardinals that there is a normal ideal${\cal I}$on$\omega _n $which satisfiesprojective antichain catching, yet${\cal I}$is not saturated. This provides a negative (...) answer to Open Question number 13 from Foreman’s chapter in the Handbook of Set Theory. (shrink)

We prove that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${(\omega_3, \omega_2) \twoheadrightarrow (\omega_2, \omega_1)}$$\end{document} implies there is an inner model with a weak repeat measure.

If [Formula: see text] is regular and [Formula: see text], then the existence of a weakly presaturated ideal on [Formula: see text] implies [Formula: see text]. This partially answers a question of Foreman and Magidor about the approachability ideal on [Formula: see text]. As a corollary, we show that if there is a presaturated ideal [Formula: see text] on [Formula: see text] such that [Formula: see text] is semiproper, then CH holds. We also show some barriers to getting the tree (...) property and a saturated ideal simultaneously on a successor cardinal from conventional forcing methods. (shrink)

We prove that various classical tree forcings—for instance Sacks forcing, Mathias forcing, Laver forcing, Miller forcing and Silver forcing—preserve the statement that every real has a sharp and hence analytic determinacy. We then lift this result via methods of inner model theory to obtain level-by-level preservation of projective determinacy (PD). Assuming PD, we further prove that projective generic absoluteness holds and no new equivalence classes are added to thin projective transitive relations by these forcings.

There is a partial order ${\mathbb{P}}$ preserving stationary subsets of ω 1 and forcing that every partial order in the ground model V that collapses a sufficiently large ordinal to ω 1 over V also collapses ω 1 over ${V^{\mathbb{P}}}$ . 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. (shrink)

Consider where κ is an uncountable regular cardinal. By a result of Shelah's we have for almost all witnessing this. Here we consider the question if there could be a similar result for. We will discuss some basic facts implying that this cannot hold in general. We will use these facts to construct an interesting example of a pseudo Prikry forcing, answering a question of Sinapova.