AN Ω-LOGIC PRIMER Introduction One Family of Results in Modern Set
AN -LOGIC PRIMER JOAN BAGARIA, NEUS CASTELLS, AND PAUL LARSON Abstract. In [12], Hugh Woodin introduced -logic, an approach to truth in the universe of sets inspired by recent work in large cardinals. Expository accounts of -logic appear in [13, 14, 1, 15, 16, 17]. In this paper we present proofs of some elementary facts about -logic, relative to the published literature, leading up to the generic invariance of - logic and the -conjecture. Introduction One family of results in modern set theory, called absoluteness results, shows that the existence of certain large cardinals implies that the truth values of certain sentences cannot be changed by forcing1. Another family of results shows that large cardinals imply that certain de¯nable sets of reals satisfy certain regularity properties, which in turn implies the existence of models satisfying other large cardinal properties. Results of the ¯rst type suggest a logic in which statements are said to be valid if they hold in every forcing extension. With some technical modi¯cations, this is Woodin's - logic, which ¯rst appeared in [12]. Results of the second type suggest that there should be a sort of internal characterization of validity in -logic. Woodin has proposed such a characterization, and the conjecture that it succeeds is called the -conjecture. Several expository papers on -logic and the -conjecture have been published [1, 13, 14, 15, 16, 17]. Here we briefly discuss the technical background of -logic, and prove some of the basic theorems in this area. This paper assumes a basic knowledge of Set Theory, including con- structibility and forcing.
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