Abstract
After giving a precise definition of the concept of decidability it is possible to show for certain predicates (properties or relations) that they are undecidable. It is easy to show the undecidability of many predicates P which are definable by the help of concepts which are directly connected with the concept of algorithm. Typical of these proofs is that they operate using a diagonal procedure.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Thus, A.: Probleme über Veränderungen von Zeichenreihen nach gegebenen Regeln. Skr. Vidensk. Selsk. I, 10, 34 pp. (1914).
Post, E. L.: Recursive Unsolvability of a Problem of Thue. J. symbolic Logic 12, 1–11 (1947).
Markov, A. A.: The impossibility of certain algorithms in the theory of associative systems [Russ.]. Dokl. Akad. Nauk SSSR. 55, 587–590 (1947).
Turing, A. M.: The Word Problem in Semi-Groups with Cancellation. Ann. Math., Princeton 52, 491–505 (1950).
Kalmár, L.: Another Proof of the Markov-Post Theorem. Acta math. Hungaricae 3, 1–25; 26–27 [Russ.]; (1952).
Novikov, P. S.: On the algorithmic unsolvability of the word problem in group theory [Russ.]. Akad. Nauk SSSR., Matém. Inst. Trudy 44, Moscow 1955. Engl. transl. by K. A. Hirsch, Amer. Math. Soc. Translations 9 (1958), 122 pp.
Boone, W. W.: Certain Simple, Unsolvable Problems of Group Theory. V. Proc. Kon. Nederl. Akad. (A) 60, 22–27 (1957).
Boone, W. W.: Certain Simple, Unsolvable Problems of Group Theory. VI. Ibid., pp. 227–232.
Boone, W. W.: The Word Problem. Ann. Math. 70, 207–265 (1959).
Britton, J. L.: The Word Problem for Groups. Proc. London math. Soc. 8, 493 to 506 (1958).
Gödel, K.: Die Vollständigkeit der Axiome des logischen Funktionenkalküls. Mb. Math. Phys. 37, 349–360 (1930).
Tarski, A.: Der Wahrheitsbegriff in den formalisierten Sprachen. Studia Philosophica 1, 261–405 (1935). Cf. also: Logic, Semantics, Metamathematics. Papers from 1923 to 1938 by A. Tarski, Translated by J. H. Woodger, Oxford: Clarendon Press 1956. pp. 152–278.
Hermes, H., and H. Scholz: Mathematische Logik. Enzyklopädie math. Wiss. I. 1 Heft 1, I. Leipzig: B. G. Teubner 1952.
Church, A.: Introduction to Mathematical Logic I. Princeton, N. J.: Princeton University Press 1956.
Scholz, H., and G. Hasenjaeger: Grundzüge der mathematischen Logik. BerlinGöttingen-Heidelberg: Springer 1961.
Church, A.: A Note on the Entscheidungsproblem. J. symbolic Logic 1, 40–41 (1936); Correction ibid. pp. 101–102.
Kalmar, L.: Ein direkter Beweis für die allgemein-rekursive Unlösbarkeit des Entscheidungsproblems des Prädikatenkalküls der ersten Stufe mit Identität. Z. math. Logik 2, 1–14 (1956).
Trachténbrot, B. A.: Impossibility of an algorithm for the decision problem in finite classes [Russ.]. Dokl. Akad. Nauk SSSR, 70, 569–572 (1950). (This shows that there exists no algorithm by the help of which we can decide whether or not an arbitrarily given formula of the predicate calculus is valid over a finite domain of individuals.)
Ackermann, W.: Solvable Cases of the Decision Problem. Amsterdam: North-Holland Publishing Company 1954.
Suranyi, J.: Reduktionstheorie des Entscheidungsproblems im Prädikatenkalkill der ersten Stufe. Budapest-Berlin: Ungarische Akademie der Wissenschaften—VEB Deutscher Verlag der Wissenschaften 1959.
Katr, A. S., E. F. Moore, and H. Wang: Entscheidungsproblem Reduced to the AEA Case. Proc. Nat. Acad. Sei., USA. 48, 365–377 (1962).
Gödel, K.: Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme. I. Mh. Math. Phys. 38, 173–198 (1931).
Gödel, K.: Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I. Mh. Math. Phys. 38, 173–198 (1931).
Gödel, K.: On Undecidable Propositions of Formal Mathematical Systems. Mimeographed. Institute for Advanced Study, Princeton, N. J. 1934. pp. 30.
Tarski, A.: Der Wahrheitsbegriff in den formalisierten Sprachen. Studia Philosophica 1, 261–405 (1935). Cf. also: Logic, Semantics, Metamathematics. Papers from 1923 to 1938 by A. Tarski, translated by J. H. Woodger, Oxford: Clarendon Press 1956. pp. 152–278.
Church, A.: An Unsolvable Problem of Elementary Number Theory. Amer. J. Math. 58, 345–363 (1936).
Rosser, B.: Extensions of Some Theorems of Gödel and Church. J. symbolic Logic 1, 87–91 (1936).
Skolem, TH.: Einfacher Beweis der Unmöglichkeit eines allgemeinen Lösungsverfahrens für arithmetische Probleme. Norske Vidensk. Selsk. Forhandi., Trondheim 13, 1–4 (1940).
Kalmar, L.: Egyszerü példa eldönthetetlen aritmetikai problémâra. (Ein einfaches Beispiel für ein unentscheidbares arithmetisches Problem.) [Hungarian, with German abstract.] Mat. fiz. Lapok 50, 1–23 (1943).
Mosrowski, A.: Sentences Undecidable in Formalized Arithmetic. Amsterdam: North-Holland Publishing Company 1952.
Tarski, A., A. Mostowsxi and R. M. Robinson: Undecidable Theories. Amsterdam: North-Holland Publishing Company 1953. (This book discusses especially the essential undecidability of the theories it deals with.)
Goodstein, R. L.: Recursive Number Theory. Amsterdam: North-Holland Publishing Company 1957.
Grzegorczyk, A.: Fonctions Récursives. Collection de Logique Mathématique, Série A. Paris — Louvain: Gauthiers-Villars — E. Nauwelaerts 1961.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 1969 Springer-Verlag Berlin · Heidelberg
About this chapter
Cite this chapter
Hermes, H. (1969). Undecidable Predicates. In: Enumerability · Decidability Computability. Die Grundlehren der mathematischen Wissenschaften, vol 127. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-46178-1_6
Download citation
DOI: https://doi.org/10.1007/978-3-642-46178-1_6
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-46180-4
Online ISBN: 978-3-642-46178-1
eBook Packages: Springer Book Archive