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 naturalness of the notion of $C^{(n)}$-cardinal is exemplified by showing that the existence of $C^{(n)}$-extendible cardinals is equivalent to simple reflection principles for classes of structures, which generalize the notions of supercompact and extendible cardinals. Moreover, building on results of \cite{BCMR}, we give new characterizations of Vopeňka's Principle in terms of $C^{(n)}$-extendible cardinals.
The mantle is the intersection of all ground models of $V$. We show that if there exists an extendible cardinal then the mantle is a ground model of $V$.
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…
We prove two general results about the preservation of extendible and $C^{(n)}$-extendible cardinals under a wide class of forcing iterations (Theorems 5.4 and 7.5). As applications we give new proofs of the preservation of Vopěnka's Principle and $C^{(n)}$-extendible cardinals under Jensen's iteration for forcing the GCH, previously obtained by Brooke-Taylor and Tsaprounis, res\-pectively. We prove that $C^{(n)}$-extendible cardinals are preserved by forcing with standard Easton-support iterations for any possible $Δ_2$-definable behaviour of the power-set function on regular cardinals. We sh…
Superstrong cardinals are never Laver indestructible. Similarly, almost huge cardinals, huge cardinals, superhuge cardinals, rank-into-rank cardinals, extendible cardinals, 1-extendible cardinals, 0-extendible cardinals, weakly superstrong cardinals, uplifting cardinals, pseudo-uplifting cardinals, superstrongly unfoldable cardinals, Σ_n-reflecting cardinals, Σ_n-correct cardinals and Σ_n-extendible cardinals (all for n>2) are never Laver indestructible. In fact, all these large cardinal properties are superdestructible: if κ exhibits any of them, with corresponding target θ, then in any forci…
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 introduce a hierarchy of large cardinals between weakly compact and measurable cardinals, that is closely related to the Ramsey-like cardinals introduced by Victoria Gitman, and is based on certain infinite filter games, however also has a range of equivalent characterizations in terms of elementary embeddings. The aim of this paper is to locate the Ramsey-like cardinals studied by Gitman, and other well-known large cardinal notions, in this hierarchy.
Many set theorists point to the linearity phenomenon in the hierarchy of consistency strength, by which natural theories tend to be linearly ordered and indeed well ordered by consistency strength. Why should it be linear? In this paper I present counterexamples, natural instances of nonlinearity and illfoundedness in the hierarchy of large cardinal consistency strength, as natural or as nearly natural as I can make them. I present diverse cautious enumerations of ZFC and large cardinal set theories, which exhibit incomparability and illfoundedness in consistency strength, and yet, I argue, ar…
We give a sharper version of a theorem of Rosicky, Trnkova and Adamek, and a new proof of a theorem of Rosicky, both about colimit preservation between categories of structures. Unlike the original proofs, which use category-theoretic methods, we use set-theoretic arguments involving elementary embeddings given by large cardinals such as alpha-strongly compact and C^(n)-extendible cardinals.
The study of inner models was initiated by Gödel's analysis of the constructible universe. Later, the study of canonical inner models with large cardinals, e.g., measurable cardinals, strong cardinals or Woodin cardinals, was pioneered by Jensen, Mitchell, Steel, and others. Around the same time, the study of infinite two-player games was driven forward by Martin's proof of analytic determinacy from a measurable cardinal, Borel determinacy from ZFC, and Martin and Steel's proof of levels of projective determinacy from Woodin cardinals with a measurable cardinal on top. First Woodin and later N…
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 $\mathr…
In this paper we introduce a new hierarchy of large cardinals between I3 and I2, the iterability hierarchy, and we prove that every step of it strongly implies the ones below.
A subset $S$ of a cardinal $κ$ is Ramsey if for every function $f:[S]^{<ω}\to κ$ with $f(a)<\min a$ for all $a\in[S]^{<ω}$, there is a set $H\subseteq S$ of cardinality $κ$ which is \emph{homogeneous} for $f$, meaning that $f\upharpoonright[H]^n$ is constant for each $n<ω$. Baumgartner proved \cite{MR0384553} that if $κ$ is a Ramsey cardinal, then the collection of non-Ramsey subsets of $κ$ is a normal ideal on $κ$. Sharpe and Welch \cite{MR2817562}, and independently Bagaria \cite{MR3894041}, extended the notion of $Π^1_n$-indescribability where $n<ω$ to that of $Π^1_ξ$-indescribability where…
One of the numerous characterizations of a Ramsey cardinal kappa involves the existence of certain types of elementary embeddings for transitive sets of size κsatisfying a large fragment of ZFC. We introduce new large cardinal axioms generalizing the Ramsey elementary embeddings characterization and show that they form a natural hierarchy between weakly compact cardinals and measurable cardinals. These new axioms serve to further our knowledge about the elementary embedding properties of smaller large cardinals, in particular those still consistent with V=L.
This paper continues the study of the Ramsey-like large cardinals. Ramsey-like cardinals are defined by generalizing the characterization of Ramsey cardinals via the existence of elementary embeddings. Ultrafilters derived from such embeddings are fully iterable and so it is natural to ask about large cardinal notions asserting the existence of ultrafilters allowing only $α$-many iterations for some countable ordinal $α$. Here we study such $α$-iterable cardinals. We show that the $α$-iterable cardinals form a strict hierarchy for $α\leqω_1$, that they are downward absolute to $L$ for $α<ω_1^L…
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…
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 fin…
We show that several dichotomy theorems concerning the second level of the Borel hierarchy are special cases of the $\aleph_0$-dimensional generalization of the open graph dichotomy, which itself follows from the usual proof(s) of the perfect set theorem. Under the axiom of determinacy, we obtain the generalizations of these results from analytic metric spaces to separable metric spaces. We also consider connections between cardinal invariants and the chromatic numbers of the corresponding dihypergraphs.
We consider classical representations of integers: Church's function iterators, cardinal equivalence classes of sets, ordinal equivalence classes of totally ordered sets. Since programs do not work on abstract entities and require formal representations of objects, we effectivize these abstract notions in order to allow them to be computed by programs. To any such effectivized representation is then associated a notion of Kolmogorov complexity. We prove that these Kolmogorov complexities form a strict hierarchy which coincides with that obtained by relativization to jump oracles and/or allowan…
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…
