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Barwise: Infinitary Logic and Admissible Sets

Published online by Cambridge University Press:  15 January 2014

H. Jerome Keisler  and
Julia F. Knight
H. Jerome Keisler
Affiliation:
Department of Mathematics, University of Wisconsin, 480 Lincoln Drive, Madison Wi 53706, U.S.A.E-mail: , [email protected]
Julia F. Knight
Affiliation:
Department of Mathematics, University of Notre Dame, 255 Hurley Hall, Notre Dame in 46556, U.S.A.E-mail: , [email protected]
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Extract

§0. Introduction. In [16], Barwise described his graduate study at Stanford. He told of his interactions with Kreisel and Scott, and said how he chose Feferman as his advisor. He began working on admissible fragments of infinitary logic after reading and giving seminar talks on two Ph.D. theses which had recently been completed: that of Lopez-Escobar, at Berkeley, on infinitary logic [46] (see also the papers [47], [48]), and that of Platek [58], at Stanford, on admissible sets.

Barwise's work on infinitary logic and admissible sets is described in his thesis [4], the book [13], and papers [5]—[16]. We do not try to give a systematic review of these papers. Instead, our goal is to give a coherent introduction to infinitary logic and admissible sets. We describe results of Barwise, of course, because he did so much. In addition, we mention some more recent work, to indicate the current importance of Barwise's ideas. Many of the central results are stated without proof, but occasionally we sketch a proof, to indicate how the ideas fit together.

Chapters 1 and 2 describe infinitary logic and admissible sets at the time Barwise began his work, circa 1965. From Chapter 3 on, we survey the developments that took place after Barwise appeared on the scene.

§1. Background on infinitary logic. In this chapter, we describe the situation in infinitary logic at the time that Barwise began his work. We need some terminology. By a vocabulary, we mean a set L of constant symbols, and relation and operation symbols with finitely many argument places. As usual, by an L-structureM, we mean a universe set M with an interpretation for each symbol of L.

Type
Research Article
Information
Bulletin of Symbolic Logic , Volume 10 , Issue 1 , March 2004 , pp. 4 - 36
Copyright
Copyright © Association for Symbolic Logic 2004

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