# Menachem MagidorHebrew University of Jerusalem | HUJI · Einstein Institute of Mathematics

Menachem Magidor

Ph.D.

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139

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Introduction

**Skills and Expertise**

## Publications

Publications (139)

We provide a model theoretical and tree property-like characterization of $\lambda $ - $\Pi ^1_1$ -subcompactness and supercompactness. We explore the behavior of these combinatorial principles at accessible cardinals.

In [Bon20], model theoretic characterizations of several established large cardinal notions were given. We continue this work, by establishing such characterizations for Woodin cardinals (and variants), various virtual large cardinals, and subtle cardinals.

We study the Galvin property. We show that various square principles imply that the cofinality of the Galvin number is uncountable (or even greater than \(\aleph _1\)). We prove that the proper forcing axiom is consistent with a strong negation of the Glavin property.

A reflection principle for Corson compacta holds in the forcing extension obtained by Levy-collapsing a supercompact cardinal to ℵ2. In this model, a compact Hausdorff space is Corson if and only if all of its continuous images of weight ℵ1 are Corson compact. We use the Gelfand–Naimark duality, and our results are stated in terms of unital abelian...

If we replace first-order logic by second-order logic in the original definition of Gödel’s inner model [Formula: see text], we obtain the inner model of hereditarily ordinal definable (HOD) sets [33]. In this paper, we consider inner models that arise if we replace first-order logic by a logic that has some, but not all, of the strength of second-...

This paper has two parts. The first is concerned with a variant of a family of games introduced by Holy and Schlicht, that we call \emph{Welch games}. Player II having a winning strategy in the Welch game of length $\omega$ on $\kappa$ is equivalent to weak compactness. Winning the game of length $2^\kappa$ is equivalent to $\kappa$ being measurabl...

In this paper we study the notion of $C^{(n)}$ -supercompactness introduced by Bagaria in [3] and prove the identity crises phenomenon for such class. Specifically, we show that consistently the least supercompact is strictly below the least $C^{(1)}$ -supercompact but also that the least supercompact is $C^{(1)}$ -supercompact (and even $C^{(n)}$...

We introduce a new inner model $C(aa)$ arising from stationary logic. We show that assuming a proper class of Woodin cardinals, or alternatively $MM^{++}$, the regular uncountable cardinals of $V$ are measurable in the inner model $C(aa)$, the theory of $C(aa)$ is (set) forcing absolute, and $C(aa)$ satisfies CH. We introduce an auxiliary concept t...

If we replace first order logic by second order logic in the original definition of G\"odel's inner model $L$, we obtain HOD. In this paper we consider inner models that arise if we replace first order logic by a logic that has some, but not all, of the strength of second order logic. Typical examples are the extensions of first order logic by gene...

THE TREE PROPERTY AT THE TWO IMMEDIATE SUCCESSORS OF A SINGULAR CARDINAL - JAMES CUMMINGS, YAIR HAYUT, MENACHEM MAGIDOR, ITAY NEEMAN, DIMA SINAPOVA, SPENCER UNGER

We prove that if $\mu$ is a singular cardinal with countable cofinality and $2^\mu=\mu^+$ then $\binom{\mu^+}{\mu}\nrightarrow\binom{\mu^+\ \aleph_2}{\mu\ \mu}$.

ITP is a combinatorial principle that is a strengthening of the tree property. For an inaccessible cardinal κ \kappa , ITP at κ \kappa holds if and only if κ \kappa is supercompact. And just like the tree property, it can be forced to hold at accessible cardinals. A broad project is obtaining ITP at many cardinals simultaneously. Past a singular ca...

We provide a model theoretical and tree property like characterization of $\lambda$-$\Pi^1_1$-subcompactness and supercompactness. We explore the behaviour of those combinatorial principles at accessible cardinals.

We introduce the large-cardinal notions of [Formula: see text]-greatly-Mahlo and [Formula: see text]-reflection cardinals and prove (1) in the constructible universe, [Formula: see text], the first [Formula: see text]-reflection cardinal, for [Formula: see text] a successor ordinal, is strictly between the first [Formula: see text]-greatly-Mahlo an...

We show that $\omega_1\nrightarrow_{\rm path}(\omega)^2_{\omega,<\omega}$ but $\omega_1\rightarrow_{\rm path}(\omega)^2_\omega$, whence we prove that if $\mu>{\rm cf}(\mu)=\omega$ and $2^\mu=\mu^+$ then $\binom{\mu^+}{\mu}\nrightarrow\binom{\mu^+\ \aleph_2}{\mu\ \mu}$.

A reflection principle for Corson compacta holds in the forcing extension obtained by Levy-collapsing a supercompact cardinal to~$\aleph_2$. In this model, a compact Hausdorff space is Corson if and only if all of its continuous images of weight~$\aleph_1$ are Corson compact. We use the Gelfand--Naimark duality, and our results are stated in terms...

DESTRUCTIBILITY OF THE TREE PROPERTY AT אω+1 - YAIR HAYUT, MENACHEM MAGIDOR

In this paper we study the notion of C (n)-supercompactness introduced by Bagaria in [Bag12] and prove the identity crises phenomenon for such classes. Specifically , we show that consistently the least supercompact is strictly below the least C (1)-supercompact but also that the least (ω 1-)strong compact is C (1)-supercompact (and even C (n)-supe...

We investigate reflection-type problems on the class SPM, of Boolean algebras carrying strictly positive finitely additive measures. We show, in particular, that in the constructible universe there is a Boolean algebra $\mathfrak A$ which is not in SPM but every subalgebra of $\mathfrak A$ of cardinality $\mathfrak c$ admits a strictly positive mea...

A remarkable result by Shelah states that if κ is a singular strong limit cardinal of uncountable cofinality, then there is a subset x of κ such that HODx contains the power set of κ. We develop a version of diagonal extender-based supercompact Prikry forcing, and use it to show that singular cardinals of countable cofinality do not in general have...

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 ^{ and there is a weakly compa...

We prove that under Martin's maximum the non-stationary ideal over $\aleph_1$ is almost weakly Laver.

We show that from infinitely many supercompact cardinals one can force a model of ZFC where both the tree property and the stationary reflection hold at א ω² + 1 .

We analyze the effect of singularizing cardinals on square properties. By work of Džamonja-Shelah and of Gitik, if you singularize an inaccessible cardinal to countable cofinality while preserving its successor, then □κ,ω holds in the bigger model. We extend this to the situation where every regular cardinal in an interval [κ, ν] is singularized, f...

In $\mathsf{ZFC}$, if there is a measurable cardinal with infinitely many Woodin cardinals below it, then for every equivalence relation $E \in L(\mathbb{R})$ on $\mathbb{R}$ with all $\mathbf{\Delta}_1^1$ classes and every $\sigma$-ideal $I$ on $\mathbb{R}$ so that the associated forcing $\mathbb{P}_I$ of $I^+$ $\mathbf{\Delta}_1^1$ subsets is pro...

Itay Neeman (Norte Dame J Form Log 55:265–298, 2014) presented a new way of iterating of proper forcings. We would like to generalize it here to semi-proper.

We construct a model in which the tree property holds in $\aleph_{\omega + 1}$ and it is destructible under $\text{Col}(\omega, \omega_1)$. On the other hand we discuss some cases in which the tree property is indestructible under small or closed forcings.

Assuming that there is a stationary set of ordinals of countable cofinality in ω2 that does not reflect, we prove that there exists a compact space which is not Corson compact and whose all continuous images of weight ≤ω1 are Eberlein compacta. This yields an example of a Banach space of density ω2 which is not weakly compactly generated but all it...

We show that the character spectrum $Sp_\chi(\lambda)$, for a singular
cardinal $\lambda$ of countable cofinality, may include any prescribed set of
regular cardinals between $\lambda$ and $2^\lambda$.

We extend a result of R. Jensen ([6]) by showing that in the constructible universe, a regular cardinal is (n + 1)-stationary if and only if it is Π
n
1-indescribable.

The present paper is about omitting types in logic of metric structures
introduced by Ben Yaacov, Berenestein, Henson and Usvyatsov. While a complete
type is omissible in a model of a complete theory if and only if it is not
principal, this is not true for incomplete types by a result of Ben Yaacov. We
prove that there is no simple test for determi...

GitikM.. All uncountable cardinals can be singular. Israel journal of mathematics, vol. 35 (1980), pp. 61–88. - Volume 49 Issue 2 - Menachem Magidor

We answer some natural questions about group radicals and torsion classes, which involve the existence of measurable cardinals, by constructing, relative to the existence of a supercompact cardinal, a model of ZFC in which the first ω 1 -strongly compact cardinal is singular.

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 of uncountable cofinality, while its successor enjoys
various combinatorial properties.
As a sample ap...

An uncountable cardinal κ is called ω 1 -strongly compact if every κ-complete ultrafilter on any set I can be extended to an ω 1 -complete ultrafilter on I. We show that the first ω 1 -strongly compact cardinal, κ 0 , cannot be a successor cardinal, and that its cofinality is at least the first measurable cardinal. We prove that the Singular Cardin...

Given a cardinal $\kappa$ that is $\lambda$-supercompact for some regular
cardinal $\lambda\geq\kappa$ and assuming $\GCH$, we show that one can force
the continuum function to agree with any function
$F:[\kappa,\lambda]\cap\REG\to\CARD$ satisfying $\forall\alpha,\beta\in\dom(F)$
$\alpha<\cf(F(\alpha))$ and $\alpha<\beta$ $\implies$ $F(\alpha)\leq...

In 1982 I. Pitowsky used Continuum Hypothesis to construct hidden variable
models for spin-1/2 and spin-1 particles in quantum mechanics. We show that the
existence of Pitowsky models is independent from ZFC.

We continue the work started in [6] and show that all monotonically normal (in short: MN) spaces are maximally resolvable if and only if all uniform ultrafilters are maximally decomposable. As a consequence we get that the existence of an MN space which is not maximally resolvable is equi-consistent with the existence of a measurable cardinal. We a...

The term ”square ” refers not just to one but to an entire family of combinatorial principles. The strongest is denoted by ”◻ ” or by ”Global ◻, ” and there are many interesting weakenings of this notion. Before introducing any particular square principle, we provide some motivating applications. In this section, the term ”square ” will serve as a...

We show that, assuming the consistency of a supercompact cardinal, the first (weakly) inaccessible cardinal can satisfy a strong form of a Löwenheim-Skolem-Tarski theorem for the equicardinality logic L(I), a logic introduced in [K. Härtig, “Über einen Quantifikator mit zwei Wirkungsbereichen”, in: Colloq. Foundations Math. Math. Machines Appl., Ti...

We analyse the influence of the forcing axiom Martin's Maximum on the existence of square sequences, with a focus on the weak square principle λ,μ .

This stimulating workshop exposed some of the most exciting recent develops in set theory, including major new results about the proper forcing axiom, stationary reflection, gaps in P(ω)/Fin, iterated forcing, the tree property, ideals and colouring numbers, as well as important new applications of set theory to C * -algebras, Ramsey theory, measur...

ContentsI Introduction to inner models 2by William J. Mitchell1 The constructible sets . . . . . . . . . . . . . . . . . . . . . . . . 31.1 Relative constructibility . . . . . . . . . . . . . . . . . . . 41.2 Measurable Cardinals . . . . . . . . . . . . . . . . . . . . 51.3 0, and sharps in general . . . . . . . . . . . . . . . . . . 91.4 Other sha...

Following the work of Gödel and Cohen we now know that it is impossible to determine the exact value of the continuum or of the power set of an arbitrary cardinal. Despite the limitations that the consistency methods impose, the quest for absolute answers as to the value of the power set (especially of singular cardinals) continues, and sometimes w...

There 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 larg...

The main aim of fine structure theory and inner model theory can be summarized as the construction of models which have a canonical inner structure (a fine structure), making it possible to analyze them in great de- tail, and which at the same time reflect important aspects of the surrounding mathematical universe, in that they satisfy certain stro...

This workshop presented recent advances in fine structure and inner model theory. There were extended tutorials on hod mice and the Mouse Set Conjecture, suitable extender sequences and their fine structure, and the construction of true K below a Woodin cardinal in ZFC. The remaining talks involved precipitous ideals, stationary set reflection, fai...

We construct a model without precipitous ideals but so that for each ¿ < @3 there is a normal ideal over @1 with generic ultrapower wellfounded up to the image of ¿.

We prove a number of consistency results complementary to the ZFC results from our paper (4). We produce examples of non-tightly sta- tionary 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 alterna...

This meeting was organised by Sy-David Friedman (University of Vienna), Menachem Magidor (Hebrew University, Jerusalem) and Hugh Woodin (University of California, Berkeley). Largely due to the generous EU support for the meeting, we were able to invite an unusually large number of young researchers (at most 10 years after the beginning of doctoral...

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...

This note proves two theorems. The first is that it is consistent to have for every n , but not have . This is done by carefully collapsing a supercompact cardinal and adding square sequences to each ω n . The crux of the proof is that in the resulting model every stationary subset of ℵ ω +1 ⋂ cof( ω ) reflects to an ordinal of cofinality ω 1 , tha...

Many systems that exhibit nonmonotonic behavior have been described and studied already in the literature. The general notion of nonmonotonic reasoning, though, has almost always been described only negatively, by the property it does not enjoy, i.e. monotonicity. We study here general patterns of nonmonotonic reasoning and try to isolate propertie...

If ω
n
has the tree property for all 2 ≤ n < ω and , then for all and n < ω. M
n
t(X) exists.

How should time be represented in models for inter-process communication? The global-time axiom implies that all events can be represented by intervals on one time-axis. Its use simplifies the analysis of protocols and allows for intuitive proofs of their properties. On the other hand,some researchers believe it is too strong an assumption which sh...

We investigate here the question of finding the minimal requirements for the registers used by n processes that solve the critical-section problem. For two processes, we show that there cannot be a solution to the critical-section problem if the two registers used are regular and of size 2 and 3. For n processes, this result generalizes to show the...

We present a generalization of the temporal propositional logic of linear time which is useful for stating and proving properties of the generic execution sequence of a parallel program or a non-deterministic program. The formal system we present is exactly that same as the third of three logics presented in [LS82], but we give it a different seman...

In this paper we propose a new form of polymorphism for object-oriented languages, called correspondence polymorphism. It lies in a different dimension than either parametric or subtype polymorphism. In correspondence polymorphism, some methods are declared to correspond to other methods, via a correspondence relation. With this relation, it is pos...

A vast and interesting family of natural semantics for belief revision is defined. Suppose one is given a distance d between any two models. One may then define the revision of a theory K by a formula ff as the theory defined by the set of all those models of ff that are closest, by d, to the set of models of K. This family is characterized by a se...

In this paper we prove the independence of $\delta^1_n$ for n $\geq$ 3. We show that $\delta^1_4$ can be forced to be above any ordinal of L using set forcing. For $\delta^1_3$ we prove that it can be forced, using set forcing, to be above any L cardinal $\kappa$ such that $\kappa$ is $\Pi_1$ definable without parameters in L. We then show that $\d...

In this paper we prove the independence of δ1n for n ≥ 3. We show that δ14 can be forced to be above any ordinal of L using set forcing. For δ13 we prove that it can be forced, using set forcing, to be above any L cardinal κ such that κ is Π1 definable without parameters in L. We then show that δ13 cannot be forced by a set forcing to be above ever...

We prove the consistency with ZFC of ``the length of an ultraproduct of Boolean algebras is smaller than the ultraproduct of the lengths''. Similarly for some other cardinal invariants of Boolean algebras.

In this paper we explicate a very weak version of the principle □ discovered by Jensen who proved it holds in the constructible universe L . This principle is strong enough to include many of the known applications of □, but weak enough that it is consistent with the existence of very large cardinals. In this section we show that this principle is...

We propose two new constructs for object oriented programming that significantly increase polymorphism. Consequently, code may be reused in ways unaccounted for by existing machinery. These constructs of type correspondence and partial inheritance are motivated from metaphors of natural language and thought. They establish correspondences between t...

In this paper we consider whether has “enough information” to contain a counterexample to the continuum hypothesis. We believe this question provides deep insight into the difficulties surrounding the continuum hypothesis. We show sufficient conditions for not to contain such a counterexample. Along the way we establish many results about nonstatio...

Starting from cardinals κ < λ where κ is 2λ supercompact and λ > κ is measurable, we construct a model for the theory “ZF + ∀n < ω[DCℵn] + ℵω + 1 is a measurable cardinal”. This is the maximum amount of dependent choice consistent with the measurability of ℵω + 1, and by a theorem of Shelah using p.c.f. theory, is the best result of this sort possi...

Assuming some large cardinals, a model of ZFC is obtained in which aleph_{omega+1} carries no Aronszajn trees. It is also shown that if lambda is a singular limit of strongly compact cardinals, then lambda^+ carries no Aronszajn trees.

We show that the construction of an almost free nonfree Abelian group can be pushed from a regular cardinal κ to ℵκ + 1. Hence there are unboundedly many almost free nonfree Abelian groups below the first cardinal fixed point. We give a sufficient condition for "κ free implies free", and then we show, assuming the consistency of infinitely many sup...

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 = ℵ 0 , and 2 κ > 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 = ℵ 0 and pp ( μ ) > λ...

Using the consistency of some large cardinals we produce a model of Set Theory in which the generalized continuum hypothesis holds and for some torsion-free abelian group G of cardinality aleph_{omega +1} and for some torsion group T, Bext^2(G,T) not =0 .

We show that in the constructible universe, the two usual definitions of Butler groups are equivalent for groups of arbitrarily
large power. We also prove that Bext2(G, T) vanishes for every torsion-free groupG and torsion groupT. Furthermore, balanced subgroups of completely decomposable groups are Butler groups. These results have been known, und...

This paper presents a logical approach to nonmonotonic reasoning based on the notion of a nonmonotonic consequence relation. A conditional knowledge base, consisting of a set of conditional assertions of the type if … then …, represents the explicit defeasible knowledge an agent has about the way the world generally behaves. We look for a plausible...

This paper presents a logical approach to nonmonotonic reasoning based on the notion of a nonmonotonic consequence relation. A conditional knowledge base, consisting of a set of conditional assertions of the type if … then …, represents the explicit defeasible knowledge an agent has about the way the world generally behaves. We look for a plausible...

We prove that König's duality theorem for infinite graphs (every graph G has a matching F such that there is a selection of one vertex from each edge in F which forms a cover of G) is inherently of very high complexity in terms of both the methods of proof it requires and the computational complexity of the covers it produces. In particular, we sho...

We introduce a generalization of the Baire property for sets of reals via the notion that a set of reals is universally Baire. We show that the universally Baire sets can be characterized in terms of their possible Souslin representations and that in the presence of large cardinals every universally Baire set is determined. We also study the connec...

Cardinal arithmetic had been one of the central themes in set theory, but in the late 60’s and early 70’s, it seemed that there were actually very few theorems that could be proved about cardinal arithmetic. For example, except for some trivial facts, the behavior of cardinal exponentiation (which is the only non-trivial operation in cardinal arith...

We show that there are stationary subsets of uncountable spaces which do not reflect.

This is a survey paper giving a self-contained account of Shelah's theory of the pcf function pcf(a)={cf(Πa/D, <D):D is an ultrafilter on a}, where a is a set of regular cardinals such that |a|<min(a). We also give several applications of the theory to cardinal arithmetic, the existence of Jonsson algebras, and partition calculus.

Many systems that exhibit nonmonotonic behavior have been described and studied already in the literature. The general notion of nonmonotonic reasoning, though, has almost always been described only negatively, by the property it does not enjoy, i.e. monotonicity. We study here general patterns of nonmonotonic reasoning and try to isolate propertie...

We establish, starting from some assumptions of the order of magnitude of a huge cardinal, the consistency of (ℵω+1,ℵω)↠(ω1,ω0), as well as of some other transfer properties of the type (κ+,κ)↠(α+,α), where κ is singular.

Let S be a Steinitz Exchange System with closure operation cl. We examine the logical complexity of the first-order theory of the lattice of cl-closed subsets of S. We show that in many cases the first-order theory of the lattice has the complexity of full second-order logic on S. The strongest results are in the special case where S is an algebrai...

Abstract Suppose a knowledge,base contains information,on how,the world generally behaves and in particular contains the information that birds, normally fly. Suppose that we obtain the information,that Tweety is a bird, why should we conclude that it is plausible that Tweety flies? The answer,to this question is unexpectedly,sophisticated since th...

We prove the following theorems. Theorem 1 ( ¬ 0 # ) (\neg {0^\# }) . Every set of ordinals which is closed under primitive recursive set functions is a countable union of sets in L L . Theorem 2. (No inner model with an Erdàs cardinal, i.e. κ → ( ω 1 ) > ω \kappa \to {({\omega _1})^{ > \omega }} .) For every ordinal β \beta , there is in K K an al...

We prove, assuming the existence of a huge cardinal, the consistency of fully non-regular ultrafilters on the successor of any regular cardinal. We also construct ultrafilters with ultraproducts of small cardinality. Part II is logically independent of Part I.

The authors present a provably strongest form of Martin's axiom, called Martin's Maximum, and show its consistency. From it we derive the solutions to several classical problems in set theory, showing that 2ℵ 0 = ℵ2, the non-stationary ideal on ω1 is ℵ2-saturated, and several other results. We show as a consequence of our techniques that there can...

We show that relative to the consistency of a supercompact cardinal does not imply . The model-theoretic transfer property ⟨ℵ1, ℵ0⟩ → ⟨ℵ
ω + 1, ℵ
ω
⟩ does not imply , and it is consistent to have an ultrafilter on ℵ
ω + 1 which is λ-indecomposible for all ω < λ < ℵ
ω
.

It has been considered desirable by many set theorists to find maximality properties which state that the universe has in some sense “many sets”. The properties isolated thus far have tended to be consistent with each other (as far as we know). For example it is a widely held view that the existence of a supercompact cardinal is consistent with the...

## Projects

Project (1)