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Menachem Magidor

Publications and source records attributed to Menachem Magidor.

At least 19 recordsLinked to original sources
arXiv2025

New inner models from second order logics

We define a new inner model C2(omega) based on the fragment of second order logic in which second order variables range over countable subsets of the domain. We compare C2(omega) to the previously studied inner model C(aa). We argue that C2(omega) appears to be a much bigger inner model than C(aa), although this cannot be literally true in ZFC alone. However, we conjecture that it follows from large cardinal assumptions. For example, assuming large cardinals, C2(omega) contains, for every n, an inner model with n Woodin cardinals, while C(aa) contains, under the same assumption, no inner model with a Woodin cardinal. As to large cardinals in C2(omega), we show that, assuming a Woodin limit of Woodin cardinals, the cardinal omega_1 of V is Mahlo in C2(omega). A stronger result is proved for the combination C2(omega, aa) of C(aa) and C2(omega). We also show that the question whether HOD1, a variant of HOD, is the same as HOD cannot be decided on the basis of ZFC even if we add the assumption that there are supercompact cardinals.

math.LO↗
arXiv2025

The tree property on long intervals of regular cardinals

In this paper we prove that the tree property can hold on regular cardinals in an interval which overlaps a strong limit cardinal. This is a crucial milestone in the long term project, tracing back to a question raised by Foreman and Magidor in the 1980s, of obtaining the tree property at every regular cardinal above the first uncountable cardinal.

math.LO↗
arXiv2024

Inner Models from Extended Logics: Part 2

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 that we call club determinacy, which simplifies the construction of $C(aa)$ greatly but may have also independent interest. Based on club determinacy, we introduce the concept of aa-mouse which we use to prove CH and other properties of the inner model $C(aa)$.

math.LO↗
arXiv2023

Games with Filters

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 $ω$ on $κ$ is equivalent to weak compactness. Winning the game of length $2^κ$ is equivalent to $κ$ being measurable. We show that for games of intermediate length $γ$, II winning implies the existence of precipitous ideals with $γ$-closed, $γ$-dense trees. The second part shows the first is not vacuous. For each $γ$ between $ω$ and $κ^+$, it gives a model where II wins the games of length $γ$, but not $γ^+$. The technique also gives models where for all $ω_1< γ\leκ$ there are $κ$-complete, normal, $κ^+$-distributive ideals having dense sets that are $γ$-closed, but not $γ^+$-closed.

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arXiv2023

Aronszajn trees and maximality

Assuming the consistency of a weakly compact cardinal above a regular uncountable cardinal $\mu$, we prove the consistency of the existence of a wide $\mu^+$-Aronszajn tree, i.e. a tree of height and cardinality $\mu^+$ with no branches of length $\mu^+$, into which every wide $\mu^+$-Aronszajn tree can be embedded.

math.LO↗
arXiv2022

Model Theoretic Characterizations of Large Cardinals Revisited

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.

math.LO↗
arXiv2020

Inner Models from Extended Logics: Part 1

If we replace first order logic by second order logic in the original definition of Gödel'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 generalized quantifiers, such as the Magidor-Malitz quantifier, the cofinality quantifier, or stationary logic. Our first set of results show that both $L$ and HOD manifest some amount of {\em formalism freeness} in the sense that they are not very sensitive to the choice of the underlying logic. Our second set of results shows that the cofinality quantifier gives rise to a new robust inner model between $L$ and HOD. We show, among other things, that assuming a proper class of Woodin cardinals the regular cardinals $>\aleph_1$ of $V$ are weakly compact in the inner model arising from the cofinality quantifier and the theory of that model is (set) forcing absolute and independent of the cofinality in question. We do not know whether this model satisfies the Continuum Hypothesis, assuming large cardinals, but we can show, assuming three Woodin cardinals and a measurable above them, that if the construction is relativized to a real, then on a cone of reals the Continuum Hypothesis is true in the relativized model.

math.LO↗
arXiv2020

Corson reflections

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 of unital abelian \cstar-algebras.

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arXiv2019

Identity crises between supercompactness and Vopenka's Principle

In this paper we study the notion of $C^{(n)}$-supercompactness introduced by Bagaria in \cite{Bag} 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 $C^{(n)}$-supercompact). Furthermore, we prove under suitable hypothesis that the ultimate identity crises is also possible. These results solve several questions posed by Bagaria and Tsaprounis.

math.LO↗
arXiv2019

Destructibility of the tree property at $\aleph_{ω+1}$

We construct a model in which the tree property holds in $\aleph_{ω+ 1}$ and it is destructible under $\text{Col}(ω, ω_1)$. On the other hand we discuss some cases in which the tree property is indestructible under small or closed forcings.

math.LO↗
arXiv2018

On Boolean algebras with strictly positive measures

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 measure. This result is essentially due to Farah and Velickovic.

math.LO↗
arXiv2017

Omitting types in logic of metric structures

This paper is about omitting types in logic of metric structures introduced by Ben Yaacov, Berenstein, Henson and Usvyatsov. While a complete type is omissible in some model of a countable complete theory if and only if it is not principal, this is not true for the incomplete types by a result of Ben Yaacov. We prove that there is no simple test for determining whether a type is omissible in a model of a theory $T$ in a countable language. More precisely, we find a theory in a countable language such that the set of types omissible in some of its models is a complete $Σ^1_2$ set and a complete theory in a countable language such that the set of types omissible in some of its models is a complete $Π^1_1$ set. Two more unexpected examples are given: (i) a complete theory $T$ and a countable set of types such that each of its finite sets is jointly omissible in a model of $T$, but the whole set is not and (ii) a complete theory and two types that are separately omissible, but not jointly omissible, in its models.

math.LO↗
arXiv2016

When an Equivalence Relation with All Borel Classes will be Borel Somewhere?

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 $\mathbfΔ_1^1$ classes and every $σ$-ideal $I$ on $\mathbb{R}$ so that the associated forcing $\mathbb{P}_I$ of $I^+$ $\mathbfΔ_1^1$ subsets is proper, there exists some $I^+$ $\mathbfΔ_1^1$ set $C$ so that $E \upharpoonright C$ is a $\mathbfΔ_1^1$ equivalence relation. In $\mathsf{ZF} + \mathsf{DC} + \mathsf{AD}_\mathbb{R} + V = L(\mathscr{P}(\mathbb{R}))$, for every equivalence relation $E$ on $\mathbb{R}$ with all $\mathbfΔ_1^1$ classes and every $σ$-ideal $I$ on $\mathbb{R}$ so that the associated forcing $\mathbb{P}_I$ is proper, there is some $I^+$ $\mathbfΔ_1^1$ set $C$ so that $E \upharpoonright C$ is a $\mathbfΔ_1^1$ equivalence relation.

math.LO↗
arXiv2016

On properties of compacta that do not reflect in small continuous images

Assuming that there is a stationary set in $ω_{2}$ of ordinals of countable cofinality that does not reflect, we prove that there exists a compact space which is not Corson compact and whose all continuous images of weight at most $ω_1$ are Eberlein compacta. This yields an example of a Banach space of density $ω_{2}$ which is not weakly compactly generated but all its subspaces of density $ω_{1}$ are weakly compactly generated. We also prove that under Martin's axiom countable functional tightness does not reflect in small continuous images of compacta.

math.GN↗
arXiv2016

A Framework for Forcing Constructions at Successors of Singular Cardinals

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 application, we prove the consistency (relative to that of ZFC plus a supercompact cardinal) of there being a strong limit singular cardinal $κ$ of uncountable cofinality where SCH fails and for which there is a collection of graphs on $κ^+$ whose size is less than $2^κ$ and such that any graph on $κ^+$ embeds into one of the graphs in the collection.

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