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The compleat 0 . (English) Zbl 0712.03044

0 is the (set of Gödel numbers of the formulae in the) theory of the structure <L[U],,κ,U,x 1 ,x 2 ,x 3 ,···,y 1 ,y 2 ,y 3 ,···> where U is a normal measure on the cardinal κ, and <x 1 ,x 2 ,x 3 ,···> and <y 1 ,y 2 ,y 3 ,···> are increasing enumerations of sets X and Y of ordinals with X<κ<Y which are indiscernibles for <L[U],,κ,U>, in the sense that the truth value of

<L[U],,κ,U>ϕ[α 1 ,···,α m ,β 1 ,···,β n ]

is independent of the choice of the increasing sequences <α 1 ,···,α m > from X and <β 1 ,···,β n > from Y·

0 was first defined by Solovay, who showed that if there were to measurable cardinals κ, λ with κ<λ then 0 exists [see A. R. D. Mathias, Period. Math. Hung. 10, 109-175 (1979; Zbl 0417.03021)]. The set 0 plays a similar role for inner models with a measurable cardinal as 0 # plays for L.

Various results about 0 have appeared in the literature. However the survey paper under review is the first detailed presentation of the theory of 0 . Section 1 defines 0 and contains the appropriate Ehrenfeucht-Mostowski theory and shows that sufficiently strong large cardinal hypotheses generate models L[U] with indiscernibles. In section 2 connections are established between classes of indiscernibles for various of these models. Finally, section 3 reviews various characterizations of the existence of 0 .

Reviewer: N.H.Williams
MSC:
03E45Constructibility, ordinal definability, and related notions
03E55Large cardinals
03E10Ordinal and cardinal arithmetic