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A method of modelling the formalism of set theory in axiomatic set theory1

Published online by Cambridge University Press:  12 March 2014

A. H. Kruse
A. H. Kruse*
Affiliation:
New Mexico State University, University Park, N.M.
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Extract

As is well known, some paradoxes arise through inadequate analysis of the meanings of terms in a language, an adequate analysis showing that the paradoxes arise through a lack of separation of an object theory and a metatheory. Under such an adequate analysis in which parts of the metatheory are modelled in the object theory, the paradoxes give way to remarkable theorems establishing limitations of the object theory.

Such a modelling is often accomplished by a Gödel numbering. Here we shall use a somewhat different technique in axiomatic set theory, from which we shall reap a few results having the effect of comparing the strength of various axiom schema of comprehension for sets and classes (cf. the numbered results of §§5–7). Similar results were obtained by A. Mostowski [7] using Gödel numbering (cf. 5.3 and 7.3 below).

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 28 , Issue 1 , March 1963 , pp. 20 - 34
Copyright
Copyright © Association for Symbolic Logic 1964

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Footnotes

1

The writer expresses thanks to the referee for several comments which have helped to round out this paper.

References

[1]Bernays, P., A system of axiomatic set theory. Parts I-VII, this Journal, vol. 2 (1937) pp. 65–77, vol. 6 (1941), pp. 1–17, vol. 7 (1942) pp. 65–89 and 133–145, vol. 8 (1943) pp. 89–106, vol. 13 (1948) pp. 65–79, vol. 19 (1954) pp. 81–96.Google Scholar
[2]Fraenkel, A. A. and Bar-Hillel, Y., Foundations of set theory, North-Holland, Amsterdam, 1950, X + 415 pp.Google Scholar
[3]Gödel, K., The consistency of the continuum hypotheses, Annals of Mathematics Studies, No. 3, Princeton University Press, Princeton, 1940, 66 pp.CrossRefGoogle Scholar
[4]Kruse, A. H., Some developments in the theory of numerations, Transactions of the American Mathematical Society, vol. 97 (1960), pp. 523–553.CrossRefGoogle Scholar
[5]Kruse, A. H., Augmenting axiomatic set theory (Preliminary report), Notices of the American Mathematical Society, Abstract 576–225, vol. 7 (1960), p. 983.Google Scholar
[6]Mostowski, A., Über die Unabhängigkeit des Wohlordungssatzes vom Ordnungsprinzip, Fundamenta mathematicae, vol. 32 (1939), pp. 201–252.CrossRefGoogle Scholar
[7]Mostowski, A., Some impredicative definitions in axiomatic set theory, Fundamenta mathematicae, vol. 37, pp. 111–124. (Correction, ibid., vol. 38 (1952), p. 238.)CrossRefGoogle Scholar
[8]Myhill, J., The hypothesis that all classes are nameable, Proceedings of the National Academy of Sciences, U.S.A., vol. 38 (1952), pp. 979–981.CrossRefGoogle ScholarPubMed
[9]Novak, I. L., A construction for models of consistent systems, Fundamenta mathematicae, vol. 37 (1951), pp. 87–110.CrossRefGoogle Scholar
[10]Robinson, R. M., On finite sequences of classes, this Journal, vol. 10 (1945), pp. 125–126.Google Scholar
[11]Rosser, J. B. and Wang, H., Non-standard models for formal logics, this Journal, vol. 15 (1950), pp. 113–129.Google Scholar
[12]Shoenfield, J. R., A relative consistency proof, this Journal, vol. 19 (1954), pp. 21–28.Google Scholar
[13]Wang, H., The non-finitizability of impredicative principles, Proceedings of the National Academy of Sciences, U.S.A., vol. 39 (1950), pp. 479–484.CrossRefGoogle Scholar