. We study some combinatorial principles intermediate between square and weak square. We construct models which distinguish various square principles, and show that a strengthened form of weak square holds in the Prikry model. Jensen proved that a large cardinal property slightly stronger than 1-extendibility is incompatible with square; we prove this is close to optimal by showing that 1-extendibility is compatible with square. 1. Introduction In this paper we study some variations on Jensen's celebrated combinatorial principle (variously pronounced as \square kappa" or \box kappa"). is a principle which is helpful in constructing objects of cardinality + ; for example Jensen showed that if holds then there is a special + -Aronszajn tree, and every stationary subset of + contains a non-reecting stationary subset. Jensen proved [Je1] that if V = L then holds for every uncountable cardinal ( ! is a trivial theorem in ZFC). In combination with Jen...

We introduce 0 |• (“zero hand-grenade”) as a sharp for an inner model with a proper class of strong cardinals. We prove the existence of the core model K in the theory “ZFC + 0 |• doesn’t exist. ” Combined with work of Woodin, Steel, and earlier work of the author, this provides the last step for determining the exact consistency strength of the assumption in the statement of the 12th Delfino problem (cf. [12]). 0 Introduction. Core models were constructed in the papers [2], [13], [7], [15] and [16], [8] (see also [23]), [27], and [28]. We refer the reader to [6], [17], and [14] for less painful introductions into core model theory. A core model is intended to be an inner model of set theory (that is, a transitive

The four authors present their speculations about the future developments of mathematical logic in the twenty-first century. The areas of recursion theory, proof theory and logic for computer science, model theory, and set theory are discussed independently.

(“zero hand-grenade”) as a sharp for an inner model with a proper class of strong cardinals. We prove the existence of the core model K in the theory “ZFC + 0 |• doesn’t exist. ” Combined with work of Woodin, Steel, and earlier work of the author, this provides the last step for determining the exact consistency strength of the assumption in the statement of the 12th Delfino problem (cf. [10]).0 Introduction. Core models were constructed in the papers [2], [11], [6], [13] and [14], [7] (see also [21]), [24], and [25]. We refer the reader to [5], [15], and [12] for less painful introductions into core model theory. A core model is intended to be an inner model of set theory (that is, a transitive