Hostname: page-component-7479d7b7d-t6hkb Total loading time: 0 Render date: 2024-07-12T09:18:27.175Z Has data issue: false hasContentIssue false
  • English
  • Français

Some proofs of independence in axiomatic set theory1

Published online by Cambridge University Press:  12 March 2014

Elliott Mendelson
Elliott Mendelson*
Affiliation:
Cornell University
Get access Rights & Permissions [Opens in a new window]

Extract

1. Gödel's theorem that sufficiently strong formal systems cannot prove their own consistency and Tarski's method for constructing truth-definitions can be combined to give several independence results in axiomatic set theory. In substance, the following theorems can be obtained: (a) The existence of inaccessible ordinals is not provable from the axioms of set theory, if these axioms are consistent, (b) The axiom of infinity is independent of the other axioms, if these other axioms are consistent, (c) The axiom of replacement is independent of the other axioms, if these other axioms are consistent. In all cases, V = L will be included as an axiom.

The result (a) concerning inaccessible ordinals already has been proved in Shepherdson [10] and Mostowski [6], but their proofs are somewhat different from the one given here. According to Mostowski [6], Kuratowski essentially had a proof of (a) in 1924. Propositions (b) and (c) have been proved, for axiomatic set theory without the axiom V = L, by Bernays [1] pp. 65–69. The method of proof used in this paper is due to Firestone and Rosser [2].

An outline of a similar proof along these lines is given in Rosser [7] pp. 60–62.

As our system G of set theory, we choose Gödel's system A, B, C, as given in [4], except for the following changes.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 21 , Issue 3 , September 1956 , pp. 291 - 303
Copyright
Copyright © Association for Symbolic Logic 1956

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.)

Footnotes

1

From a thesis in partial fulfillment of the requirements for the degree of M. A. in the Department of Mathematics of Cornell University, written while the author was an Erastus Brooks Fellow in 1952–1953. I should like to thank Professor J. Barkley Rosser for many valuable suggestions concerning the subject of this paper.

References

BIBLIOGRAPHY

[1]Bernays, P., A system of axiomatic set theory — Part VI, this Journal, vol. 13 (1948), pp. 65–79.Google Scholar
[2]Firestone, C. D. and Rosser, J. B., The consistency of the hypothesis of accessibility (abstract), this Journal, vol. 14 (1949), p. 79.Google Scholar
[3]Gödel, K., Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I, Monatshefte für Mathematik und Physik, vol. 38 (1931), pp. 173–198.CrossRefGoogle Scholar
[4]Gödel, K., The consistency of the continuum hypothesis, revised ed., Princeton, N.J. (Princeton University Press), 1951.Google Scholar
[5]Kuratowski, C., Sur l'état actuel de l'axiomatique de la théorie des ensembles, Annates de la Société Polonaise de Mathématique, vol. 3 (1924), pp. 146–147.Google Scholar
[6]Mostowski, A., An undecidable arithmetic statement, Fundamenta mathematicae, vol. 36 (1949), pp. 143–164.CrossRefGoogle Scholar
[7]Rosser, J. Barkley, Deux esquisses de logique, Paris (Gauthier-Villars), 1955.Google Scholar
[8]Rosser, J. Barkley, Logic for mathematicians, New York (McGraw-Hill), 1953.Google Scholar
[9]Rosser, J. Barkley and Wang, Hao, Non-standard models for formal logics, this Journal, vol. 15 (1950), pp. 113–129.Google Scholar
[10]Shepherdson, J. C., Inner models for set theory, Part I, this Journal, vol. 16 (1951), pp. 161–190; Part II, Inner models for set theory, Part I, this Journal, vol. 17 (1952), pp. 225–237.Google Scholar
[11]Sierpinski, W., Leçons sur les nombres transfinis, Paris, 1928.Google Scholar
[12]Tarski, A., Der Wahrheistbegriff in den formalisierten Sprachen, Studia philosophica, vol. 1 (1936), pp. 261–405.Google Scholar
[13]Tarski, A., Unerreichbare Kardinalzahlen, Fundamenta mathematicae, vol. 30 (1938), pp. 66–89.CrossRefGoogle Scholar
[14]Wang, Hao, Truth-definitions and consistency proofs, Transactions of the American Mathematical Society, vol. 73 (1952), pp. 243–275.CrossRefGoogle Scholar