Large cardinal

Abstract: The decidability of the word problem for the free left distributive law is proved by introducing a structure group which describes the underlying identities. This group is closely connected with Artin's braid group B. . Braid colourings associated with free left distributive structures are used to show the existence of a unique ordering on the braids which is compatible with left translation and such that every generator ai is preponderant over all ak with k > i . This ordering is a linear ordering. The first goal of the present paper is to give a proof of the following result, which had been conjectured for several years: Theorem. There is an effective algorithm for deciding whether a given identity is or is not a consequence of the left distributivity identity x(yz) = (xy)(xz). Until recently this question has had a rather unusual status. Conditional solutions were given independently in [6] and [24], where the decision problem was reduced to a specific algebraic hypothesis, one which had been shown by Richard Laver to be a consequence of a very strong set-theoretical axiom, of a type which certainly cannot be derived from the usual axioms of set theory, namely, a large cardinal axiom. The question as to whether a strong axiom of this type was actually needed remained open. Opinions were in fact divided: a connection between large cardinals and a purely finitistic problem of this caliber would seem paradoxical, but it is well known that problems of a combinatorial type can embody surprisingly strong proof principles (see for instance [28]), and some work on free distributive structures has shown that they do give rise to intrinsically complex objects, typically nonprimitive recursive ones (cf. [12, 13]). We will show that in the present case a solution which is purely algebraic in terms of the methods employed and the spirit of the argument can in fact be given. In particular, no unusual set-theoretical axioms are required for this argument. The decision method described in [6] was fully effective, but the proof of the correctness of the algorithm amounted to a direct invocation of this specific algebraic hypothesis which followed from a large cardinal axiom, with no hint of a direct proof. We shall refine this decision method below, introducing uniqueness at each step of the process, and the correctness will then be seen to follow very naturally. Received by the editors February 5, 1993. 1991 Mathematics Subject Classification. Primary 20F36, 1 7A30, 1 7A50.

Abstract: Since the work of Godel and Cohen, which showed that Hilbert's First Problem (the Continuum Hypothesis) was independent of the usual assumptions of mathematics (axiomatized by Zermelo–Fraenkel Set Theory with the Axiom of Choice, ZFC), 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 (a fundamental notion in PCF theory), and on consistency results between relatively strong forms of square and stationary set reflection.

Abstract: This paper outlines the basic theory of canonical inner models satisfying large cardinal hypotheses. It begins with the definition of the models, and their fine structural analysis modulo iterability assumptions. It then outlines how to construct canonical inner models, and prove their iterability, in roughly the greatest generality in which it is currently known how to do this. The paper concludes with some applications: genericity iterations, proofs of generic absoluteness, and a proof that the hereditarily ordinal definable sets of L(ℝ) constitute a canonical inner model.