A Complete First-Order Logic with Infinitary Predicates by H. J. KeislerReview by: Carol KarpThe Journal of Symbolic Logic, Vol. 31, No. 2 (Jun., 1966), p. 269Published by: Association for Symbolic LogicStable URL: http://www.jstor.org/stable/2269833 .

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REVIEWS 269

H. J. KEISLER. A complete first-order logic with infinitary predicates. Funda- menta mathematicae, vol. 52 (1963), pp. 177-203.

The formal systems considered in this paper are adaptations of ordinary first-order predicate logic to languages with infinitary predicates, but finitary formation rules. The languages have a set V of variables and have p(p)-placed predicate symbols p, p (p) & V. For a given cardinal m < V+, formulas of L(V, ,u, m) are formed by the ordinary propositional connectives and by quantifications (VW)O where W is a set of variables of power < m. The axioms and rules of inference are modifications of those of first-order predicate logic, substitution rules are added. Proofs are finite in length.

The main theorem is the Extended Completeness Theorem: Every consistent set of formulas of L(V, ,i, m) is satisfiable in a structure of type s. Moreover if the number of formulas is ? n and it = n/i for all predicate symbols p, then the structure may be taken to be of power n. The theorem is extended to cover languages with infinitary terms as well. The method of proof is a refinement of Henkin's completeness proof.

The languages L(V, y, m) are an almost perfect match for the Halmos polyadic algebras. The appendix of the paper contains the author's proof that the extended completeness theorem implies the representation theorem for polyadic algebras of infinite degree: Every such algebra is homomorphic to a subalgebra of a two-valued functional polyadic algebra. An algebraic proof of this result, found independently by Daigneault, appears in the article by Daigneault and Monk which immediately precedes the present article (XXIX 148).

The compactness of the languages is an immediate consequence of the main theorem. However, the compactness is destroyed by allowing equality, or by admitting in- finitary propositional connectives, or, it seems, by extending the languages in any way that makes it possible to express interesting properties of infinitary predicates. None- theless, they are a useful technical tool. CAROL KARP

S. FEFERMAN. Arithmetization of metamathematics in a general setting. Ibid., vol. 49 no. 1 (1960), PP. 35-92.

This paper treats such questions as the possibility of proving in theory d that theory . is consistent, and the existence of a relative interpretation of a into d. In doing so it naturally discusses and extends G6del's underivability theorems, and touches on many important results in this area discovered since G6del's work. Here important new results and ideas are added to this body of ideas. In developing his material the author gives a very clear exposition of results (often in sharpened form) which are strewn through a considerable and not always lucid literature. This review will not attempt to give a complete summary of the paper.

A first-order theory a? is specified by the language K and the set of axioms A, -= <A, K>. A set B of non-negative numbers is numerated by a formula P of d if k e B if and only if FRY P(k) , where A is the number symbol for h. G6del's second underiv- ability theorem is sometimes loosely stated as saying that if .1 is a consistent theory with recursively enumerable axiom system A, containing a sufficient portion of arithmetic, then the arithmetized consistency assertion Consf is not provable in d. The construction of the formula Cons involves one important ambiguity: it involves the construction of a formula a numerating A in d. Thus one should write Con. rather than ConW,. It turns out that sometimes one must impose restrictions on a: to avoid unnatural a. the author introduces the class of RE-formulas. (A formula a

is equivalent over the predicate calculus to an RE-formula if and only if it is equi- valent to a formula obtained from a quantifier-free one by prefixing existential and bounded universal quantifiers.) A correct version of Gsidel's second underivability

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