Gabriel Goldberg

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.

In the first part of the manuscript, we establish several consistency results concerning Woodin's HOD hypothesis and large cardinals around the level of extendibility. First, we prove that the first extendible cardinal can be the first strongly compact in HOD. We extend a former result of Woodin by showing that under the HOD hypothesis the first extendible cardinal is C⁽¹⁾-supercompact in HOD. We also show that the first cardinal-correct extendible may not be extendible, thus answering a question by Gitman and Osinski. In the second part of the manuscript, we discuss the extent to which weak covering can fail below the first supercompact cardinal δ in a context where the HOD hypothesis holds. Answering a question of Cummings et al., we show that under the HOD hypothesis there are many singulars κ < δ where cf^HOD(κ) = cf(κ) and κ^+HOD = κ⁺. In contrast, we also show that the HOD hypothesis is consistent with δ carrying a club of HOD-regular cardinals κ such that κ^+HOD < κ⁺. Finally, we close the manuscript with a discussion about the HOD hypothesis and ω-strong measurability.

In the 1990s, Steel and Woodin showed that under large cardinal hypotheses, the HOD of L(ℝ) admits a fine-structural analysis. Although this theorem sheds light on various problems in descriptive set theory, the fine-structural representations of many fundamental objects of determinacy theory are still unknown. For example, Woodin asked whether the ultrapower of HOD by the closed unbounded filter on ω₁ is given by an iteration tree on HOD according to its fine-structural extender sequence and canonical iteration strategy. In this paper, we give a positive answer to Woodin's question, not only for the closed unbounded filter but for any ultrafilter on an ordinal. The key tool that enables the solution of Woodin's problem is a recent advance in inner model theory: the Steel–Schlutzenberg theory of normalizing iteration trees, which allows us to represent HOD and its ultrapowers as normal iterates of a single countable mouse. Despite our results, the precise structure of the iteration trees that lead from HOD into its ultrapowers remains a mystery.

We settle a question of Woodin motivated by the philosophy of potentialism in set theory. A Σ₂-sentence φ is possible if for any ordinal α, φ holds in a forcing extension W of the universe of sets V such that V and W contain the same sets of rank less than α. We show that it is consistent relative to a supercompact cardinal that every possible sentence is true; this is Woodin's Σ₂-potentialist principle. We accomplish this by generalizing Gitik's method of iterating distributive forcings by embedding them into Prikry-type forcings. Finally, using the concept of mutual stationarity, we show that the Σ₂-potentialist principle implies the consistency of a Woodin cardinal.

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.

Under large cardinal hypotheses beyond the Kunen inconsistency — hypotheses so strong as to contradict the Axiom of Choice — we solve several variants of the generalized continuum problem and identify structural features of the levels V_α of the cumulative hierarchy of sets that are eventually periodic, alternating according to the parity of the ordinal α. For example, if there is an elementary embedding from the universe of sets to itself, then for sufficiently large ordinals α, the supremum of the lengths of all wellfounded relations on V_α is a strong limit cardinal if and only if α is odd.

We prove two compactness theorems for HOD. First, if κ is a strong limit singular cardinal with uncountable cofinality and for stationarily many δ<κ, (δ⁺)^HOD=δ⁺, then (κ⁺)^HOD=κ⁺. Second, if κ is a singular cardinal with uncountable cofinality and stationarily many δ<κ are singular in HOD, then κ is singular in HOD. We also discuss the optimality of these results and show that the first theorem does not extend from HOD to other ω-club amenable inner models.

Stationary logic is the extension of first-order logic with a quantifier expressing that "almost all" countable subsets of a structure have a given property. This paper studies C(aa), the smallest model of ZF containing the ordinal numbers and closed under the satisfaction predicate for stationary logic. This model was constructed by Kennedy–Magidor–Väänänen as a generalization of Gödel's constructible universe L, obtained by iterating definability in stationary logic rather than first-order logic. Unlike L, however, C(aa) can contain large cardinals far beyond a measurable cardinal. We show in this paper that nevertheless, assuming large cardinals in V, the inner model C(aa) shares many of the nice properties of L. In particular, we prove that C(aa) satisfies the Generalized Continuum Hypothesis, the Ultrapower Axiom, and the axiom V = HOD.

We continue the study of the Galvin property, deepening the connection between certain diamond-like principles and non-Galvin ultrafilters. We also show that any Dodd sound non p-point ultrafilter is non-Galvin. We use these ideas to formulate what appears to be the optimal large cardinal hypothesis implying the existence of a non-Galvin ultrafilter, improving on a result of Benhamou–Dobrinen. Finally, we use a strengthening of the Ultrapower Axiom to prove that in all the known canonical inner models, a κ-complete ultrafilter has the Galvin property if and only if it is an iterated sum of p-points.

Much of the theory of large cardinals beyond a measurable cardinal concerns the structure of elementary embeddings of the universe of sets into inner models. This paper seeks to answer the question of whether the inner model uniquely determines the elementary embedding.

An elementary embedding j:M→N between two inner models of ZFC is cardinal preserving if M and N correctly compute the class of cardinals. We look at the case N=V and show that there is no nontrivial cardinal preserving elementary embedding from M into V, answering a question of Caicedo.

We give new, purely combinatorial characterizations of several kinds of large cardinals, such as strongly C⁽ⁿ⁾-compact and C⁽ⁿ⁾-extendible, in terms of reflecting measures. We then study the key property of tightness of elementary embeddings that witness strong C⁽ⁿ⁾-compactness, which prompts the introduction of the new large-cardinal notion of tightly C⁽ⁿ⁾-compact cardinal. Then we prove, assuming the Ultrapower Axiom, that a cardinal is tightly C⁽ⁿ⁾-compact if and only if it is either C^(n-1)-extendible or a measurable limit of C^(n-1)-extendible cardinals. In the last section, we also give new characterizations of Σ_n-strong cardinals in terms of reflecting extenders.

Answering a question of Usuba, we show that an extendible cardinal can be preserved by a set forcing that is not a small forcing.

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 omega-Jonsson algebra. We show that if the HOD hypothesis holds and HOD satisfies the Ultrapower Axiom, then every supercompact cardinal is supercompact in HOD.

We prove that if there is an elementary embedding from the universe to itself, then there is a proper class of measurable successor cardinals.

Assuming choiceless large cardinal axioms, we show that for sufficiently large ordinals λ, the structure of V_λ depends on the parity of λ.

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.

This paper establishes a conjecture of Steel regarding the structure of elementary embeddings from a level of the cumulative hierarchy into itself. Steel’s question is related to the Mitchell order on these embeddings, which has been studied by Laver, Neeman, Steel, and others. Although this order is known to be illfounded, Steel conjectured that it has certain large wellfounded suborders, which is what we establish. The proof relies on a simple and general analysis of the much broader class of extender embeddings and a variant of the Mitchell order called the internal relation.

We prove the consistency of the theory ZFC + there is a strongly compact cardinal from the existence of a cardinal preserving embedding from the universe into an inner model. The proof almost shows that under SCH, every cardinal preserving embedding preserves the cofinality and continuum functions.

A cardinal is weakly Reinhardt if it is the critical point of an elementary embedding from the universe of sets into a model that contains the double powerset of every ordinal. This note establishes the equiconsistency of a proper class of weakly Reinhardt cardinals with a proper class of Reinhardt cardinals in the context of second-order set theory without the Axiom of Choice.

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 + I0. By a recent result of Schlutzenberg, an elementary embedding from V_λ+2 to V_λ+2 does not suffice.

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​-HOD consisting of sets that are hereditarily ordinal definable from a κ​-complete ultrafilter over an ordinal; κ​-HOD seems to be the first nontrivial example of a ground of V whose definition does not involve forcing.

We prove that the Generalized Continuum Hypothesis holds above a supercompact cardinal assuming the Ultrapower Axiom, an abstract comparison principle motivated by inner model theory at the level of supercompact cardinals.

We study a partial order on countably complete ultrafilters introduced by Ketonen as a generalization of the Mitchell order. The following are our main results: the order is wellfounded; its linearity is equivalent to the Ultrapower Axiom, a principle introduced in the author’s dissertation; finally, assuming the Ultrapower Axiom, the Ketonen order coincides with Lipschitz reducibility in the sense of generalized descriptive set theory.

We study the structure of the Rudin-Frolík 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, one can prove much more about the Rudin-Frolík order than is possible in ZFC alone. Our main theorem is that under the Ultrapower Axiom, a countably complete ultrafilter has at most finitely many predecessors in the Rudin-Frolík order. In other words, any wellfounded ultrapower (of the universe) is the ultrapower of at most finitely many ultrapowers.

We show from an abstract comparison principle (the Ultrapower Axiom) that the Mitchell order is linear on sufficiently strong ultrafilters: normal ultrafilters, Dodd solid ultrafilters, and assuming GCH, generalized normal ultrafilters. This gives a conditional answer to the well-known question of whether a 2^κ-supercompact cardinal κ must carry more than one normal measure of order 0. Conditioned on a very plausible iteration hypothesis, the answer is no, since the Ultrapower Axiom holds in the canonical inner models at the finite levels of supercompactness.

Notes from a summer school on UA.

Notes from the Vienna Inner Model Theory conference, including results with David Casey on weak covering in HOD.

Notes from the Irvine Inner Model Theory conference, including a sketch of Woodin's proof that UA holds in the HOD of a natural model of real game determinacy.

We discuss some mathematical results indicating a tension between large cardinal axioms and maximality principles such as forcing axioms and the Axiom of Choice. Topics will include the failure of the Ground Axiom in natural models, the optimality of Usuba's theorem, the analogy between large cardinal axioms and determinacy principles, and the role of forcing in theory selection.

In his work on the projective ordinals, Jackson initiated a detailed analysis of the structure of the cardinal numbers in the inner model L(R). To complete this analysis seems to require the development of a global theory of measures on ordinals in L(R). We present some results in this direction which are inspired by the unexplained analogy between the Axiom of Determinacy and the strongest large cardinal hypotheses.

I'll prove that if there is a Reinhardt cardinal, there is a proper class of regular cardinals and discuss further results on measurable cardinals.

The program to build canonical models for large cardinal axioms has hit an impasse below the level of supercompact cardinals. The Ultrapower Axiom (UA), a simple combinatorial principle that is expected to hold in all canonical models, enables one to develop a detailed structure theory for supercompact cardinals, apparently providing the first glimpse of the canonical models at this level. Assuming they exist, canonical models with supercompact cardinals are expected to exert a strong influence on the structure of the universe of sets itself, motivating the prediction that certain consequences of UA are actually provable outright in ZFC or from large cardinal axioms. This talk surveys some attempts to confirm this prediction.

Sample results: any two elementary embeddings from the universe of sets to the same inner model agree on the ordinals (a conjecture of Woodin); any regular cardinal above the first strongly compact that carries a uniform indecomposable ultrafilter is measurable (a question of Silver); assuming a proper class of strongly compact cardinals, there is no elementary embedding from the universe into a cardinal correct inner model (a conjecture of Caicedo); if κ is extendible, the ultrapower of the universe by a κ⁺-complete ultrafilter admits no non-trivial self-embeddings (a question of van Name); the Mitchell order is wellfounded on elementary embeddings from V_λ to V_λ with critical points bounded strictly below λ (a conjecture of Steel). Applications to countably incomplete ultrafilters, set-theoretic geology, the HOD conjecture, and the theory of large cardinals beyond choice will also be discussed.

I'll discuss some ideas relating strong compactness, club filters, and inner models, with two applications: first, a generalization of Woodin's HOD dichotomy that applies to any inner model with access to the ω-club filter on each ordinal of uncountable cofinality, and second, an analysis of strong compactness in the HODs of models satisfying either the Axiom of Determinacy or choiceless large cardinal assumptions.

Woodin's HOD conjecture asserts that in the context of very large cardinals, the inner model HOD closely approximates the universe of sets in the same way Gödel's constructible universe does assuming 0^# does not exist. The subject of these two talks is the relationship between Woodin's conjecture and certain constraints on the structure of elementary embeddings of the universe of sets. For example, in the second talk, we will prove that any two elementary embeddings of the universe of sets into the same inner model agree on HOD, while if a local version of this theorem held, then the HOD conjecture would follow.

Roughly speaking, Woodin's HOD conjecture asserts that assuming the existence of very large cardinals, arbitrary sets can be closely approximated by definable ones. This talk outlines an approach to the conjecture based on an analysis of the uniqueness properties of ultrafilters and elementary embeddings, which has a number of applications: for example, a proof of a variant of the HOD conjecture for sets definable from ultrafilters, a proof of Woodin's HOD dichotomy theorem from a single strongly compact cardinal, and a proof that past an extendible cardinal, elementary embeddings of the universe of sets are uniquely determined by their codomains.

Jensen's covering lemma states that either every uncountable set of ordinals is covered by a constructible set of ordinals of the same size or else there is an elementary embedding from the constructible universe to itself. This talk takes up the question of whether there could be an analog of this theorem with constructibility replaced by ordinal definability. For example, we answer a question posed by Woodin: assuming the HOD conjecture and a strongly compact cardinal, there is no nontrivial elementary embedding from HOD to HOD.

The Ultrapower Axiom is a structural principle for ultrafilters expected to hold in the canonical inner model for a supercompact cardinal if it exists. The existence of such a model should also imply the HOD Conjecture, which asserts that under large cardinal hypotheses, HOD correctly computes successors of singular cardinals. The proof that the HOD Conjecture follows from UA motivates the study of definability from ultrafilters under large cardinal assumptions. Combined with techniques from set theoretic geology, this investigation leads to some new structural properties of large cardinals related to both UA and the HOD Conjecture. For example, while it is consistent that there is an inner model M ​​that admits 2^2κ​​-many elementary embeddings j : V → M​​ where κ​​ is the least measurable cardinal, we show that if there is an extendible cardinal δ​​ and a proper class of strongly compact cardinals, then any inner model M​​ admits at most one elementary embedding j : V → M​​ with critical point greater than δ​​.

The Burali-Forti paradox suggests that the transfinite cardinals “go on forever,” surpassing any conceivable bound one might try to place on them. The traditional Zermelo-Frankel axioms for set theory fall into a hierarchy of axiomatic systems formulated by reasserting this intuition in increasingly elaborate ways: the large cardinal hierarchy. Or so the story goes. A serious problem for this already naive account of large cardinal set theory is the Kunen inconsistency theorem, which seems to impose an upper bound on the extent of the large cardinal hierarchy itself. If one drops the Axiom of Choice, Kunen’s proof breaks down and a new hierarchy of choiceless large cardinal axioms emerges. These axioms, if consistent, represent a challenge for those “maximalist” foundational stances that take for granted both large cardinal axioms and the Axiom of Choice. This talk concerns some recent advances in our understanding of the weakest of the choiceless large cardinal axioms and the prospect, as yet unrealized, of establishing their consistency and reconciling them with the Axiom of Choice.

This talk will explore the connection between Woodin's HOD Conjecture and certain uniqueness properties of elementary embeddings of models of set theory, which will lead to some consequences of the failure of the HOD Conjecture reminiscent of the consequences of choiceless large cardinal axioms.

Countably complete ultrafilters are the combinatorial manifestation of strong large cardinal axioms, but many of their basic properties are undecidable no matter the large cardinal axioms one is willing to adopt. The Ultrapower Axiom (UA) is a set theoretic principle that permits the development of a much clearer picture of countably complete ultrafilters and, consequently, the large cardinals from which they derive. It is not known whether UA is (relatively) consistent with very large cardinals, but assuming there is a canonical inner model with a supercompact cardinal, the answer should be yes: this inner model should satisfy UA and yet inherit all large cardinals present in the universe of sets. The predicted resemblance between the large cardinal structure of this model and that of the universe itself is so extreme as to suggest that certain consequences of UA must in fact be provable outright from large cardinal axioms. While the inner model theory of supercompact cardinals remains a major open problem, this talk will describe a technique that already permits a number of consequences of UA to be replicated from large cardinals alone. Still, the technique rests on the existence of inner models that absorb large cardinals, but instead of building canonical inner models, one takes ultrapowers.

Expanding on ideas due to Hamkins in the context of forcing and Woodin in the context of inner model theory, we prove several theorems which suggest that the large cardinal structure of the universe of sets above a strongly compact cardinal is more tractable than the ubiquity of independence results at this level would suggest.

I outlined the consequences of the Ultrapower Axiom for the theory of supercompactness and strong compactness.

I outlined the consequences of the Ultrapower Axiom for the theory of supercompactness and strong compactness.

I tried to argue for the Ground Axiom on the basis of Usuba's Theorem and the Ultimate L Conjecture on the basis of the Ultrapower Axiom.

The Ultrapower Axiom (UA) is a set theoretic principle motivated by inner model theory. Under UA, it is possible to develop certain parts of large cardinal theory that are intractable under the standard ZFC axioms. Is UA consistent with all large cardinal axioms? Since UA is necessarily compatible with any large cardinal axiom that admits a canonical inner model theory, this question is likely to be intertwined with the longstanding open problem of constructing canonical inner models with very large cardinals. In this talk, we present two natural generalizations of UA that are not motivated by inner model theory. The goal is to refute these principles from large cardinal axioms. We view this as a way of identifying interesting complex structure at key levels of the large cardinal hierarchy. It also serves to corroborate the intuition that the (apparent) consistency of UA with supercompact cardinals is evidence that inner model theory extends to this level.

In 2017, I presented a proof that the Ultrapower Axiom implies GCH above a supercompact cardinal. In 2018, I sketched a proof that the Ultrapower Axiom implies the equivalence of strong compactness and supercompactness (modulo a class of counterexamples identified by Menas in the 1970s).