arXiv subjects

Philipp Lücke

Publications and source records attributed to Philipp Lücke.

At least 19 recordsLinked to original sources

arXiv2025

Large cardinals, structural reflection, and the HOD Conjecture

We introduce exacting cardinals and a strengthening of these, ultraexacting cardinals. These are natural large cardinals defined equivalently as weak forms of rank-Berkeley cardinals, strong forms of Jónsson cardinals, or in terms of principles of structural reflection. However, they challenge commonly held intuition on strong axioms of infinity. We prove that ultraexacting cardinals are consistent with Zermelo-Fraenkel Set Theory with the Axiom of Choice (ZFC) relative to the existence of an I0 embedding. However, the existence of an ultraexacting cardinal below a measurable cardinal implies the consistency of ZFC with a proper class of I0 embeddings, thus challenging the linear--incremental picture of the large cardinal hierarchy. We show that the existence of an exacting cardinal implies that V is not equal to HOD (Gödel's universe of Hereditarily Ordinal Definable sets), showing that these cardinals surpass the current hierarchy of large cardinals consistent with ZFC. Moreover, we prove that the existence of an exacting cardinal above an extendible cardinal implies the "V is far from HOD" alternative of Woodin's HOD Dichotomy. In particular, it follows that the consistency of ZFC with an exacting cardinal above an extendible cardinal would refute Woodin's HOD Conjecture and Ultimate-L Conjecture. Finally, we show that the consistency of ZF with certain large cardinals beyond choice implies the consistency of ZFC with the existence of an exacting cardinal above an extendible cardinal.

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arXiv2025

Large cardinals beyond HOD

Exacting and ultraexacting cardinals are large cardinal numbers compatible with the Zermelo-Fraenkel axioms of set theory, including the Axiom of Choice. In contrast with standard large cardinal notions, their existence implies that the set-theoretic universe V is not equal to Gödel's subuniverse of Hereditarily Ordinal Definable (HOD) sets. We prove that the existence of an ultraexacting cardinal is equiconsistent with the well-known axiom I0; moreover, the existence of ultraexacting cardinals together with other standard large cardinals is equiconsistent with generalizations of I0 for fine-structural models of set theory extending $L(V_{λ+1})$. We prove tight bounds on the strength of exacting cardinals, placing them strictly between the axioms I3 and I2. The argument extends to show that I2 implies the consistency of Vopěnka's Principle together with an exacting cardinal and the HOD Hypothesis. In particular, we obtain the following result: the existence of an extendible cardinal above an exacting cardinal does not refute the HOD Hypothesis. We also give several new characterizations of exacting and ultraexacting cardinals; first in terms of strengthenings of the axioms I3 and I1 with the addition of Ordinal Definable predicates, and finally also in terms of principles of Structural Reflection which characterize exacting and ultraexacting cardinals as natural two-cardinal forms of strong unfoldability.

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arXiv2025

Weak compactness cardinals for strong logics and subtlety properties of the class of ordinals

Motivated by recent work of Boney, Dimopoulos, Gitman and Magidor, we characterize the existence of weak compactness cardinals for all abstract logics through combinatorial properties of the class of ordinals. This analysis is then used to show that, in contrast to the existence of strong compactness cardinals, the existence of weak compactness cardinals for abstract logics does not imply the existence of a strongly inaccessible cardinal. More precisely, it is proven that the existence of a proper class of subtle cardinals is consistent with the axioms of ZFC if and only if it is not possible to derive the existence of strongly inaccessible cardinals from the existence of weak compactness cardinals for all abstract logics. Complementing this result, it is shown that the existence of weak compactness cardinals for all abstract logics implies that unboundedly many ordinals are strongly inaccessible in the inner model HOD of all hereditarily ordinal definable sets.

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arXiv2025

Outward compactness

We introduce and study a new type of compactness principle for strong logics that, roughly speaking, infers the consistency of a theory from the consistency of its small fragments in certain outer models of the set-theoretic universe. We refer to this type of compactness property as outward compactness, and we show that instances of this type of principle for second-order logic can be used to characterize various large cardinal notions between measurability and extendibility, directly generalizing a classical result of Magidor that characterizes extendible cardinals as the strong compactness cardinals of second-order logic. In addition, we generalize a result of Makowsky that shows that Vopěnka's Principle is equivalent to the existence of compactness cardinals for all abstract logics by characterizing the principle "Ord is Woodin" through outward compactness properties of abstract logics.

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arXiv2024

Descriptive properties of I2-embeddings

We contribute to the study of generalizations of the Perfect Set Property and the Baire Property to subsets of spaces of higher cardinalities, like the power set $P(λ)$ of a singular cardinal $λ$ of countable cofinality or products $\prod_{i<ω}λ_i$ for a strictly increasing sequence $\langleλ_i ~ \vert ~ i<ω\rangle$ of cardinals. We consider the question under which large cardinal hypotheses classes of definable subsets of these spaces possess such regularity properties, focusing on rank-into-rank axioms and classes of sets definable by $Σ_1$-formulas with parameters from various collections of sets. We prove that $ω$-many measurable cardinals, while sufficient to prove the Perfect Set Property of all $Σ_1$-definable sets with parameters in $V_λ\cup\{V_λ\}$, are not enough to prove it if there is a cofinal sequence in $λ$ in the parameters. For this conclusion, the existence of an I2-embedding is enough, but there are parameters in $V_{λ+1}$ for which I2 is still not enough. The situation is similar for the Baire Property: under I2 all sets that are $Σ_1$-definable using elements of $V_λ$ and a cofinal sequence as parameters have the Baire property, but I2 is not enough for some parameter in $V_{λ+1}$. Finally, the existence of an I0-embedding implies that all sets that are $Σ^1_n$-definable with parameters in $V_{λ+1}$ have the Baire property.

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arXiv2024

On $Σ_1$-Definable Closed Unbounded Sets

Definable stationary sets, and specifically, ordinal definable ones, play a significant role in the study of canonical inner models of set theory and the class HOD of hereditarily ordinal definable sets. Fixing a certain notion of definability and an uncountable cardinal, one can consider the associated family of definable closed unbounded sets. In this paper, we study the extent to which such families can approximate the full closed unbounded filter, and their dependence on the defining complexity. Focusing on closed unbounded subsets of a cardinal $κ$ which are $Σ_1$-definable in parameters from $H_κ$ and ordinal parameters, we show that the ability of such closed unbounded sets to well approximate the closed unbounded filter on $κ$ can highly vary, and strongly depends on key properties of the underlying universe of set theory.

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arXiv2024

The complexity of non-stationary ideals

We present an overview of results on the question of whether the non-stationary ideal of an uncountable regular cardinal $κ$ can be defined by a $Π_1$-formula using parameters of hereditary cardinality at most $κ$. These results show that this question is deeply connected to several central topics of current research in set theory.

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arXiv2023

Huge Reflection

We study Structural Reflection beyond Vopěnka's Principle, at the level of almost-huge cardinals and higher, up to rank-into-rank embeddings. We identify and classify new large cardinal notions in that region that correspond to some form of what we call Exact Structural Reflection ($\mathrm{ESR}$). Namely, given cardinals $κ<λ$ and a class $\mathcal{C}$ of structures of the same type, the corresponding instance of $\mathrm{ESR}$ asserts that for every structure $A$ in $\mathcal{C}$ of rank $λ$, there is a structure $B$ in $\mathcal{C}$ of rank $κ$ and an elementary embedding of $B$ into $A$. Inspired by the statement of Chang's Conjecture, we also introduce and study sequential forms of $\mathrm{ESR}$, which, in the case of sequences of length $ω$, turn out to be very strong. Indeed, when restricted to $Π_1$-definable classes of structures they follow from the existence of $I1$-embeddings, while for more complicated classes of structures, e.g., $Σ_2$, they are not known to be consistent. Thus, these principles unveil a new class of large cardinals that go beyond $I1$-embeddings, yet they may not fall into Kunen's Inconsistency.

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arXiv2023

Patterns of Structural Reflection in the large-cardinal hierarchy

We unveil new patterns of Structural Reflection in the large-cardinal hierarchy below the first measurable cardinal. Namely, we give two different characterizations of strongly unfoldable and subtle cardinals in terms of a weak form of the principle of Structural Reflection, and also in terms of weak product structural reflection. Our analysis prompts the introduction of the new notion of $C^{(n)}$-strongly unfoldable cardinal for every natural number $n$, and we show that these cardinals form a natural hierarchy between strong unfoldable and subtle cardinals analogous to the known hierarchies of $C^{(n)}$-extendible and $Σ_n$-strong cardinals. These results show that the relatively low region of the large-cardinal hierarchy comprised between the first strongly unfoldable and the first subtle cardinals is completely analogous to the much higher region between the first strong and the first Woodin cardinals, and also to the much further upper region of the hierarchy ranging between the first supercompact and the first Vopěnka cardinals.

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arXiv2023

$Σ_1$-definability at higher cardinals: Thin sets, almost disjoint families and long well-orders

Given an uncountable cardinal $κ$, we consider the question of whether subsets of the power set of $κ$ that are usually constructed with the help of the Axiom of Choice are definable by $Σ_1$-formulas that only use the cardinal $κ$ and sets of hereditary cardinality less than $κ$ as parameters. For limits of measurable cardinals, we prove a perfect set theorem for sets definable in this way and use it to generalize two classical non-definability results to higher cardinals. First, we show that a classical result of Mathias on the complexity of maximal almost disjoint families of sets of natural numbers can be generalized to measurable limits of measurables. Second, we prove that for a limit of countably many measurable cardinals, the existence of a simply definable well-ordering of subsets of $κ$ of length at least $κ^+$ implies the existence of a projective well-ordering of the reals. In addition, we determine the exact consistency strength of the non-existence of $Σ_1$-definitions of certain objects at singular strong limit cardinals. Finally, we show that both large cardinal assumptions and forcing axioms cause analogs of these statements to hold at the first uncountable cardinal $ω_1$.

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arXiv2023

Continuous images of closed sets in generalized Baire spaces

Let $κ$ be an uncountable cardinal with $κ=κ^{{<}κ}$. Given a cardinal $μ$, we equip the set ${}^κμ$ consisting of all functions from $κ$ to $μ$ with the topology whose basic open sets consist of all extensions of partial functions of cardinality less than $κ$. We prove results that allow us to separate several classes of subsets of ${}^κκ$ that consist of continuous images of closed subsets of spaces of the form ${}^κμ$. Important examples of such results are the following: (i) there is a closed subset of ${}^κκ$ that is not a continuous image of ${}^κκ$; (ii) there is an injective continuous image of ${}^κκ$ that is not $κ$-Borel (i.e. that is not contained in the smallest algebra of sets on ${}^κκ$ that contains all open subsets and is closed under $κ$-unions); (iii) the statement "every continuous image of ${}^κκ$ is an injective continuous image of a closed subset of ${}^κκ$" is independent of the axioms of $\mathrm{ZFC}$; and (iv) the axioms of $\mathrm{ZFC}$ do not prove that the assumption "$2^κ>κ^+$'' implies the statement "every closed subset of ${}^κκ$ is a continuous image of ${}^κ(κ^+)$'' or its negation.

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arXiv2022

Forcing axioms and the complexity of non-stationary ideals

We study the influence of strong forcing axioms on the complexity of the non-stationary ideal on $ω_2$ and its restrictions to certain cofinalities. Our main result shows that the strengthening $MM^{++}$ of Martin's Maximum does not decide whether the restriction of the non-stationary ideal on $ω_2$ to sets of ordinals of countable cofinality is $Δ_1$-definable by formulas with parameters in $H(ω_3)$. The techniques developed in the proof of this result also allow us to prove analogous results for the full non-stationary ideal on $ω_2$ and strong forcing axioms that are compatible with CH. Finally, we answer a question of S. Friedman, Wu and Zdomskyyshow by showing that the $Δ_1$-definability of the non-stationary ideal on $ω_2$ is compatible with arbitrary large values of the continuum function at $ω_2$.

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arXiv2021

Structural reflection, shrewd cardinals and the size of the continuum

Motivated by results of Bagaria, Magidor and Väänänen, we study characterizations of large cardinal properties through reflection principles for classes of structures. More specifically, we aim to characterize notions from the lower end of the large cardinal hierarchy through the principle $\mathrm{SR}^-$ introduced by Bagaria and Väänänen. Our results isolate a narrow interval in the large cardinal hierarchy that is bounded from below by total indescribability and from above by subtleness, and contains all large cardinals that can be characterized through the validity of the principle $\mathrm{SR}^-$ for all classes of structures defined by formulas in a fixed level of the Lévy hierarchy. Moreover, it turns out that no property that can be characterized through this principle can provably imply strong inaccessibility. The proofs of these results rely heavily on the notion of "shrewd cardinals", introduced by Rathjen in a proof-theoretic context, and embedding characterizations of these cardinals that resembles Magidor's classical characterization of supercompactness. In addition, we show that several important weak large cardinal properties, like weak inaccessibility, weak Mahloness or weak $Π^1_n$-indescribability, can be canonically characterized through localized versions of the principle $\mathrm{SR}^-$. Finally, the techniques developed in the proofs of these characterizations also allow us to show that Hamkin's "weakly compact embedding property" is equivalent to Lévy's notion of weak $Π^1_1$-indescribability.

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arXiv2021

Strong unfoldability, shrewdness and combinatorial consequences

We show that the notions of "strongly unfoldable cardinals", introduced by Villaveces in his model-theoretic studies of models of set theory, and "shrewd cardinals", introduced by Rathjen in a proof-theoretic context, coincide. We then proceed by using ideas from the proof of this equivalence to establish the existence of "ordinal anticipating Laver functions" for strong unfoldability. With the help of these functions, we show that the principle $\Diamond_κ(\mathrm{Reg})$ holds at every strongly unfoldable cardinal $κ$ with the property that there exists a subset $z$ of $κ$ such that every subset of $κ$ is ordinal definable from $z$. While a result of Džamonja and Hamkins shows that $\Diamond_κ(\mathrm{Reg})$ can consistently fail at a strongly unfoldable cardinal $κ$, this implication can be used to prove that various canonical extensions of the axioms of ZFC are either compatible with the assumption that $\Diamond_κ(\mathrm{Reg})$ holds at every strongly unfoldable cardinal $κ$ or outright imply this statement. Finally, we will also use our methods to contribute to the study of strong chain conditions of partials orders and their productivity.

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arXiv2021

Non-absoluteness of Hjorth's Cardinal Characterization

In [5], Hjorth proved that for every countable ordinal $α$, there exists a complete $\mathcal{L}_{ω_1,ω}$-sentence $ϕ_α$ that has models of all cardinalities less than or equal to $\aleph_α$, but no models of cardinality $\aleph_{α+1}$. Unfortunately, his solution does not yield a single $\mathcal{L}_{ω_1,ω}$-sentence $ϕ_α$, but a set of $\mathcal{L}_{ω_1,ω}$-sentences, one of which is guaranteed to work. It was conjectured in [9] that it is independent of the axioms of ZFC which of these sentences has the desired property. In the present paper, we prove that this conjecture is true. More specifically, we isolate a diagonalization principle for functions from $ω_1$ to $ω_1$ which is a consequence of the Bounded Proper Forcing Axiom (BPFA) and then we use this principle to prove that Hjorth's solution to characterizing $\aleph_2$ in models of BPFA is different than in models of CH. In addition, we show that large cardinals are not needed to obtain this independence result by proving that our diagonalization principle can be forced over models of CH.

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arXiv2021

Descriptive properties of higher Kurepa trees

We use generalizations of concepts from descriptive set theory to study combinatorial objects of uncountable regular cardinality, focussing on higher Kurepa trees and the representation of the sets of cofinal branches through such trees as continuous images of function spaces. For different types of uncountable regular cardinals $κ$, our results provide a complete picture of all consistent scenarios for the representation of sets of cofinal branches through $κ$-Kurepa trees as retracts of the generalized Baire space ${}^κκ$ of $κ$. In addition, these results can be used to determine the consistency of most of the corresponding statements for continuous images of ${}^κκ$.

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arXiv2020

Closure properties of measurable ultrapowers

We study closure properties of measurable ultrapowers with respect to Hamkin's notion of "freshness" and show that the extent of these properties highly depends on the combinatorial properties of the underlying model of set theory. In one direction, a result of Sakai shows that, by collapsing a strongly compact cardinal to become the double successor of a measurable cardinal, it is possible to obtain a model of set theory in which such ultrapowers possess the strongest possible closure properties. In the other direction, we use various square principles to show that measurable ultrapowers of canonical inner models only possess the minimal amount of closure properties. In addition, the techniques developed in the proofs of these results also allow us to derive statements about the consistency strength of the existence of measurable ultrapowers with non-minimal closure properties.

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arXiv2020

Small models, large cardinals, and induced ideals

We show that many large cardinal notions up to measurability can be characterized through the existence of certain filters for small models of set theory. This correspondence will allow us to obtain a canonical way in which to assign ideals to many large cardinal notions. This assignment coincides with classical large cardinal ideals whenever such ideals had been defined before. Moreover, in many important cases, relations between these ideals reflect the ordering of the corresponding large cardinal properties both under direct implication and consistency strength.

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