. Starting with GCH and a P3-hypermeasurable cardinal, a model is produced in which 2 = + if is a successor cardinal and 2 = ++ if is a limit cardinal. The proof uses a Reverse Easton extension followed by a modified Radin forcing. 1. INTRODUCTION 1. Historical remarks and statement of the main result. The continuum problem is an old one, dating back to Cantor and his statement of the Continuum Hypothesis in [Ca]. Put in a modern form which might have puzzled Cantor, the problem is to determine which behaviours of the continuum function 7\Gamma! 2 are consistent with ZFC. Throughout this paper ZFC will be the base set theory, though as we see below strong set-theoretic hypotheses will play an essential role in the result. Before Godel progress on the continuum problem was made by the descriptive set theorists, who showed that certain easily definable sets of reals could not be counterexamples to CH. Godel [G] took the major step forward of showing that in a certain ...

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Abstract. Our main results are: 1) every countably certified extender that coheres with the core model K is on the extender sequence of K, 2)Kcomputes successors of weakly compact cardinals correctly, 3) every model on the maximal 1-small construction is an iterate of K, 4) (joint with W. J. Mitchell) K�κ is universal for mice of height ≤ κ whenever κ ≥ℵ2,5)ifthereisaκ such that κ is either a singular countably closed cardinal or a weakly compact cardinal, and � <ω κ fails, then there are inner models with Woodin cardinals, and 6) an ω-Erdös cardinal suffices to develop the basic theory of K. 1.

When modern set theory is applied to conventional mathematical problems, it has a disconcerting tendency to produce independence results rather than theorems in the usual sense. The resulting preoccupation with “consistency ” rather than “truth ” may be felt to give the subject an air of unreality. Even elementary questions about the basic arithmetical operations of exponentiation in the context of infinite cardinalities, like the value of 2 ℵ0, cannot be settled on the basis of the usual axioms of set theory (ZFC). Although much can be said in favor of such independence results, rather than undertaking to challenge such prejudices, we have a more modest goal; we wish to point out an area of contemporary set theory in which theorems are abundant, although the conventional wisdom views the subject as dominated by independence results, namely, cardinal arithmetic. To see the subject in this light it will be necessary to carry out a substantial shift in our point of view. To make a very rough analogy with another generalization of ordinary arithmetic, the natural response to the loss of unique factorization caused

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

Some years ago A. L. S. Corner proved that every countable and cotorsion-free ring can be realized as the endomorphism ring of some torsion-free abelian group. This result has many interesting consequences for abelian groups. Using a set-theoretic axiom Vk., which follows for instance from V = L, we can drop the countability condition in Corner's theorem.

The objective of this paper is to assay several forms of the axiom of choice that have been deemed constructive. In addition to their deductive relationships, the paper will be concerned with metamathematical properties effected by these choice principles and also with some of their classical models.

The Singular Cardinal Hypothesis (SCH) asserts that if κ is any singular strong limit cardinal then 2 κ = κ +. It is known to be consistent that the SCH fails: Prikry [Pr] obtains a model of ¬SCH from a model in which the GCH fails at a measurable cardinal κ, and Silver in turn (see [KM]) obtains the failure of the GCH at a

Introduction The fact that small cardinals (for example @ 1 and @ 2 ) can consistently have properties similar to those of large cardinals (for example measurable or supercompact cardinals) is a recurring theme in set theory. In these notes I discuss three examples of this phenomenon; stationary reflection, saturated ideals and the tree property. These notes represent approximately the contents of a series of expository lectures given during the Set Theory meeting at CRM Barcelona in June 1996. None of the results discussed here is due to me unless I say so explicitly. I would like to express my thanks to Joan Bagaria and Adrian Mathias for organising a very enjoyable meeting. 1 2 Large cardinals and elementary embeddings We begin by reviewing the formulation of large cardinal properties in terms of elementary embeddings. See [40], [22] or [21] for more on this topic. We will write "j : V<F14.4

Let (At | i e I) be an indexed family of nonempty intervals of a linearly ordered set {L, <). Let (/, E) be the intersection graph of (^4, |IG/), that is {i,j}eE if and only if At n At J = 0. In [6], R. Rado considered the following sentence R(K), where K is a cardinal number. If Chr(J, E n |V] 2) ^ K for all;£/, |J | ^ K +, then Chr(/, E) < K. He proved (see [6, Theorem 2]) that R(K) holds for every finite K, and conjectured (see [6, Conjecture 1]) that R(K) holds for every cardinal K. In this note we show that if /?(X0) holds, then there is an inner model of set theory with many measurable cardinals. On the other hand, using consistency of the existence of a supercompact cardinal, we prove that /?(X0) is consistent with the usual axioms of set theory. We also prove a few results about the intersection graph of (/I, | i e /). 1. Notation and definitions Let (L, <) be a linearly ordered set. An interval of L is a nonempty subset A of L such that if x < y < z and x,z e A, then ye A. Let (A { | i e I) be a family of intervals of L. The intersection graph of (At \ie I) is the graph (/, E), where E = {{ij} E [/] 2 | A, n Aj ± 0}. If ^ = (V, F) is a graph, then by Chr (^) we denote the chromatic number of ^, that is the least cardinal K such that there is a representation