Hostname: page-component-8448b6f56d-sxzjt Total loading time: 0 Render date: 2024-04-19T22:08:30.546Z Has data issue: false hasContentIssue false
  • English
  • Français

Degrees of models

Published online by Cambridge University Press:  12 March 2014

J. R. Shoenfield
J. R. Shoenfield*
Affiliation:
Duke University
Get access Rights & Permissions [Opens in a new window]

Extract

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].

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 25 , Issue 3 , September 1960 , pp. 233 - 237
Copyright
Copyright © Association for Symbolic Logic 1960

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Craig, W., On axiomatizdbility within a system, this Journal, vol. 18 (1953), pp. 30–32.Google Scholar
[2]Gödel, K., Die Vollständigkeit der Axiome des logischen Funktionenkalküls, Monatshefte für Mathematik und Physik, vol. 37 (1930), 349–360.CrossRefGoogle Scholar
[3]Hasenjaeger, G., Eine Bemerkung zu Henkin's Beweis für die Vollständigkeit der Prädikatenkalküls der ersten Stufe, this Journal, vol. 18 (1953), pp. 42–48.Google Scholar
[4]Henkin, L., The completeness of the first-order functional calculus, this Journal, vol. 14 (1949), pp. 159–166.Google Scholar
[5]Kleene, S. C., Introduction to metamathematics, Amsterdam (North Holland), Groningen (Noordhoff), Toronto and New York (Van Nostrand), 1952.Google Scholar
[6]Kleene, S. C., Arithmetical predicates and function quantifiers, Transactions of the American Mathematical Society, vol. 79 (1955), pp. 312–340.CrossRefGoogle Scholar
[7]Kleene, S. C., On the forms of the predicates in the theory of constructive ordinals (second paper), American journal of mathematics, vol. 77 (1955), pp. 405–428.CrossRefGoogle Scholar
[8]Kleene, S. C. and Post, E. L., The upper semi-lattice of degrees of recursive unsolvability, Annals of mathematics, vol. 59 (1954), 379–407.CrossRefGoogle Scholar
[9]Kreisel, G., A variant to Hubert's theory of the foundations of arithmetic, British journal for the philosophy of science, vol. 4 (1953), pp. 107–129.CrossRefGoogle Scholar
[10]Kreisel, G., Note on arithmetic models for consistent formulae of the predicate calculus II, Proceedings of the Xlth International Congress of Philosophy (Brussels, August 20–26,1953), Amsterdam (North Holland) and Louvain (E. Nauwelaerts) 1953, vol. XIV, pp. 39–49.Google Scholar
[11]Mostowski, A., A formula with no recursively enumerable model, Fundamenta matnematicae, vol. 42 (1955), pp. 125–140.CrossRefGoogle Scholar
[12]Putnam, H., Arithmetic models for consistent formulae of quantification theory (abstract), this Journal, vol. 22 (1957), pp. 110.Google Scholar
[13]Shoenfield, J. R., On degrees on unsolvability, Annals of mathematics, vol. 69 (1959), pp. 644–653.CrossRefGoogle Scholar
[14]Tarski, A., Logic, semantics, metamathematics, Oxford (Clarendon), 1956.Google Scholar