¹ The terminology of this paper is that of Undecidable theories, Tarski, Mostowski, and Robinson, North Holland Publishing Co., Amsterdam (1953)Google Scholar. This work will be cited as “U.T.”.

A solution to problem (1) has erroneously been announced by Myhill, (Solution of a problem of Tarski's, this Journal, vol. 21 (1956), pp. 49–51)Google Scholar. Actually Myhill has solved a related problem; but an answer to (1) does not follow from what he proves, as he asserts. (Cf. section 7., below).

(Except for section 7) attention in the present paper is confined to theories with a finite number of constants. Kreisel has previously solved problem (1) for theories which use an infinite number of constants. A statement of Kreisel's proof appears in his review of the article by Myhill just cited; see Mathematical reviews, vol. 17, no. 8 (09 1956), p. 815Google Scholar.

² This is easily seen to be a reformulation of a problem in U.T., p. 18. (Cf. the discussion below). From now on, when the term ‘extension’ is used, ‘with the same constants’ will be understood.

³ This problem was suggested by G. Kreisel whose interest and infectious enthusiasm have led me to work on these questions.

⁴ U.T., p. 64. Also, a decision method for the elementary theory of the successor function and an arbitrary one-place predicate is given in section 3. below. This can, of course, be used to decide sentences of R.

⁵ Logical signs are used as names of themselves throughout the present paper, and not in their object-language use.

⁶ Craig's method is as follows (On axiomatizability within a system, this Journal, vol. 18 (1953) pp. 30–32Google Scholar): let {A₁, A₂, …} be a recursively enumerable but not recursive set of axioms for a theory T. Replace each axiom A_i by the conjunctions whose members are A_i repeated n times, for each n such that n is the number of a proof that A_i belongs to the set {A₁, A₂, …} (or rather that the Gödel number of A_t belongs to the corresponding recursively enumerable set of integers). The recursiveness of the new axiom set follows from the fact that the set of pairs (n, A) such that A belongs to {A₁, A₂, …} and n is the number of a proof that this is the case (in a suitable formalism) is recursive.

⁷ ⊤ and ⊥ (“tee” and “eet”) are Quine's symbols for truth and falsity (Methods of logic, Harvard (1950Google ScholarPubMed)). They are always eliminated by elementary reduction techniques; or else the whole formula reduces to ⊤ or ⊥. The logical notation in the present paper follows Quine (e.g., in using juxtaposition for conjunction; in using both ~ and − for negation), except that more dots than are strictly necessary for punctuation are sometimes used to facilitate reading. Following U.T., identity is included in each elementary theory as a logical constant.

⁸ In the sequel, a single formula will also be called a “conjunction” (with one member).

^8a Beiträge zur Algebra der Logik, insbesondere zum Entscheidungsproblem, Mathematische Annalen, vol. 9, pp. 1–36Google Scholar.

⁹ “Prime formulas” are formulas of the forms P(x + m), (x + m), x + m = y + n, x + m ≠ y + n, where m, n, may be positive, negative, or zero.

¹⁰ If the procedure eventuates in a formula of the form (∃x)^m(x ≠y), this is reduced to ⊤. Formulas of the forms (∃x)(x = y + m) and x = x + m are also eliminated.

^10a “Integers n, n + 1, n + 2, … n + 4 realize PPPP” if and only if the integer n satisfies the formula PPPP; i.e., if and only if P(n)P(n+ 1)(n + 2)P(n+3)P(n+4) is true. — And similarly in similar cases.

¹¹ O rekursivnoj otdélimosti, Doklady Akadémii Nauh USSR, vol. 88 (1953), pp. 953–956Google Scholar.

¹² At the referee's suggestion, I stress the informal character of this question. The terms ‘immediately stronger,’ ‘just weak enough,’ etc., at the beginning of section 6. are also used informally.

¹³ U.T., esp. pp. 77–80.

¹⁴ By Theorem 6 of U.T. (given below in section 7.).

¹⁵ The method is in fact contained in the proof of Lemma 1 of section 3.; the only modification is as follows: formulas of the form (∃x)^m(x ≠ y) are not reduced to ⊤ (as in fn. 10 above), but are instead replaced by (∃x)^m+1(x = x).

¹⁶ In U.T. only the weaker statement is given (p. 49) that a theory is essentially undecidable if every recursive function is definable in it.