Large Cardinal Hypothesis Research Articles

Strong reducibilities and set theory

We study Medvedev reducibility in the context of set theory — specifically, forcing and large cardinal hypotheses. Answering a question of Hamkins and Li [7], we show that the Medvedev degrees of countable ordinals are far from linearly ordered in multiple ways, our main result here being that there is a club of ordinals which is an antichain with respect to Medvedev reducibility. We then generalize these results to arbitrary “reasonably-definable” reducibilities, under appropriate set-theoretic hypotheses.We then turn from ordinals to general structures. We show that some of the results above yield characterizations of counterexamples to Vaught's conjecture; another applies to all situations, assigning an ordinal to any reasonable class of structures and “measure” on that class. We end by discussing some directions for future research.

Approachable free subsets and fine structure derived scales

Shelah showed that the existence of free subsets over internally approachable subalgebras follows from the failure of the PCF conjecture on intervals of regular cardinals. We show that a stronger property called the Approachable Bounded Subset Property can be forced from the assumption of a cardinal λ for which the set of Mitchell orders {o(μ)|μ<λ} is unbounded in λ. Furthermore, we study the related notion of continuous tree-like scales, and show that such scales must exist on all products in canonical inner models. We use this result, together with a covering-type argument, to show that the large cardinal hypothesis from the forcing part is optimal.

Strongly compact cardinals and ordinal definability

This paper explores several topics related to Woodin’s HOD conjecture. We improve the large cardinal hypothesis of Woodin’s HOD dichotomy theorem from an extendible cardinal to a strongly compact cardinal. We show that assuming there is a strongly compact cardinal and the HOD hypothesis holds, there is no elementary embedding from HOD to HOD, settling a question of Woodin. We show that the HOD hypothesis is equivalent to a uniqueness property of elementary embeddings of levels of the cumulative hierarchy. We prove that the HOD hypothesis holds if and only if every regular cardinal above the first strongly compact cardinal carries an ordinal definable [Formula: see text]-Jónsson algebra. We show that if the HOD hypothesis holds and HOD satisfies the Ultrapower Axiom, then every supercompact cardinal is supercompact in HOD.

On -Strongly Measurable Cardinals

Abstract We prove several consistency results concerning the notion of $\omega $ -strongly measurable cardinal in $\operatorname {\mathrm {HOD}}$ . In particular, we show that is it consistent, relative to a large cardinal hypothesis weaker than $o(\kappa ) = \kappa $ , that every successor of a regular cardinal is $\omega $ -strongly measurable in $\operatorname {\mathrm {HOD}}$ .

Simultaneously vanishing higher derived limits without large cardinals

A question dating to Mardešić and Prasolov’s 1988 work [S. Mardešić and A. V. Prasolov, Strong homology is not additive, Trans. Amer. Math. Soc. 307(2) (1988) 725–744], and motivating a considerable amount of set theoretic work in the years since, is that of whether it is consistent with the ZFC axioms for the higher derived limits [Formula: see text] [Formula: see text] of a certain inverse system [Formula: see text] indexed by [Formula: see text] to simultaneously vanish. An equivalent formulation of this question is that of whether it is consistent for all [Formula: see text]-coherent families of functions indexed by [Formula: see text] to be trivial. In this paper, we prove that, in any forcing extension given by adjoining [Formula: see text]-many Cohen reals, [Formula: see text] vanishes for all [Formula: see text]. Our proof involves a detailed combinatorial analysis of the forcing extension and repeated applications of higher-dimensional [Formula: see text]-system lemmas. This work removes all large cardinal hypotheses from the main result of [J. Bergfalk and C. Lambie-Hanson, Simultaneously vanishing higher derived limits, Forum Math. Pi 9 (2021) e4] and substantially reduces the least value of the continuum known to be compatible with the simultaneous vanishing of [Formula: see text] for all [Formula: see text].

The Combinatorics and Absoluteness of Definable Sets of Real Numbers

AbstractThis thesis divides naturally into two parts, each concerned with the extent to which the theory of $L(\mathbf {R})$ can be changed by forcing.The first part focuses primarily on applying generic-absoluteness principles to how that definable sets of reals enjoy regularity properties. The work in Part I is joint with Itay Neeman and is adapted from our paper Happy and mad families in $L(\mathbf {R})$ , JSL, 2018. The project was motivated by questions about mad families, maximal families of infinite subsets of $\omega $ of which any two have only finitely many members in common. We begin, in the spirit of Mathias, by establishing (Theorem 2.8) a strong Ramsey property for sets of reals in the Solovay model, giving a new proof of Törnquist’s theorem that there are no infinite mad families in the Solovay model.In Chapter 3 we stray from the main line of inquiry to briefly study a game-theoretic characterization of filters with the Baire Property.Neeman and Zapletal showed, assuming roughly the existence of a proper class of Woodin cardinals, that the boldface theory of $L(\mathbf {R})$ cannot be changed by proper forcing. They call their result the Embedding Theorem, because they conclude that in fact there is an elementary embedding from the $L(\mathbf {R})$ of the ground model to that of the proper forcing extension. With a view toward analyzing mad families under $\mathsf {AD}^+$ and in $L(\mathbf {R})$ under large-cardinal hypotheses, in Chapter 4 we establish triangular versions of the Embedding Theorem. These are enough for us to use Mathias’s methods to show (Theorem 4.5) that there are no infinite mad families in $L(\mathbf {R})$ under large cardinals and (Theorem 4.9) that $\mathsf {AD}^+$ implies that there are no infinite mad families. These are again corollaries of theorems about strong Ramsey properties under large-cardinal assumptions and $\mathsf {AD}^+$ , respectively. Our first theorem improves the large-cardinal assumption under which Todorcevic established the nonexistence of infinite mad families in $L(\mathbf {R})$ . Part I concludes with Chapter 5, a short list of open questions.In the second part of the thesis, we undertake a finer analysis of the Embedding Theorem and its consistency strength. Schindler found that the the Embedding Theorem is consistent relative to much weaker assumptions than the existence of Woodin cardinals. He defined remarkable cardinals, which can exist even in L, and showed that the Embedding Theorem is equiconsistent with the existence of a remarkable cardinal. His theorem resembles a theorem of Harrington–Shelah and Kunen from the 1980s: the absoluteness of the theory of $L(\mathbf {R})$ to ccc forcing extensions is equiconsistent with a weakly compact cardinal. Joint with Itay Neeman, we improve Schindler’s theorem by showing that absoluteness for $\sigma $ -closed $\ast $ ccc posets—instead of the larger class of proper posets—implies the remarkability of $\aleph _1^V$ in L. This requires a fundamental change in the proof, since Schindler’s lower-bound argument uses Jensen’s reshaping forcing, which, though proper, need not be $\sigma $ -closed $\ast $ ccc in that context. Our proof bears more resemblance to that of Harrington–Shelah than to Schindler’s.The proof of Theorem 6.2 splits naturally into two arguments. In Chapter 7 we extend the Harrington–Shelah method of coding reals into a specializing function to allow for trees with uncountable levels that may not belong to L. This culminates in Theorem 7.4, which asserts that if there are $X\subseteq \omega _1$ and a tree $T\subseteq \omega _1$ of height $\omega _1$ such that X is codable along T (see Definition 7.3), then $L(\mathbf {R})$ -absoluteness for ccc posets must fail.We complete the argument in Chapter 8, where we show that if in any $\sigma $ -closed extension of V there is no $X\subseteq \omega _1$ codable along a tree T, then $\aleph _1^V$ must be remarkable in L.In Chapter 9 we review Schindler’s proof of generic absoluteness from a remarkable cardinal to show that the argument gives a level-by-level upper bound: a strongly $\lambda ^+$ -remarkable cardinal is enough to get $L(\mathbf {R})$ -absoluteness for $\lambda $ -linked proper posets.Chapter 10 is devoted to partially reversing the level-by-level upper bound of Chapter 9. Adapting the methods of Neeman, Hierarchies of forcing axioms II, we are able to show that $L(\mathbf {R})$ -absoluteness for $\left |\mathbf {R}\right |\cdot \left |\lambda \right |$ -linked posets implies that the interval $[\aleph _1^V,\lambda ]$ is $\Sigma ^2_1$ -remarkable in L.Abstract prepared by Zach Norwood.E-mail: zachnorwood@gmail.com

Incompleteness and jump hierarchies

This paper is an investigation of the relationship between Gödel’s second incompleteness theorem and the well-foundedness of jump hierarchies. It follows from a classic theorem of Spector that the relation { ( A , B ) ∈ R 2 : O A ≤ H B } \{(A,B) \in \mathbb {R}^2 : \mathcal {O}^A \leq _H B\} is well-founded. We provide an alternative proof of this fact that uses Gödel’s second incompleteness theorem instead of the theory of admissible ordinals. We then derive a semantic version of the second incompleteness theorem, originally due to Mummert and Simpson, from this result. Finally, we turn to the calculation of the ranks of reals in this well-founded relation. We prove that, for any A ∈ R A\in \mathbb {R} , if the rank of A A is α \alpha , then ω 1 A \omega _1^A is the ( 1 + α ) (1 + \alpha ) th admissible ordinal. It follows, assuming suitable large cardinal hypotheses, that, on a cone, the rank of X X is ω 1 X \omega _1^X .

Local Ramsey theory: an abstract approach

AbstractGiven a topological Ramsey space , we extend the notion of semiselective coideal to sets and study conditions for that will enable us to make the structure a Ramsey space (not necessarily topological) and also study forcing notions related to which will satisfy abstract versions of interesting properties of the corresponding forcing notions in the realm of Ellentuck's space (cf. [][]). This extends results from [][] to the most general context of topological Ramsey spaces. As applications, we prove that for every topological Ramsey space , under suitable large cardinal hypotheses every semiselective ultrafilter is generic over ; and that given a semiselective coideal , every definable subset of is ‐Ramsey. This generalizes the corresponding results for the case when is equal to Ellentuck's space (cf. [][]).

Indestructibility and destructible measurable cardinals

Say that $${\kappa}$$?'s measurability is destructible if there exists a $${\kappa}$$?. It then follows that $${A_{1} = \{\delta < \kappa \mid \delta}$$A1={?<??? is measurable, ? is not a limit of measurable cardinals, ? is not ?+ strongly compact, and ?'s measurability is destructible when forcing with partial orderings having rank below ??} is unbounded in $${\kappa}$$?. On the other hand, under the same hypotheses, $${A_{2} = \{\delta < \kappa \mid \delta}$$A2={?<??? is measurable, ? is not a limit of measurable cardinals, ? is not ?+ strongly compact, and ??s measurability is indestructible when forcing with either Add(?, 1) or Add(?, ?+)} is unbounded in $${\kappa}$$? as well. The large cardinal hypothesis on ? is necessary, as we further demonstrate by constructing via forcing two distinct models in which either $${A_{1} = \emptyset}$$A1=? or $${A_{2} = \emptyset}$$A2=?. In each of these models, both of which have restricted large cardinal structures above $${\kappa}$$?, every measurable cardinal ? which is not a limit of measurable cardinals is ?+ strongly compact, and there is an indestructibly supercompact cardinal $${\kappa}$$?. In the model in which $${A_{1} = \emptyset}$$A1=?, every measurable cardinal ? which is not a limit of measurable cardinals is <?? strongly compact and has its <?? strong compactness (and hence also its measurability) indestructible when forcing with ?-directed closed partial orderings having rank below ??. The choice of the least beth fixed point above ? is arbitrary, and other values of ?? are also possible.

The least weakly compact cardinal can be unfoldable, weakly measurable and nearly $${\theta}$$ θ -supercompact

We prove from suitable large cardinal hypotheses that the least weakly compact cardinal can be unfoldable, weakly measurable and even nearly $${\theta}$$?-supercompact, for any desired $${\theta}$$?. In addition, we prove several global results showing how the entire class of weakly compactcardinals, a proper class, can be made to coincide with the class of unfoldable cardinals, with the class of weakly measurable cardinals or with the class of nearly $${\theta_\kappa}$$??-supercompact cardinals $${\kappa}$$?, for nearly any desired function $${\kappa\mapsto\theta_\kappa}$$????. These results answer several questions that had been open in the literature and extend to these large cardinals the identity-crises phenomenon, first identified by Magidor with the strongly compact cardinals.

Foundational aspects of singular integrals

We investigate integration of classes of real-valued continuous functions on (0,1]. Of course difficulties arise if there is a non-L1 element in the class, and the Hadamard finite part integral (p.f.) does not apply. Such singular integrals arise naturally in many contexts including PDEs and singular ODEs.The Lebesgue integral as well as p.f., starting at zero, obey two fundamental conditions: (i) they act as antiderivatives and, (ii) if f=g on (0,a), then their integrals from 0 to x coincide for any x∈(0,a).We find that integrals from zero with the essential properties of p.f., plus positivity, exist by virtue of the Axiom of Choice (AC) on all functions on (0,1] which are L1((ε,1]) for all ε>0. However, this existence proof does not provide a satisfactory construction. Without some regularity at 0, the existence of general antiderivatives which satisfy only (i) and (ii) above on classes with a non-L1 element is independent of ZF (the usual ZFC axioms for mathematics without AC), and even of ZFDC (ZF with the Axiom of Dependent Choice). Moreover we show that there is no mathematical description that can be proved (within ZFC or even extensions of ZFC with large cardinal hypotheses) to uniquely define such an antiderivative operator.Such results are precisely formulated for a variety of sets of functions, and proved using methods from mathematical logic, descriptive set theory and analysis. We also analyze p.f. on analytic functions in the punctured unit disk, and make the connection to singular initial value problems.

The Nonabsoluteness of Model Existence in Uncountable Cardinals for Lω1,ω

For sentences ϕ of Lω1,ω, we investigate the question of absoluteness of ϕ having models in uncountable cardinalities. We first observe that having a model in ℵ1 is an absolute property, but having a model in ℵ2 is not as it may depend on the validity of the continuum hypothesis. We then consider the generalized continuum hypothesis (GCH) context and provide sentences for any α∈ω1∖{0,1,ω} for which the existence of a model in ℵα is nonabsolute (relative to large cardinal hypotheses). Finally, we present a complete sentence for which model existence in ℵ3 is nonabsolute.

A definable failure of the singular cardinal hypothesis

We show first that it is consistent that κ is a measurable cardinal where the GCH fails, while there is a lightface definable wellorder of H(κ+). Then with further forcing we show that it is consistent that GCH fails at ℵω, ℵω strong limit, while there is a lightface definable wellorder of H(ℵω+1) (“definable failure” of the singular cardinal hypothesis at ℵω). The large cardinal hypothesis used is the existence of a κ++-strong cardinal, where κ is κ++-strong if there is an embedding j: V → M with critical point κ such that H(κ++) ⊆ M. By work of M. Gitik and W. J. Mitchell [12], [20], our large cardinal assumption is almost optimal. The techniques of proof include the “tuning-fork” method of [10] and [3], a generalisation to large cardinals of the stationary-coding of [4] and a new “definable-collapse” coding based on mutual stationarity. The fine structure of the canonical inner model L[E] for a κ++-strong cardinal is used throughout.

Indestructibility, measurability, and degrees of supercompactness

AbstractSuppose that κ is indestructibly supercompact and there is a measurable cardinal λ &gt; κ. It then follows that A1 = {δ &lt; κ∣δ is measurable, δ is not a limit of measurable cardinals, and δ is not δ+ supercompact} is unbounded in κ. If in addition λ is 2λ supercompact, then A2 = {δ &lt; κ∣δ is measurable, δ is not a limit of measurable cardinals, and δ is δ+ supercompact} is unbounded in κ as well. The large cardinal hypotheses on λ are necessary, as we further demonstrate by constructing via forcing two distinct models in which either \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$A_1 = \varnothing$\end{document} or \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$A_2 = \varnothing$\end{document}. In each of these models, there is an indestructibly supercompact cardinal κ, and a restricted large cardinal structure above κ. If we weaken the indestructibility requirement on κ to indestructibility under partial orderings which are both κ‐directed closed and (κ+, ∞)‐distributive, then it is possible to construct a model containing a supercompact cardinal κ witnessing this degree of indestructibility in which every measurable cardinal δ &lt; κ is (at least) δ+ supercompact.

MRP, tree properties and square principles

AbstractWe show that MRP + MA implies that ITP(λ,ω2) holds for all cardinal λ ≥ ω2. This generalizes a result by Weiβ who showed that PFA implies that ITP(λ, ω2) holds for all cardinal λ ≥ ω2. Consequently any of the known methods to prove MRP + MA consistent relative to some large cardinal hypothesis requires the existence of a strongly compact cardinal. Moreover if one wants to force MRP + MA with a proper forcing, it requires at least a supercompact cardinal. We also study the relationship between MRP and some weak versions of square. We show that MRP implies the failure of □(λ, ω) for all λ ≥ ω2 and we give a direct proof that MRP + MA implies the failure of □(λ, ω1) for all λ ≥ ω2.

Indestructibility, HOD, and the Ground Axiom

Let φ1 stand for the statement V = HOD and φ2 stand for the Ground Axiom. Suppose Ti for i = 1, …, 4 are the theories “ZFC + φ1 + φ2,” “ZFC + ¬φ1 + φ2,” “ZFC + φ1 + ¬φ2,” and “ZFC + ¬φ1 + ¬φ2” respectively. We show that if κ is indestructibly supercompact and λ > κ is inaccessible, then for i = 1, …, 4, Ai = df{δ < κ∣δ is an inaccessible cardinal which is not a limit of inaccessible cardinals and Vδ⊨Ti} must be unbounded in κ. The large cardinal hypothesis on λ is necessary, as we further demonstrate by constructing via forcing four models in which Ai = ∅︁ for i = 1, …, 4. In each of these models, there is an indestructibly supercompact cardinal κ, and no cardinal δ > κ is inaccessible. We show it is also the case that if κ is indestructibly supercompact, then Vκ⊨T1, so by reflection, B1 = df{δ < κ∣δ is an inaccessible limit of inaccessible cardinals and Vδ⊨T1} is unbounded in κ. Consequently, it is not possible to construct a model in which κ is indestructibly supercompact and B1 = ∅︁. On the other hand, assuming κ is supercompact and no cardinal δ > κ is inaccessible, we demonstrate that it is possible to construct a model in which κ is indestructibly supercompact and for every inaccessible cardinal δ < κ, Vδ⊨T1. It is thus not possible to prove in ZFC that Bi = df{δ < κ∣δ is an inaccessible limit of inaccessible cardinals and Vδ⊨Ti} for i = 2, …, 4 is unbounded in κ if κ is indestructibly supercompact. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

Diagonal Prikry extensions

§1. Introduction. It is a well-known phenomenon in set theory that problems in infinite combinatorics involving singular cardinals and their successors tend to be harder than the parallel problems for regular cardinals. Examples include the behaviour of cardinal exponentiation, the extent of the tree property, the extent of stationary reflection, and the existence of non-free almost-free abelian groups. The explanation for this phenomenon lies in inner model theory, in particular core models and covering lemmas. If W is an inner model of V then1. W strongly covers V if every uncountable set of ordinals is covered by a set of the same V -cardinality lying in W.2. W weakly covers V if W computes the successor of every V-singular cardinal correctly.Strong covering implies weak covering.In inner model theory there are many theorems of the general form “if there is no inner model of large cardinal hypothesis X then there is an L-like inner model Kx which Y covers V”. Here the L-like properties of Kx always include GCH and Global Square. Examples include1. X is “0# exists”, Kx is L, Y is “strongly”.2. X is “there is a measurable cardinal”, Kx is the Dodd-Jensen core model, Y is “strongly”.3. X is “there is a Woodin cardinal”, Kx is the core model for a Woodin cardinal, Y is “weakly”.

Indestructibility, instances of strong compactness, and level by level inequivalence

Suppose λ > κ is measurable. We show that if κ is either indestructibly supercompact or indestructibly strong, then A = {δ < κ | δ is measurable, yet δ is neither δ + strongly compact nor a limit of measurable cardinals} must be unbounded in κ. The large cardinal hypothesis on λ is necessary, as we further demonstrate by constructing via forcing two models in which \({A = \emptyset}\). The first of these contains a supercompact cardinal κ and is such that no cardinal δ > κ is measurable, κ’s supercompactness is indestructible under κ-directed closed, (κ +, ∞)-distributive forcing, and every measurable cardinal δ < κ is δ + strongly compact. The second of these contains a strong cardinal κ and is such that no cardinal δ > κ is measurable, κ’s strongness is indestructible under < κ-strategically closed, (κ +, ∞)-distributive forcing, and level by level inequivalence between strong compactness and supercompactness holds. The model from the first of our forcing constructions is used to show that it is consistent, relative to a supercompact cardinal, for the least cardinal κ which is both strong and has its strongness indestructible under κ-directed closed, (κ +, ∞)-distributive forcing to be the same as the least supercompact cardinal, which has its supercompactness indestructible under κ-directed closed, (κ +, ∞)-distributive forcing. It further follows as a corollary of the first of our forcing constructions that it is possible to build a model containing a supercompact cardinal κ in which no cardinal δ > κ is measurable, κ is indestructibly supercompact, and every measurable cardinal δ < κ which is not a limit of measurable cardinals is δ + strongly compact.

Strong Logics of First and Second Order

AbstractIn this paper we investigate strong logics of first and second order that have certain absoluteness properties. We begin with an investigation of first order logic and the strong logics ω-logic and β-logic, isolating two facets of absoluteness, namely, generic invariance and faithfulness. It turns out that absoluteness is relative in the sense that stronger background assumptions secure greater degrees of absoluteness. Our aim is to investigate the hierarchies of strong logics of first and second order that are generically invariant and faithful against the backdrop of the strongest large cardinal hypotheses. We show that there is a close correspondence between the two hierarchies and we characterize the strongest logic in each hierarchy. On the first-order side, this leads to a new presentation of Woodin's Ω-logic. On the second-order side, we compare the strongest logic with full second-order logic and argue that the comparison lends support to Quine's claim that second-order logic is really set theory in sheep's clothing.

The number of normal measures

AbstractThere have been numerous results showing that a measurable cardinal κ can carry exactly α normal measures in a model of GCH. where α is a cardinal at most κ++. Starting with just one measurable cardinal, we have [9] (for α = 1), [10] (for α = α++, the maximum possible) and [1] (for α = κ+, after collapsing κ++). In addition, under stronger large cardinal hypotheses, one can handle the remaining cases: [12] (starting with a measurable cardinal of Mitchell order α), [2] (as in [12], but where κ is the least measurable cardinal and α is less than κ, starting with a measurable of high Mitchell order) and [11] (as in [12], but where κ is the least measurable cardinal, starting with an assumption weaker than a measurable cardinal of Mitchell order 2). In this article we treat all cases by a uniform argument, starting with only one measurable cardinal and applying a cofinality-preserving forcing. The proof uses κ-Sacks forcing and the “tuning fork” technique of [8]. In addition, we explore the possibilities for the number of normal measures on a cardinal at which the GCH fails.