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A note on the Entscheidungsproblem

  • Alonzo Church (a1)

In a recent paper the author has proposed a definition of the commonly used term “effectively calculable” and has shown on the basis of this definition that the general case of the Entscheidungsproblem is unsolvable in any system of symbolic logic which is adequate to a certain portion of arithmetic and is ω-consistent. The purpose of the present note is to outline an extension of this result to the engere Funktionenkalkul of Hilbert and Ackermann.

In the author's cited paper it is pointed out that there can be associated recursively with every well-formed formula a recursive enumeration of the formulas into which it is convertible. This means the existence of a recursively defined function a of two positive integers such that, if y is the Gödel representation of a well-formed formula Y then a(x, y) is the Gödel representation of the xth formula in the enumeration of the formulas into which Y is convertible.

Consider the system L of symbolic logic which arises from the engere Funktionenkalkül by adding to it: as additional undefined symbols, a symbol 1 for the number 1 (regarded as an individual), a symbol = for the propositional function = (equality of individuals), a symbol s for the arithmetic function x+1, a symbol a for the arithmetic function a described in the preceding paragraph, and symbols b1, b2, …, bk for the auxiliary arithmetic functions which are employed in the recursive definition of a; and as additional axioms, the recursion equations for the functions a, b1, b2, …, bk (expressed with free individual variables, the class of individuals being taken as identical with the class of positive integers), and two axioms of equality, x = x, and x = y →[F(x)→F(y)].

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1 An unsolvable problem of elementary number theory, American journal of mathematics, vol. 58 (1936).

2 Grundzüge der theoretischen Logik, Berlin 1928.

3 Definitions of the terms well-formed formula and convertible are given in the cited paper.

4 Cf. Ackermann, Wilhelm, Begründung des “tertium non datur” mittels der Hilbertschen Theorie der Widerspruchsfreiheit, Mathemaiische Annalen, vol. 93 (19241925), pp. 1136; Neumann, J. v., Zur Hilbertschen Beweistheorie, Mathematische Zeitschrift, vol. 26 (1927), pp. 146; Herbrand, Jacques, Sur la non-contradiction de l'arithmétique, Journal für die reine und angewandte Mathematik, vol. 166 (19311932), pp. 18.

5 In lectures at Princeton, N. J., 1936. The methods employed are those of existing consistency proofs.

6 By the Entscheidungsproblem of a system of symbolic logic is here understood the problem to find an effective method by which, given any expression Q in the notation of the system, it can be determined whether or not Q is provable in the system. Hilbert and Ackermann (loc. cit.) understand the Entscheidungsproblem of the engere Funktionenkalkül in a slightly different sense. But the two senses are equivalent in view of the proof by Kurt Gödel of the completeness of the engere Funktionenkalkiil (Monatshefte für Mathematik und Physik, vol. 37 (1930), pp. 349360).

7 From this follows further the unsolvability of the particular case of the Entscheidungsproblem of the engere Funktionenkalkül which concerns the provability of expressions of the form (Ex 1)(Ex 2)(Ex 3)(y 1)(y 2) …(y n)P, where P contains no quantifiers and no individual variables except x1, x2, x3, y1, y2, …, yn. Cf. Gödel, Kurt, Zum Entscheidungsproblem des logischen Funktionenkalküls, Monatshefte für Mathematik und Physik, vol. 40 (1933), pp. 433–143.

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The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
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