AXIOM I0 AND HIGHER DEGREE THEORY

In this paper, we analyze structures of Zermelo degrees via a list of four degree theoretic questions (see §2) in various fine structure extender models, or under large cardinal assumptions. In particular we give a detailed analysis of the structures of Zermelo degrees in the Mitchell model for ω many measurable cardinals. It turns out that there is a profound correlation between the complexity of the degree structures at countable cofinality singular cardinals and the large cardinal strength of the relevant cardinals. The analysis applies to general degree notions, Zermelo degree is merely the author’s choice for illustrating the idea.I0(λ) is the assertion that there is an elementary embedding j : L(Vλ+1) → L(Vλ+1) with critical point < λ. We show that under I0(λ), the structure of Zermelo degrees at λ is very complicated: it has incomparable degrees, is not dense, satisfies Posner–Robinson theorem etc. In addition, we show that I0 together with a mild condition on the critical point of the embedding implies that the degree determinacy for Zermelo degrees at λ is false in L(Vλ+1). The key tool in this paper is a generic absoluteness theorem in the theory of I0, from which we obtain an analogue of Perfect Set Theorem for “projective” subsets of Vλ+1, and the Posner–Robinson follows as a corollary. Perfect Set Theorem and Posner–Robinson provide evidences supporting the analogy between $$AD$$ over L(ℝ) and I0 over L(Vλ+1), while the failure of degree determinacy is one for disanalogy. Furthermore, we conjecture that the failure of degree determinacy for Zermelo degrees at any uncountable cardinal is a theorem of $$ZFC$$.

Inverse limit reflection and the structure of L(Vλ+1)

We extend the results of Laver on using inverse limits to reflect large cardinals of the form, there exists an elementary embedding Lα(Vλ+1) → Lα(Vλ+1). Using these inverse limit reflection embeddings directly and by broadening the collection of U(j)-representable sets, we prove structural results of L(Vλ+1) under the assumption that there exists an elementary embedding j : L(Vλ+1) → L(Vλ+1). As a consequence we show the impossibility of a generalized inverse limit X-reflection result for X ⊆ Vλ+1, thus focusing the study of L(ℝ) generalizations on L(Vλ+1).

Ramsey-like cardinals

One of the numerous characterizations of a Ramsey cardinal κ involves the existence of certain types of elementary embeddings for transitive sets of size κ satisfying 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.

Regular embeddings of the stationary tower and Woodin's maximality theorem

We present Woodin's proof that if there exists a measurable Woodin cardinal δ then there is a forcing extension satisfying all sentences ϕ such that CH + ϕ holds in a forcing extension of V by a partial order in Vδ. We also use some of the techniques from this proof to show that if there exists a stationary limit of stationary limits of Woodin cardinals, then in a homogeneous forcing extension there is an elementary embedding j: V → M with critical point such that M is countably closed in the forcing extension.

Forbidden intervals

Many classical statements of set theory are settled by the existence of generic elementary embeddings that are analogous the elementary embeddings posited by large cardinals. [2] The embeddings analogous to measurable cardinals are determined by uniform, κ-complete precipitous ideals on cardinals κ. Stronger embeddings, analogous to those originating from supercompact or huge cardinals are encoded by normal fine ideals on sets such as [κ]<λ or [κ]λ.The embeddings generated from these ideals are limited in ways analogous to conventional large cardinals. Explicitly, if j: V → M is a generic elementary embedding with critical point κ and λ supnЄωjn(κ) and the forcing yielding j is λ-saturated then j“λ+ ∉ M. (See [2].)Ideals that yield embeddings that are analogous to strongly compact cardinals have more puzzling behavior and the analogy is not as straightforward. Some natural ideal properties of this kind have been shown to be inconsistent:Theorem 1 (Kunen). There is no ω2-saturated, countably complete uniform ideal on any cardinal in the interval [ℵω, ℵω).Generic embeddings that arise from countably complete, ω2-saturated ideals have the property that sup . So the Kunen result is striking in that it apparently allows strong ideals to exist above the conventional large cardinal limitations. The main result of this paper is that it is consistent (relative to a huge cardinal) that such ideals exist.

Perfect trees and elementary embeddings

An important technique in large cardinal set theory is that of extending an elementary embedding j: M → N between inner models to an elementary embedding j* : M[G] → N[G*] between generic extensions of them. This technique is crucial both in the study of large cardinal preservation and of internal consistency. In easy cases, such as when forcing to make the GCH hold while preserving a measurable cardinal (via a reverse Easton iteration of α-Cohen forcing for successor cardinals α), the generic G* is simply generated by the image of G. But in difficult cases, such as in Woodin's proof that a hypermeasurable is sufficient to obtain a failure of the GCH at a measurable, a preliminary version of G* must be constructed (possibly in a further generic extension of M[G]) and then modified to provide the required G*. In this article we use perfect trees to reduce some difficult cases to easy ones, using fusion as a substitute for distributivity. We apply our technique to provide a new proof of Woodin's theorem as well as the new result that global domination at inaccessibles (the statement that d(κ) is less than 2κ for inaccessible κ, where d(κ) is the dominating number at κ) is internally consistent, given the existence of 0#.

Iterates of the core model

Let N be a transitive model of ZFC such that “N ⊂ N and P(ℝ) ⊂ N. Assume that both V and N satisfy “the core model K exists.” Then KN is an iterate of K, i.e., there exists an iteration tree F on K such that F has successor length and . Moreover, if there exists an elementary embedding π: V → N then the iteration map associated to the main branch of F equals π յ K. (This answers a question of W. H. Woodin, M. Gitik, and others.) The hypothesis that P(ℝ) ⊂ N is not needed if there does not exist a transitive model of ZFC with infinitely many Woodin cardinals.

Jonsson-like partition relations and j: V → V

Working in the theory ”ZF + There is a nontrivial elementary embedding j : V → V“, we show that a final segment of cardinals satisfies certain square bracket finite and infinite exponent partition relations. As a corollary to this, we show that this final segment is composed of Jonsson cardinals. We then show how to force and bring this situation down to small alephs. A prototypical result is the construction of a model for ZF in which every cardinal μ ≥ ℵ2 satisfies the square bracket infinite exponent partition relation . We conclude with a discussion of some consistency questions concerning different versions of the axiom asserting the existence of a nontrivial elementary embedding j: V → V. By virtue of Kunen's celebrated inconsistency result, we use only a restricted amount of the Axiom of Choice.

On elementary embeddings from an inner model to the universe

We consider the following question of Kunen:Does Con(ZFC + ∃M a transitive inner model and a non-trivial elementary embedding j: M → V)imply Con(ZFC + ∃ a measurable cardinal)?We use core model theory to investigate consequences of the existence of such a j: M → V. We prove, amongst other things, the existence of such an embedding implies that the core model K is a model of “there exists a proper class of almost Ramsey cardinals”. Conversely, if On is Ramsey, then such a j. M are definable.We construe this as a negative answer to the question above. We consider further the consequences of strengthening the closure assumption on j to having various classes of fixed points.