We conclude that T is axiomatizable. In fact, it proves to be possible, by selecting a suitable set of logical axioms, to use the operation of detachment as the only operation of inference in formulating adequate definitions of derivability and logical validity. If fP is a sentence of B', then fP is valid in T P and hence derivable from the set of all sentences p P where P is valid in T. Eundamentale Begriffe der Methodologie der deduktiven Wissenschaften I. Bulletin of the American Mathematical Society, vol.
Some subtheories of F are known which are also essentially undecidable, e. We assume that the set of logical axioms has been selected in such a way that the operation of detachment suffices as the only operation of inference cf. The discussion in this section will further contribute to widening the applicability of our methods and results. Consider now the sentences Da: 3 L1,. Compatible theories , 12, 18f. Conditions 1 and 2 have thus been established.
As is well known, the decision problem is one of the central problems of contemporary metamathematics. Ii We assume that in this theory an infinite sequence of terms. To obtain 3 , we consider the following four sentences of T: 9 , Cf. Even without this assumption, however, the omission of condition ii would not invalidate the proof of the essential results of this section, i. Clearly Til is a complete and consistent extension of T' and has the same constants as T'. Part I, Journal of Symbolic Logic, vol. Clearly, l' is a subset of G.
This definition oftP P could be formulated more precisely with the help of a recursive procedure, starting with atomic formulas and passing through all the operations by means of which compound formulas are constructed from simpler ones. We also assume that under this definition every sentence which is logically derivable from a set of valid sentences is itself valid, and that consequently every logically valid sentence is valid; this is the only condition imposed upon the definition of validity. T is constructed as a theory with standard formalization in which the set of non-logical constants consists of six symbols, 0, e, I, 1, I. Finally, we denote by I' the relation which holds between any two functions I,g E l' if and only if g is an iteration of I. Two general theorems on undefinability and undecidability. On the other hand, various interesting extensions of G are known to be undecidable, though not essentially undecidable. We express these results briefly by saying that Peano's arithmetic is, not only undecidable, but also essentially undecidable.
Thus, if p - En, we have n - m 2 + p whenever p ~ 2·m; therefore, by Qa, all the formulas. Monatshefte fiir Mathematik und Physik, vol. Thus there is a recursive set B' of valid sentences of T P such that every sentence valid in T P is derivable from it. Consequently, conclusion ii also holds, and the proof is complete. For many extensions of G, e. Metamathematics, 3, 12 Method in proofs of undecidability , see: Direct Indirect - of eliminating quantifiers, 19, 63 Model, Ilf. From these assumptions it follows, in particular, that the function D is also recursive.
Recursiveness and definability· in subtheories of arithmetic. An extension T2 of T1 is referred to as a finite extension if there is a finite set A of valid sentences of T2 such that every valid sentence of T2 is derivable from a set of sentences which are valid in T10r belong to A. It suffices for the proof of Theorem 3 and Corollary 4 outlined below; however, condition iii will be essentially involved in the proof of Theorem 1 and Corollary 2. Two further notions, those of logical derivability and logical validity, are involved in the metamathematical discussion of any theory T. Let now F be any function definable in T, and let rp be a formula defining F. Journal of Symbolic Logic, vol. Consequently every sentence of A' is valid in T.
Technical symbol, 6 Tenth problem of Hilbert, 35 Term, 7, 42ff. Journal of Symbolic Logic, vol. For instance, let be a binary operation symbol occurring in T, and let fP be the sentence Ax Ay Yz z x y. A sentence is valid in T7 if it is satisfied in the system N, 0, B, +,. Since only few theories turn out to be decidable 3, most endeavors are directed toward a negative solution. Hence, by including all tautological sentences in the set of logical axioms, we obtain a simple characterization of derivability in terms of logical axioms. A remark concerning decidability of complete theories.
Hence, using 7 , we finally conclude that 6 is valid. The new definition is weaker than the original one and differs from the latter by the absence of condition iii. It can be shown that Theory Q, i. For the time being we make no other assumptions regarding the nature of this correspondence. Robinson in 1950 to arbitrary integers.