One of the standard ways of postulating large cardinal axioms is to consider elementary embeddings,j, from the universe,V, into some transitive submodel,M. See Reinhardt–Solovay [7] for more details. Ifjis not the identity, andκis the first ordinal moved byj, thenκis a measurable cardinal. Conversely, Scott [8] showed that wheneverκis measurable, there is suchjandM. If we had assumed, in addition, that, thenκwould be theκth measurable cardinal; in general, the wider we assumeMto be, the largerκmust be.

If the Wholeness Axiom wa $_0$ is itself consistent, then it is consistent with v=hod. A consequence of the proof is that the various Wholeness Axioms are not all equivalent. Additionally, the theory zfc+wa $_0$ is finitely axiomatizable.

Applying the seed concept to Prikry tree forcing P μ , I investigate how well P μ preserves the maximality property of ordinary Prikry forcing and prove that P μ Prikry sequences are maximal exactly when μ admits no non-canonical seeds via a finite iteration. In particular, I conclude that if μ is a strongly normal supercompactness measure, then P μ Prikry sequences are maximal, thereby proving, for a large class of measures, a conjecture of W. Hugh Woodin's.

A model in which strongness ofκ is indestructible under κ+ -weakly closed forcing notions satisfying the Prikry condition is constructed. This is applied to solve a question of Hajnal on the number of elements of {λ δ |2 δ <λ}.

In this paper we introduce the Wholeness Axiom , which asserts that there is a nontrivial elementary embedding from V to itself. We formalize the axiom in the language {∈, j } , adding to the usual axioms of ZFC all instances of Separation, but no instance of Replacement, for j -formulas, as well as axioms that ensure that j is a nontrivial elementary embedding from the universe to itself. We show that WA has consistency strength strictly between I 3 (...) and the existence of a cardinal that is super- n -huge for every n . ZFC + WA is used as a background theory for studying generalizations of Laver sequences. We define the notion of Laver sequence for general classes E consisting of elementary embeddings of the form i :V β → M , where M is transitive, and use five globally defined large cardinal notions – strong, supercompact, extendible, super-almost-huge, superhuge – for examples and special cases of the main results. Assuming WA at the beginning, and eventually refining the hypothesis as far as possible, we prove the existence of a strong form of Laver sequence for a broad range of classes E that include the five large cardinal types mentioned. We show that if κ is globally superstrong, if E is Laver-closed at beth fixed points, and if there are superstrong embeddings i with critical point κ and arbitrarily large targets such that E is weakly compatible with i , then our standard constructions are E -Laver at κ . , there is an extendible Laver sequence at κ .) In addition, in most cases our Laver sequences can be made special if E is upward λ -closed for sufficiently many λ. (shrink)

In 1970, K. Kunen, working in the context of Kelley–Morse set theory, showed that the existence of a nontrivial elementary embedding j:V→V is inconsistent. In this paper, we give a finer analysis of the implications of his result for embeddings V→V relative to models of ZFC. We do this by working in the extended language , using as axioms all the usual axioms of ZFC , along with an axiom schema that asserts that j is a nontrivial elementary embedding. Without (...) additional axiomatic assumptions on j, we show that that the resulting theory is weaker than an ω-Erdös cardinal, but stronger than n-ineffables. We show that natural models of ZFC+BTEE give rise to Schindler’s remarkable cardinals. The approach to inconsistency from ZFC+BTEE forks into two paths: extensions of ZFC+BTEE+Cofinal Axiom and ZFC+BTEE+¬Cofinal Axiom, where Cofinal Axiom asserts that the critical sequence is cofinal in the ordinals. We describe near-minimal inconsistent extensions of each of these theories. The path toward inconsistency from ZFC+BTEE+¬Cofinal Axiom is paved with a sequence of theories of increasing large cardinal strength. Indeed, the extensions of the theory ZFC +“j is a nontrivial elementary embedding” form a hierarchy of axioms, ranging in strength from Con to the existence of a cardinal that is super-n-huge for every n, to inconsistency. This hierarchy is parallel to the usual hierarchy of large cardinal axioms, and can be used in the same way. We also isolate several intermediate-strength axioms which, when added to ZFC+BTEE, produce theories having strengths in the vicinity of a measurable cardinal of high Mitchell order, a strong cardinal, ω Woodin cardinals, and n-huge cardinals. We also determine precisely which combinations of axioms, of the form result in inconsistency. (shrink)

Versions of Laver sequences are known to exist for supercompact and strong cardinals. Assuming very strong axioms of infinity, Laver sequences can be constructed for virtually any globally defined large cardinal not weaker than a strong cardinal; indeed, under strong hypotheses, Laver sequences can be constructed for virtually any regular class of embeddings. We show here that if there is a regular class of embeddings with critical point κ, and there is an inaccessible above κ, then it is consistent for (...) there to be a regular class that admits no Laver sequence. We also show that extendible cardinals are Laver-generating, i.e., that assuming only that κ is extendible, there is an extendible Laver sequence at κ. We use the method of proof to answer a question about Laver-closure of extendible cardinals at inaccessibles. Finally, we consider Laver sequences for super-almost-huge cardinals. Assuming slightly more than super-almost-hugeness, we show that there are super-almost-huge Laver sequences, improving the previously known upper bound for such Laver sequences. We also describe conditions under which the canonical construction of a Laver sequence fails for super-almost-huge cardinals. (shrink)

The Wholeness Axiom (WA) is an axiom schema that can be added to the axioms of ZFC in an extended language $\{\in,j\}$ , and that asserts the existence of a nontrivial elementary embedding $j:V\to V$ . The well-known inconsistency proofs are avoided by omitting from the schema all instances of Replacement for j-formulas. We show that the theory ZFC + V = HOD + WA is consistent relative to the existence of an $I_1$ embedding. This answers a question about the (...) existence of Laver sequences for regular classes of set embeddings: Assuming there is an $I_1$ -embedding, there is a transitive model of ZFC +WA + “there is a regular class of embeddings that admits no Laver sequence.”. (shrink)