We show that every sufficiently iterable countable mouse has a unique iteration strategy whose associated iteration maps are lexicographically minimal. This enables us to extend the results of [3] on the good behavior of the standard parameter from tame mice to arbitrary mice.

We show that every sufficiently iterable countable mouse has a unique iteration strategy whose associated iteration maps are lexicographically minimal. This enables us to extend the results of [3] on the good behavior of the standard parameter from tame mice to arbitrary mice.

We prove that in all Mitchell-Steel core models, □k holds for all k. From this we obtain new consistency strength lower bounds for the failure of □k if k is either singular and countably closed, weakly compact, or measurable. Jensen introduced a large cardinal property that we call subcompactness; it lies between superstrength and supercompactness in the large cardinal hierarchy. We prove that in all Jensen core models, □k holds iff k is not subcompact.

We prove that in all Mitchell-Steel core models, □ κ holds for all κ. (See Theorem 2.). From this we obtain new consistency strength lower bounds for the failure of □ κ if κ is either singular and countably closed, weakly compact, or measurable. (Corallaries 5, 8, and 9.) Jensen introduced a large cardinal property that we call subcompactness; it lies between superstrength and supercompactness in the large cardinal hierarchy. We prove that in all Jensen core models, □ κ holds (...) iff κ is not subcompact. (See Theorem 15; the only if direction is essentially due to Jensen.). (shrink)

We study the fine structure of the core model for one Woodin cardinal, building of the work of Mitchell and Steel on inner models of the form . We generalize to some combinatorial principles that were shown by Jensen to hold in L. We show that satisfies the statement: “□κ holds whenever κ the least measurable cardinal λ of order λ++”. We introduce a hierarchy of combinatorial principles □κ, λ for 1 λ κ such that □κ□κ, 1 □κ, λ □κ, (...) κ□κ*. We prove that if holds in V. As an application, we show that ZFC + PFA Con. We also obtain one Woodin cardinal as a lower bound on the consistency strength of stationary reflection at κ+ for a singular, countably closed limit cardinal κ such that # exists; likewise for the failure of □κ* at such a κ. (shrink)

We extend the construction of Mitchell and Steel to produce iterable fine structure models which may contain Woodin limits of Woodin cardinals, and more. The precise level reached is that of a cardinal which is both a Woodin cardinal and a limit of cardinals strong past it.

Since the work of Gödel and Cohen, which showed that Hilbert's First Problem was independent of the usual assumptions of mathematics, there have been a myriad of independence results in many areas of mathematics. These results have led to the systematic study of several combinatorial principles that have proven effective at settling many of the important independent statements. Among the most prominent of these are the principles diamond and square discovered by Jensen. Simultaneously, attempts have been made to find suitable (...) natural strengthenings of ZFC, primarily by Large Cardinal or Reflection Axioms. These two directions have tension between them in that Jensen's principles, which tend to suggest a rather rigid mathematical universe, are at odds with reflection properties. A third development was the discovery by Shelah of "PCF Theory", a generalization of cardinal arithmetic that is largely determined inside ZFC. In this paper we consider interactions between these three theories in the context of singular cardinals, focusing on the various implications between square and scales, and on consistency results between relatively strong forms of square and stationary set reflection. (shrink)