Critical points in an algebra of elementary embeddings

For each natural number $n$, let $C^{(n)}$ be the closed and unbounded proper class of ordinals $\alpha$ such that $V_\alpha$ is a $\Sigma_n$ elementary substructure of $V$. We say that $\kappa$ is a \emph{$C^{(n)}$-cardinal} if it is the critical point of an elementary embedding $j:V\to M$, $M$ transitive, with $j(\kappa)$ 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 a...

The following pcf results are proved: 1. Assume that kappa > aleph_0 is a weakly compact cardinal. Let mu > 2^kappa be a singular cardinal of cofinality kappa. Then for every regular lambda < pp^+_{Gamma(kappa)} (mu) there is an increasing sequence (lambda_i | i < kappa) of regular cardinals converging to mu such that lambda = tcf(prod_{i < kappa} lambda_i, <_{J^{bd}_kappa}). 2. Let mu be a strong limit cardinal and theta a cardinal above mu. Suppose that at least one of them...

The critical point between varieties A and B of algebras is defined as the least cardinality of the semilattice of compact congruences of a member of A but of no member of B, if it exists. The study of critical points gives rise to a whole array of problems, often involving lifting problems of either diagrams or objects, with respect to functors. These, in turn, involve problems that belong to infinite combinatorics. We survey some of the combinatorial problems and results th...

I analyze the hierarchy of large cardinals between a supercompact cardinal and an almost-huge cardinal. Many of these cardinals are defined by modifying the definition of a high-jump cardinal. A high-jump cardinal is defined as the critical point of an elementary embedding $j: V \to M$ such that $M$ is closed under sequences of length $\sup\set{j(f)(\kappa) \st f: \kappa \to \kappa}$. Some of the other cardinals analyzed include the super-high-jump cardinals, almost-high-jump...

We study the notion of non-trivial elementary embeddings $j : V \rightarrow V$ under the assumption that $V$ satisfies $ZFC$ without Power Set but with the Collection Scheme. We show that no such embedding can exist under the additional assumption that it is cofinal and either $V_{\textrm{crit}(j)}$ is a set or that the Dependent Choice Schemes holds. We then study failures of instances of collection in symmetric submodels of class forcings.

We analyze the notion of guessing model, a way to assign combinatorial properties to arbitrary regular cardinals. Guessing models can be used, in combination with inaccessibility, to characterize various large cardinals axioms, ranging from supercompactness to rank-to-rank embeddings. The majority of these large cardinals properties can be defined in terms of suitable elementary embeddings j\colon V_\gamma \to V_\lambda. One key observation is that such embeddings are uniquel...

In this paper, we provide a positive answer to a question by Matthews whether $\mathsf{ZF}^-$ is consistent with a non-trivial cofinal Reinhardt elementary embedding $j\colon V\to V$. The consistency follows from $\mathsf{ZFC} + I_0$, and more precisely, Schultzenberg's model of $\mathsf{ZF}$ with an elementary embedding $k\colon V_{\lambda+2}\to V_{\lambda+2}$.

Suppose $\kappa$ is $\lambda$-supercompact witnessed by an elementary embedding $j:V\rightarrow M$ with critical point $\kappa$, and further suppose that $F$ is a function from the class of regular cardinals to the class of cardinals satisfying the requirements of Easton's theorem: (1) $\forall\alpha$ $\alpha<\textrm{cf}(F(\alpha))$ and (2) $\alpha<\beta$ $\Longrightarrow$ $F(\alpha)\leq F(\beta)$. In this article we address the question: assuming GCH, what additional assumpt...

We investigate the structure of rank-to-rank elementary embeddings, working in ZF set theory without the Axiom of Choice. Recall that the levels $V_\alpha$ 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 $j:V_{\alpha+1}\to V_{\alpha+1}$ is a (non-trivial) elementary embedding, we show that the structure of $V_\alpha$ is fundamentally different to that o...

We investigate large set axioms defined in terms of elementary embeddings over constructive set theories, focusing on $\mathsf{IKP}$ and $\mathsf{CZF}$. Most previously studied large set axioms, notably the constructive analogues of large cardinals below $0^\sharp$, have proof-theoretic strength weaker than full Second-order Arithmetic. On the other hand, the situation is dramatically different for those defined via elementary embeddings. We show that by adding to $\mathsf{IK...