By J.W., Leon Henkin, and Alfred Tarski (eds). Addison

Amsterdam 1965 North-Holland. 8vo., 494pp. Hardcover. VG, no DJ.

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Extra resources for The Theory of Models. Proceedings of the 1963 International Symposium at Berkeley

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Let öc>ß in 351. Assume that some element Fy in 31 sets up a univalent mapping ψ from β onto oc. , let R(ß') be the set of all y such that for some P, P forces ψ to be univalent and ordinal-valued and ψ(β') = γ. If for each such y we pick just one associated P, we obtain a set S(ß') of conditions P such that if Pi, P2 belong to S(ß'), Pi and P2 are incompatible in the sense that Pi U P 2 is not an admissible set of conditions. We shall prove below that such a set of conditions is at most countable (in 3R).

Combining some of our earlier results with Theorems 14, 15, 16, and 17, we can obtain purely algebraic characterizations of these classes, without the assumption t h a t J f G E C J . A theorem analogous to the above four theorems and which does not appear to carry over to the general case is the previously mentioned result of Keisler on Horn classes and reduced products. This problem is quite interesting because there is a proper generalization of the theorem in the logic I (see Chang [61]). Symposium on the Theory of Models ; North-Holland Publ.

In $1, the continuum has cardinality χ Τ or χ τ + ι according as χ τ is not or is co-final with ω. In this connection should be mentioned the work of Solovay who showed t h a t the set of statements 2Kn = Xi7(W) is consistent if g(n) is an increasing integer-valued function. He also obtained more general results and recently William Easton has announced a theorem which would seem to be the most general in this direction, allowing essentially any set of equalities which are not prohibited by a suitable generalization of König's lemma.

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