In this paper we consider whether L(R) has “enough information” to contain a counterexample to the continuum hypothesis. We believe this question provides deep insight into the difficulties surrounding the continuum hypothesis. We show sufficient conditions for L(R) not to contain such a counterexample. Along the way we establish many results about nonstationary towers, non-reflecting stationary sets, generalizations of proper and semiproper forcing and Chang's conjecture.

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)

It is consistent that, for every n ≥ 2, every stationary subset of ω n consisting of ordinals of cofinality ω k, where k = 0 or k ≤ n - 3, reflects fully in the set of ordinals of cofinality ω n - 1. We also show that this result is best possible.

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 present a general construction of a □κ-sequence in Jensen's fine structural extender models. This construction yields a local definition of a canonical □κ-sequence as well as a characterization of those cardinals κ, for which the principle □κ fails. Such cardinals are called subcompact and can be described in terms of elementary embeddings. Our construction is carried out abstractly, making use only of a few fine structural properties of levels of the model, such as solidity and condensation.

I develop a translation procedure between λ-structures, which correspond to premice in the Friedman–Jensen indexing convention on the one hand and s-structures, which are essentially the same as premice in the Mitchell–Steel indexing scheme.

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 show that either of the following hypotheses imply that there is an inner model with a proper class of strong cardinals and a proper class of Woodin cardinals. 1) There is a countably closed cardinal k ≥ N₃ such that □k and □(k) fail. 2) There is a cardinal k such that k is weakly compact in the generic extension by Col(k, k⁺). Of special interest is 1) with k = N₃ since it follows from PFA by theorems of (...) Todorcevic and Velickovic. Our main new technical result, which is due to the first author, is a weak covering theorem for the model obtained by stacking mice over $K^c ||k.$. (shrink)

We prove that the statement "For every pair A, B, stationary subsets of ω 2 , composed of points of cofinality ω, there exists an ordinal α such that both A ∩ α and $B \bigcap \alpha$ are stationary subsets of α" is equiconsistent with the existence of weakly compact cardinal. (This completes results of Baumgartner and Harrington and Shelah.) We also prove, assuming the existence of infinitely many supercompact cardinals, the statement "Every stationary subset of ω ω + 1 (...) has a stationary initial segment.". (shrink)

Continuing the work of Fuchs [1], I show that the translation functions developed previously map iterable λ-structures to iterable s-structures and vice versa. To this end, I analyse how the translation functions interact with the formation of extender ultrapowers and normal iterations. This analysis makes it possible to translate iterations, and, in a last step, iteration strategies, thus arriving at the result.

We investigate both iteration hypotheses and extender models at the level of one supercompact cardinal. The HOD Conjecture is introduced and shown to be a key conjecture both for the Inner Model Program and for understanding the limits of the large cardinal hierarchy. We show that if the HOD Conjecture is true then this provides strong evidence for the existence of an ultimate version of Gödel's constructible universe L. Whether or not this "ultimate" L exists is now arguably the central (...) issue for the Inner Model Program. (shrink)

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)

We investigate large cardinal axioms beyond the level of ω-huge in context of the universality of the suitable extender models of [Suitable Extender Models I, J. Math. Log.10 101–339]. We show that there is an analog of ADℝ at the level of ω-huge, more precisely the construction of the minimum model of ADℝ generalizes to the level of Vλ+1. This allows us to formulate the indicated generalization of ADℝ and then to prove that if the axiom holds in V at (...) a proper class of λ then in every suitable extender model, the axiom holds at a proper class of λ. (shrink)

It is consistent that, for every $n \geq 2$, every stationary subset of $\omega_n$ consisting of ordinals of cofinality $\omega_k$, where $k = 0$ or $k \leq n - 3$, reflects fully in the set of ordinals of cofinality $\omega_{n - 1}$. We also show that this result is best possible.

We investigate how weak square principles are denied by Chang’s Conjecture and its generalizations. Among other things we prove that Chang’s Conjecture does not imply the failure of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\square_{\omega_1, 2}}$$\end{document}, i.e. Chang’s Conjecture is consistent with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\square_{\omega_1, 2}}$$\end{document}.