R. Sherman Lehman has called my attention to an error which vitiates the proofs of two theorems in the paper mentioned in the title (this Journal, vol. 22, pp. 19–35, 113–129). New proofs are here presented.

Proof of Lemma, p. 27. Induction Step: Assume as hypothesis of induction that the lemma holds for formulas P with n—1 or fewer free restricted variables and consider a P having n (n>0) such variables.

This paper presents several results whose common element is their connection with Hubert's tenth problem.1 The one that seems most striking (perhaps because it can be stated with a minimum of technical apparatus) is this: there does not exist an algorithm for determining whether or not a polynomial (in n variables) represents every integer.2

In addition, the present paper (i) gives a simple characterization of the diophantine sets (of positive, non-negative, and arbitrary integers), and (ii) gives a rigorous proof of Myhill's theorem [6] that there is no decision method for statements of the form

According to Gödel's completeness theorem, every consistent theory1 has a model whose domain is a set of natural numbers. The objects of the model corresponding to the predicate symbols of the theory are then predicates of natural numbers. Kleene [5] p. 398 showed that, if the theory is axiomatizable,2 then the model can be chosen so that these predicates are in both two-quantifier forms, i.e., they can be expressed in both the forms (x)(Ey)R and (Ex)(y)S with R and S recursive. An alternative proof has been given by Hasen jaeger [3].

Let the prepositional calculus (PC) be cast in the following form: (a) The primitive signs of PC are to be a denumerably infinite hst of propositional letters, the two connectives ‘∼’ and ‘&’, and the two parentheses ‘(‘and’)’; (b) The formulas of PC, referred to hereafter by ‘A’, ‘B’, ‘C’, and ‘D,’ are to be all finite sequences of primitive signs of PC; (c) The well-formed formulas (wffs) of PC are to be all propositional letters, all formulas of the form ∼A, where A is a wff of PC, and all formulas of the form (A&B), where A and B are wffs of PC; (d) (A⊃B) is to be short for ∼(A&∼B), (AνB) short for ∼(∼A&∼B), and (A ≡ B) short for (((A ⊃B)&(B⊃A)).