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.
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 sub…
We introduce the notion of weakly extendible cardinals and show that these cardinals are characterized in terms of weak compactness of second order logic. The consistency strength and largeness of weakly extendible cardinals are located strictly between that of strongly unfoldable (i.e. shrewd) cardinals, and strongly uplifting cardinals. Weak compactness of many other logics can be connected to certain variants of the notion of weakly extendible cardinals. We also show that, under V=L, a cardinal $κ$ is the weak compactness number of ${\cal L}^{\aleph_0,II}_{stat,κ,ω}$ if and only if it is th…
Strong reflection principles with the reflection cardinal $\leq\aleph_1$ or $<2^{\aleph_0}$ imply that the size of the continuum is either $\aleph_1$ or $\aleph_2$ or very large. Thus, the stipulation, that a strong reflection principle should hold, seems to support the trichotomy on the possible size of the continuum. In this article, we examine the situation with the reflection principles and related notions of generic large cardinals.
Full reflection of stationary sets at regular cardinals
一个正则不可数基数Kappa的平稳子集S在正则基数上全反射，如果对每个由正则基数组成的高阶平稳集Tsubseteq Kappa，T中存在一个α，使得S cap α是α的平稳子集。我们证明了全反射公理，即每个平稳集在正则基数上全反射，以及n-Mahlo基数的存在性与Pi^1_n-不可描述基数的存在性是等相容的.我们还对极大Mahlo基数给出了适当的推广。
After discussing the limitations inherent to all set-theoretic reflection principles akin to those studied by A. Lévy et. al. in the 1960's, we introduce new principles of reflection based on the general notion of \emph{Structural Reflection} and argue that they are in strong agreement with the conception of reflection implicit in Cantor's original idea of the unknowability of the \emph{Absolute}, which was subsequently developed in the works of Ackermann, Lévy, Gödel, Reinhardt, and others. We then present a comprehensive survey of results showing that different forms of the new principles of…
A cardinal $λ$ satisfies a property P robustly if, whenever $\mathbb{Q}$ is a forcing poset and $|\mathbb{Q}|^+ < λ$, $λ$ satisfies P in $V^{\mathbb{Q}}$. We study the extent to which certain reflection properties of large cardinals can be satisfied robustly by small cardinals. We focus in particular on stationary reflection and the tree property, both of which can consistently hold but fail to be robust at small cardinals. We introduce natural strengthenings of these principles which are always robust and which hold at sufficiently large cardinals, consider the extent to which these strengthe…
We prove a weakened version of the reflection of Reinhardt cardinals by super Reinhardt cardinals: Let $M=(V^M,P)$ be a countable model of second order set theory $\mathsf{ZF}_2$ (with universe $V^M$ and classes $P$) which models "$κ$ is super Reinhardt". We show that there are unboundedly many $μ<κ$ such that there is $j$ such that $(V^M,j)$ models $\mathsf{ZF}(j)+$"$μ$ is Reinhardt, as witnessed by $j$". In particular, $j\upharpoonright X\in V^M$ for all $X\in V^M$ (but we allow $j\notin P$).
We recently formulated a new large-cardinal axiom of strength intermediate between a totally indescribable cardinal and an $ω$-Erdős cardinal, positing the existence of what we called an "extremely reflective cardinal", and we showed that the property of being extremely reflective was in fact equivalent to the property of being remarkable, and we sought to argue that this axiom should be seen as intrinsically justified. This built on related earlier work in which the notion of an $α$-reflective cardinal was formulated. Then Welch and Roberts put forward a family of reflection principles, Welch…
We obtain an array of consistency results concerning trees and stationary reflection at double successors of regular cardinals $κ$, updating some classical constructions in the process. This includes models of $\mathsf{CSR}(κ^{++})\wedge \mathsf{TP}(κ^{++})$ (both with and without $\mathsf{AP}(κ^{++})$) and models of the conjunctions $\mathsf{SR}(κ^{++}) \wedge \mathsf{wTP}(κ^{++}) \wedge \mathsf{AP}(κ^{++})$ and $\neg \mathsf{AP}(κ^{++}) \wedge \mathsf{SR}(κ^{++})$ (the latter was originally obtained in joint work by Krueger and the first author \cite{GilKru:8fold}, and is here given using di…
The weakly compact reflection principle $\text{Refl}_{\text{wc}}(κ)$ states that $κ$ is a weakly compact cardinal and every weakly compact subset of $κ$ has a weakly compact proper initial segment. The weakly compact reflection principle at $κ$ implies that $κ$ is an $ω$-weakly compact cardinal. In this article we show that the weakly compact reflection principle does not imply that $κ$ is $(ω+1)$-weakly compact. Moreover, we show that if the weakly compact reflection principle holds at $κ$ then there is a forcing extension preserving this in which $κ$ is the least $ω$-weakly compact cardinal.…
Inspired by Zermelo's quasi-categoricity result characterizing the models of second-order Zermelo-Fraenkel set theory $\text{ZFC}_2$, we investigate when those models are fully categorical, characterized by the addition to $\text{ZFC}_2$ either of a first-order sentence, a first-order theory, a second-order sentence or a second-order theory. Thus we mount an analysis of the categorical large cardinals. This mathematical analysis leads naturally to philosophical issues concerning structuralism and realism, including especially the tension between categoricity and reflection. Ultimately we ident…
Bounded stationary reflection at a cardinal $λ$ is the assertion that every stationary subset of $λ$ reflects but there is a stationary subset of $λ$ that does not reflect at arbitrarily high cofinalities. We produce a variety of models in which bounded stationary reflection holds. These include models in which bounded stationary reflection holds at the successor of every singular cardinal $μ> \aleph_ω$ and models in which bounded stationary reflection holds at $μ^+$ but the approachability property fails at $μ$.
William Tait and Peter Koellner have written on the topic of which reflection principles are intrinsically justified. Phillip Welch and Sam Roberts have recently sought to motivate much stronger reflection principles. This work also had some similarities with previous work of Victoria Marshall's on reflection principles, as may be seen from considerations which we will seek to articulate in what follows. The statement of our three philosophical theses shall be given in the Introduction. We will present a set of technical results in support of three philosophical theses which will be articulate…
Bagaria and Väänänen developed a framework for studying the large cardinal strength of downwards Löwenheim-Skolem theorems and related set theoretic reflection properties. The main tool was the notion of symbiosis, originally introduced by the third author. Symbiosis provides a way of relating model theoretic properties of strong logics to definability in set theory. In this paper we continue the systematic investigation of symbiosis and apply it to upwards Löwenheim-Skolem theorems and reflection principles. To achieve this, we need to adapt the notion of symbiosis to a new form, called bound…
Reflection principles (or dually speaking, compactness principles) often give rise to combinatorial guessing principles. Uniformization properties, on the other hand, are examples of anti-guessing principles. We discuss the tension and the compatibility between reflection principles and uniformization properties at the level of the second uncountable cardinal.
We show that the tree property, stationary reflection and the failure of approachability at $κ^{++}$ are consistent with $\mathfrak{u}(κ) = κ^+ < 2^κ$, where $κ$ is a singular strong limit cardinal with the countable or uncountable cofinality. As a by-product, we show that if $λ$ is a regular cardinal, then stationary reflection at $λ^+$ is indestructible under all $λ$-cc forcings (out of general interest, we also state a related result for the preservation of club stationary reflection).
Starting from infinitely many supercompact cardinals, we force a model of ZFC where $\aleph_{ω^2+1}$ satisfies simultaneously a strong principle of reflection, called $Δ$-reflection, and a version of the square principle, denoted $\square(\aleph_{ω^2+1}).$ Thus we show that $\aleph_{ω^2+1}$ can satisfy simultaneously a strong reflection principle and an anti-reflection principle.
Continuing the previous paper, we study the Strong Downward Löwenheim-Skolem Theorems (SDLSs) of the stationary logic and their variations. It has been shown that the SDLS for the ordinary stationary logic with weak second-order parameters down to $<\aleph_2$ is equivalent to the conjunction of CH and Cox's Diagonal Reflection Principle for internally clubness. We show that the SDLS for the stationary logic without weak second-order parameters down to $<2^{\aleph_0}$ implies that the size of the continuum is $\aleph_2$. In contrast, an internal interpretation of the stationary logic can satisf…
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.
We introduce reflection properties of cardinals in which the attributes that reflect are expressible by infinitary formulas whose lengths can be strictly larger than the cardinal under consideration. This kind of generalized reflection principle leads to the definitions of $L_{κ^+,κ^+}$-indescribability and $Π^1_ξ$-indescribability of a cardinal $κ$ for all $ξ<κ^+$. In this context, universal $Π^1_ξ$ formulas exist, there is a normal ideal associated to $Π^1_ξ$-indescribability and the notions of $Π^1_ξ$-indescribability yield a strict hierarchy below a measurable cardinal. Additionally, given…
For each natural number $n$, let $C^{(n)}$ be the closed and unbounded proper class of ordinals $α$ such that $V_α$ is a $Σ_n$ elementary substructure of $V$. We say that $κ$ is a \emph{$C^{(n)}$-cardinal} if it is the critical point of an elementary embedding $j:V\to M$, $M$ transitive, with $j(κ)$ in $C^{(n)}$. By analyzing the notion of $C^{(n)}$-cardinal at various levels of the usual hierarchy of large cardinal principles we show that, starting at the level of superstrong cardinals and up to the level of rank-into-rank embeddings, $C^{(n)}$-cardinals form a much finer hierarchy. The natur…
We investigate questions involving Aronszajn trees, square principles, and stationary reflection. We first consider two strengthenings of $\square(κ)$ introduced by Brodsky and Rinot for the purpose of constructing $κ$-Souslin trees. Answering a question of Rinot, we prove that the weaker of these strengthenings is compatible with stationary reflection at $κ$ but the stronger is not. We then prove that, if $μ$ is a singular cardinal, $\square_μ$ implies the existence of a special $μ^+$-tree with a $\mathrm{cf}(μ)$-ascent path, thus answering a question of Lücke.
Motivated by a characterization of weakly compact cardinals due to Todorcevic, we introduce a new cardinal characteristic, the C-sequence number, which can be seen as a measure of the compactness of a regular uncountable cardinal. We prove a number of ZFC and independence results about the C-sequence number and its relationship with large cardinals, stationary reflection, and square principles. We then introduce and study the more general C-sequence spectrum and uncover some tight connections between the C-sequence spectrum and the strong coloring principle U(...), introduced in Part I of this…
Covering matrices were introduced by Viale in his proof that the Singular Cardinals Hypothesis follows from the Proper Forcing Axiom. In the course of his work and in subsequent work with Sharon, he isolated two reflection principles, $\mathrm{CP}$ and $\mathrm{S}$, which may hold of covering matrices. In this paper, we continue previous work of the author investigating connections between failures of $\mathrm{CP}$ and $\mathrm{S}$ and variations on Jensen's square principle. We prove that, for a regular cardinal $λ> ω_1$, assuming large cardinals, $\square(λ, 2)$ is consistent with $\mathrm{C…
For $n<ω$, we say that the $Π^1_n$-reflection principle holds at $κ$ and write $\text{Refl}_n(κ)$ if and only if $κ$ is a $Π^1_n$-indescribable cardinal and every $Π^1_n$-indescribable subset of $κ$ has a $Π^1_n$-indescribable proper initial segment. The $Π^1_n$-reflection principle $\text{Refl}_n(κ)$ generalizes a certain stationary reflection principle and implies that $κ$ is $Π^1_n$-indescribable of order $ω$. We define a forcing which shows that the converse of this implication can be false in the case $n=1$. Moreover, we prove that if $κ$ is $(α+1)$-weakly compact where $α<κ^+$, then ther…
It is well-known that the square principle $\square_λ$ entails the existence of a non-reflecting stationary subset of $λ^+$, whereas the weak square principle $\square^*_λ$ does not. Here we show that if $μ^{\mathrm{cf}(λ)} < λ$ for all $μ< λ$, then $\square^*_λ$ entails the existence of a non-reflecting stationary subset of $E^{λ^+}_{\mathrm{cf}(λ)}$ in the forcing extension for adding a single Cohen subset of $λ^+$. It follows that indestructible forms of simultaneous stationary reflection entail the failure of weak square. We demonstrate this by settling a question concerning the subcomplet…
We investigate the interaction between compactness principles and guessing principles in the Radin forcing extensions. In particular, we show that in any Radin forcing extension with respect to a measure sequence on $κ$, if $κ$ is weakly compact, then $\diamondsuit(κ)$ holds. This provides contrast with a well-known theorem of Woodin, who showed that in a certain Radin extension over a suitably prepared ground model relative to the existence of large cardinals, the diamond principle fails at a strongly inaccessible Mahlo cardinal. Refining the analysis of the Radin extensions, we consistently …
We establish the consistency of the failure of the diamond principle on a cardinal $κ$ which satisfies a strong simultaneous reflection property. The result is based on an analysis of Radin forcing, and further leads to a characterization of weak compactness of $κ$ in a Radin generic extension.
We introduce and consider the inner-model reflection principle, which asserts that whenever a statement $φ(a)$ in the first-order language of set theory is true in the set-theoretic universe $V$, then it is also true in a proper inner model $W\subsetneq V$. A stronger principle, the ground-model reflection principle, asserts that any such $φ(a)$ true in $V$ is also true in some non-trivial ground model of the universe with respect to set forcing. These principles each express a form of width reflection in contrast to the usual height reflection of the Lévy-Montague reflection theorem. They are…
In Part I of this series, we introduced a class of notions of forcing which we call Sigma-Prikry, and showed that many of the known Prikry-type notions of forcing that center around singular cardinals of countable cofinality are Sigma-Prikry. We showed that given a Sigma-Prikry poset P and a P-name for a non-reflecting stationary set T, there exists a corresponding Sigma-Prikry poset that projects to P and kills the stationarity of T. In this paper, we develop a general scheme for iterating Sigma-Prikry posets and, as an application, we blow up the power of a countable limit of Laver-indestruc…
