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A system of axiomatic set theory - Part VII84

Published online by Cambridge University Press:  12 March 2014

Paul Bernays
Paul Bernays*
Affiliation:
Zurich
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Extract

The reader of Part VI will have noticed that among the set-theoretic models considered there some models were missing which were announced in Part II for certain proofs of independence. These models will be supplied now.

Mainly two models have to be constructed: one with the property that there exists a set which is its own only element, and another in which the axioms I–III and VII, but not Va, are satisfied. In either case we need not satisfy the axiom of infinity. Thereby it becomes possible to set up the models on the basis of only I–III, and either VII or Va, a basis from which number theory can be obtained as we saw in Part II.

On both these bases the Π0-system of Part VI, which satisfies the axioms I–V and VII, but not VI, can be constructed, as we stated there. An isomorphic model can also be obtained on that basis, by first setting up number theory as in Part II, and then proceeding as Ackermann did.

Let us recall the main points of this procedure.

For the sake of clarity in the discussion of this and the subsequent models, it will be necessary to distinguish precisely between the concepts which are relative to the basic set-theoretic system, and those which are relative to the model to be defined.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 19 , Issue 2 , June 1954 , pp. 81 - 96
Copyright
Copyright © Association for Symbolic Logic 1954

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Footnotes

84

Parts I–VI appeared in this Journal, vol. 2 (1937), pp. 65–77; vol.6 (1941), pp. 1–17; vol. 7 (1942), pp. 65–89, 133–145; vol. 8 (1943), pp. 89–106; vol. 13 (1948), pp. 65–79.

References

85 Ackermann, W., Die Widerspruchsfreiheit der allgemeinen Mengenlehre, Math. Ann. vol. 114 (1937), pp. 305–315CrossRefGoogle Scholar.

86 In the sense of Part I, pp. 70–71; for the proof see Part II, pp. 11–14.

87 Cf. Part II, p. 9.

88 This has been recently shown by another method by Ernst Specker in his Habilitationsschrift of 1952, soon to appear.

89 Cf. Part II, p. 10, bottom.

90 We use here “↔” instead of “if and only if”.

91 Stenius, E.; Das Interpretationsproblem der formalisierten Zahlentheorie und ihre formate Widerspruchsjreiheit, Acta Academiae Aboensis (Math. et Phys.) vol. 18, no. 3, Åbo Akademi, Åbo 1952, 102 ppGoogle Scholar.

92 Novak, I., A construction for models of consistent systems, Fundamenta mathematicae, vol. 37 (1950), pp. 87–110CrossRefGoogle Scholar.

93 Henkin, L., The completeness of the first-order functional calculus, this Journal, vol. 14 (1949), pp. 159–166Google Scholar. An essential step in Henkin's procedure is the construction of a complete set of formulas from any consistent set of formulas, on some logical basis. This method goes back to A. Lindenbaum; see Tarski, A., Fundamentale Begriffe der Methodologie der deduktiven Wissenschaften I, Monatshefte für Mathematik und Physik, vol. 37 (1930), Satz I. 56, p. 394Google Scholar. We shall therefore refer to this method, which (in the form in which we have to use it) will be explained, as the “Lindenbaum completion process”.

94 Hasenjäger, G., Eine Bemerkung zu Henkin's Beweis für die Vollständigkeit des Prädikatenkalküls der ersten Stufe, this Journal, vol. 18 (1953), pp. 42–48Google Scholar.

95 Gödel, K., The consistency of the continuum hypothesis, Annals of mathematics studies, No. 3 (1940), axiom B5Google Scholar.

96 This Journal, vol. 2 (1937), pp. 75–76.

97 With regard to a remark made by Gödel (l.c., footnote 94, p . 7), it may be noticed that as a consequence of our preceding reasoning Gödel's axioms B7 and B8 are derivable from the axioms III without using b(1).

In fact Gödel's axioms B7 and B8 stand not only for our axiom III c(3), but also for c(2), as results from the derivability of B6 from B4, B5 and B8 stated by Markov, A. A. in On the dependence of axiom B6 on the other axioms of the Bernays-Gödel system, Izvéstiyá Akadémii Nauk. SSSR, ser. mat., vol. 12 (1948), pp. 569–570Google Scholar; cf. Mathematical reviews, vol. 10 (1949), p. 421Google Scholar. On the other hand, it may be observed that upon assuming B6, which is the same as our c(2), Gödel's B7 is equivalent to our c(3); thus it follows tha t B8 is provable from B1–B7.

98 This Journal, vol. 13 (1948), pp. 220–221.

99 Fundamenta mathematicae, vol. 16 (1930), pp. 29–47, especially pp. 37–38CrossRefGoogle Scholar.