A general tool for consistency results related to I1

In this paper we provide a general tool to prove the consistency of $I1(\lambda)$ with various combinatorial properties at $\lambda$ typical at settings with $2^\lambda>\lambda^+$, that does not need a profound knowledge of the forcing notions involved. Examples of such properties are the first failure of GCH, a very good scale and the negation of the approachability property, or the tree property at $\lambda^+$ and $\lambda^{++}$

A partially non-proper ordinal beyond L(V \u3bb+1)

In his recent work, Woodin has defined new axioms stronger than I0 (the existence of an elementary embedding from to itself), that involve elementary embeddings between slightly larger models. There is a natural correspondence between I0 and Determinacy, but to extend this correspondence in the new framework we must insist that these elementary embeddings are proper. Previous results validated the definition, showing that there exist elementary embeddings that are not proper, but it was still open whether properness was determined by the structure of the underlying model or not. This paper proves that this is not the case, defining a model that generates both proper and non-proper elementary embeddings, and compare this new model to the older ones

I0 and rank-into-rank axioms

This is a survey about I0 and rank-into-rank axioms, with some previously unpublished proofs

Generic I0 at \u2135\u3c9

In this paper we introduce a generic large cardinal akin to I0, together with the consequences of \u2135\u3c9being s uch ageneric large cardinal. In this case \u2135\u3c9is J\ub4onsson, and in a choiceless inner model many properties hold that arein contrast with pcf theory in ZFC

Totally non-proper ordinals beyond L(V \u3bb+1)

In recent work Woodin has defined new axioms stronger than I0 (the existence of an elementary embedding j from L(V\u3bb+1) to itself), that involve elementary embeddings between slightly larger models. There is a natural correspondence between I0 and determinacy, but to extend this correspondence in this new framework we must insist that these elementary embeddings are proper. While at first this seemed to be a common property, in this paper will be provided a model in which all such elementary embeddings are not proper. This result fills a gap in a theorem by Woodin and justifies the definition of propernes

The iterability hierarchy above I3

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

Rank-into-rank hypotheses and the failure of GCH

In this paper we are concerned about the ways GCH can fail in relation to rank-into-rank hypotheses, i.e., very large cardinals usually denoted by I3, I2, I1 and I0. The main results are a satisfactory analysis of the way the power function can vary on regular cardinals in the presence of rank-into-rank hypotheses and the consistency under I0 of the existence of j:V\u3bb+1 7aV\u3bb+1 with the failure of GCH at \u3bb

The 17-Prikry condition

In this paper we isolate a property for forcing notions, the *-Prikry condition, that is similar to the Prikry condition but that is topological: A forcing P satisfies it iff for every p 08 P and for every open dense D 86 P, there are n 08 \u3c9 and q 64 17 p such that for any r 64 q with l(r) = l(q) + n, r 08 D, for some length notion l. This is implicit in many proofs in literature. We prove this for the tree Prikry forcing and the long extender Prikry forcing