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Some proofs of independence in axiomatic set theory1

Published online by Cambridge University Press:  12 March 2014

Elliott Mendelson
Elliott Mendelson*
Affiliation:
Cornell University
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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

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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

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