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Gabriel Goldberg

Publications and source records attributed to Gabriel Goldberg.

35 records · Page 2Linked to original sources
arXiv2021

Even ordinals and the Kunen inconsistency

This paper contributes to the theory of large cardinals beyond the Kunen inconsistency, or choiceless large cardinal axioms, in the context where the Axiom of Choice is not assumed. The first part of the paper investigates a periodicity phenomenon: assuming choiceless large cardinal axioms, the properties of the cumulative hierarchy turn out to alternate between even and odd ranks. The second part of the paper explores the structure of ultrafilters under choiceless large cardinal axioms, exploiting the fact that these axioms imply a weak form of the author's Ultrapower Axiom. The third and final part of the paper examines the consistency strength of choiceless large cardinals, including a proof that assuming DC, the existence of an elementary embedding from $V_{λ+3}$ to $V_{λ+3}$ implies the consistency of ZFC + $I_0$. By a recent result of Schlutzenberg, an elementary embedding from $V_{λ+2}$ to $V_{λ+2}$ does not suffice.

math.LO↗
arXiv2021

On products of ultrafilters

Assuming the Generalized Continuum Hypothesis, this paper answers the question: when is the tensor product of two ultrafilters equal to their Cartesian product? It is necessary and sufficient that their Cartesian product is an ultrafilter; that the two ultrafilters commute in the tensor product; that for all cardinals $λ$, one of the ultrafilters is both $λ$-indecomposable and $λ^+$-indecomposable; that the ultrapower embedding associated to each ultrafilter restricts to a definable embedding of the ultrapower of the universe associated to the other.

math.LO↗
arXiv2020

Periodicity in the cumulative hierarchy

We investigate the structure of rank-to-rank elementary embeddings, working in ZF set theory without the Axiom of Choice. Recall that the levels $V_α$ of the cumulative hierarchy are defined via iterated application of the power set operation, starting from $V_0=\emptyset$, and taking unions at limit stages. Assuming that $j:V_{α+1}\to V_{α+1}$ is a (non-trivial) elementary embedding, we show that the structure of $V_α$ is fundamentally different to that of $V_{α+1}$. We show that $j$ is definable from parameters over $V_{α+1}$ iff $α+1$ is an odd ordinal. Moreover, if $α+1$ is odd then $j$ is definable over $V_{α+1}$ from the parameter $j`` V_α=\{j(x)\bigm|x\in V_α\}$, and uniformly so. This parameter is optimal in that $j$ is not definable from any parameter which is an element of $V_α$. In the case that $α=β+1$, we also give a characterization of such $j$ in terms of ultrapower maps via certain ultrafilters. Assuming $λ$ is a limit ordinal, we prove that if $j:V_λ\to V_λ$ is $Σ_1$-elementary, then $j$ is not definable over $V_λ$ from parameters, and if $β<λ$ and $j:V_β\to V_λ$ is fully elementary and $\in$-cofinal, then $j$ is likewise not definable; note that this last result is relevant to embeddings of much lower consistency strength than rank-to-rank. If there is a Reinhardt cardinal, then for all sufficiently large ordinals $α$, there is indeed an elementary $j:V_α\to V_α$, and therefore the cumulative hierarchy is eventually periodic (with period 2).

math.LO↗
arXiv2020

Some combinatorial properties of Ultimate L and V

This paper establishes a number of constraints on the structure of large cardinals under strong compactness assumptions. These constraints coincide with those imposed by the Ultrapower Axiom, a principle that is expected to hold in Woodin's hypothesized Ultimate \(L\), providing some evidence for the Ultimate \(L\) Conjecture. We show that every regular cardinal above the first strongly compact that carries an indecomposable ultrafilter is measurable, answering a question of Silver for large enough cardinals. We show that any successor almost strongly compact cardinal of uncountable cofinality is strongly compact, making progress on a question of Boney, Unger, and Brooke-Taylor. We show that if there is a proper class of strongly compact cardinals then there is no nontrivial cardinal preserving elementary embedding from the universe of sets into an inner model, answering a question of Caicedo granting large cardinals. Finally, we show that if \(κ\) is strongly compact, then \(V\) is a set forcing extension of the inner model \(κ\text{-HOD}\) consisting of sets that are hereditarily ordinal definable from a \(κ\)-complete ultrafilter over an ordinal; \(κ\text{-HOD}\) seems to be the first nontrivial example of a ground of \(V\) whose definition does not involve forcing.

math.LO↗
arXiv2020

The Ultrapower Axiom

The inner model problem for supercompact cardinals, one of the central open problems in modern set theory, asks whether there is a canonical model of set theory with a supercompact cardinal. The problem is closely related to the more precise question of the equiconsistency of strongly compact cardinals and supercompact cardinals. This dissertation approaches these two problems abstractly by introducing a principle called the Ultrapower Axiom which is expected to hold in all known canonical models of set theory. By investigating the consequences of the Ultrapower Axiom under the hypothesis that there is a supercompact cardinal, we provide evidence that the inner model problem can be solved. Moreover, we establish that under the Ultrapower Axiom, strong compactness and supercompactness are essentially equivalent.

math.LO↗
arXiv2020

Structure theorems above a strongly compact cardinal

While many inner model theoretic combinatorial principles are incompatible with large cardinal axioms, on some rare occasions, large cardinals actually imply that the structure of the universe of sets is analogous to the canonical inner models. This note provides two new instances of this phenomenon.

math.LO↗
arXiv2018

The internal relation

This paper explores various generalizations of the Mitchell order focusing mostly on a generalization called the internal relation. The internal relation lacks the implicit strength requirement in the definition of the Mitchell order, and therefore can fail to be wellfounded. We establish some constraints on the illfoundedness of the internal relation, which leads to a proof of a conjecture of Steel regarding rank-to-rank cardinals.

math.LO↗
arXiv2018

The seed order

We study various orders on countably complete ultrafilters on ordinals that coincide and are wellorders under a hypothesis called the Ultrapower Axiom. Our main focus is on the relationship between the Ultrapower Axiom and the linearity of these orders.

math.LO↗
arXiv2018

The Rudin-Frolik order and the Ultrapower Axiom

We study the structure of the Rudin-Frolik order on countably complete ultrafilters under the assumption that this order is directed. This assumption, called the Ultrapower Axiom, holds in all known canonical inner models. It turns out that assuming the Ultrapower Axiom, much more about the Rudin-Frolik order can be determined. Our main theorem is that under the Ultrapower Axiom, a countably complete ultrafilter has at most finitely many predecessors in the Rudin-Frolik order. In other words, any wellfounded ultrapower (of the universe) is the ultrapower of at most finitely many ultrapowers.

math.LO↗
arXiv2018

Fréchet cardinals

An infinite cardinal $λ$ is called Fréchet if the Fréchet filter on $λ$ extends to a countably complete ultrafilter. We investigate the relationship between Fréchet cardinals and strongly compact cardinals under a hypothesis called the Ultrapower Axiom.

math.LO↗
arXiv2018

The Ultrapower Axiom and the equivalence between strong compactness and supercompactness

The relationship between the large cardinal notions of strong compactness and supercompactness cannot be determined under the standard ZFC axioms of set theory. Under a hypothesis called the Ultrapower Axiom, we prove that the notions are equivalent except for a class of counterexamples identified by Menas. This is evidence that strongly compact and supercompact cardinals are equiconsistent.

math.LO↗
arXiv2018

Compactness and Comparison

Assuming an abstract comparison principle called the Ultrapower Axiom, which is motivated by the comparison process of inner model theory and generalizes the statement that the Mitchell order is linear on normal ultrafilters, we characterize strongly compact cardinals in terms of supercompactness: a strongly compact cardinal is either supercompact or a limit of supercompact cardinals. Assuming the Ultrapower Axiom and the GCH, we also prove a local result that roughly states that every countably complete ultrafilter factors as a finite iteration of supercompact ultrafilters.

math.LO↗
arXiv2017

The Linearity of the Mitchell Order

We show from a weak comparison principle (the Ultrapower Axiom) that the Mitchell order is linear on certain kinds of ultrafilters: normal ultrafilters, Dodd solid ultrafilters, and assuming GCH, generalized normal ultrafilters. In the process we prove a generalization of Solovay's lemma to singular cardinals.

math.LO↗
arXiv2017

The Seed Order

This paper introduces the seed order, a partial order of the class of uniform countably complete ultrafilters that generalizes the Mitchell order on normal measures. Like that order, the seed order is consistently a linear ordering even under strong large cardinal assumptions. In fact, the linearity of the seed order is a feature of all known canonical inner models for large cardinal axioms. We develop the theory of the seed order under the assumption that it is linear. Augmented by large cardinal hypotheses currently out of reach of inner model theory (namely supercompactness), this linearity assumption, which we call the Ultrapower Axiom, has surprisingly strong consequences reminiscent of the structure theory of the canonical inner models.

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