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A system of axiomatic set theory—Part VI62

Published online by Cambridge University Press:  12 March 2014

Paul Bernays
Paul Bernays*
Affiliation:
Zurich
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Extract

Comparability of classes. Till now we tried to get along without the axioms Vc and Vd. We found that this is possible in number theory and analysis as well as in general set theory, even keeping in the main to the usual way of procedure.

For the considerations of the present section application of the axioms Vc, Vd is essential. Our axiomatic basis here consists of the axioms I—III, V*, Vc, and Vd. From V*, as we know, Va and Vb are derivable. We here take axiom V* in order to separate the arguments requiring the axiom of choice from the others. Instead of the two axioms V* and Vc, as was observed in Part II, V** may be taken as well.

Type
Articles
Information
The Journal of Symbolic Logic , Volume 13 , Issue 2 , June 1948 , pp. 65 - 79
Copyright
Copyright © Association for Symbolic Logic 1948

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Footnotes

62

Parts I-V 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.

References

63 Cf. Part II, §4, p. 3.

64 Cf. Part IV, §11, p. 137.

65 See Part II, §5, lemma 3, p. 7 and pp. 8, 9.

66 See Part IV, §12, p. 143.

67 Cf. Part V, §13, p. 91.

68 Cf. §13, p. 94.

69 Our class Π corresponds to von Neumann's “Bereich Π” (loc. cit. p. 237), our Ψ to his function ψ (p. 236); however, our definition of Ψ is somewhat simpler than von Neumann's definition of ψ.

70 This observation, mentioned already in Part II (§4, p. 6), was communicated to the author by Gödel in July 1939.

71 Cf. Part IV, §11, p. 136.

72 See §10, p. 86.

73 It is the axiom IV2, cf. 2992, p. 225, and 2995, p. 675.

74 Cf. Part III, p. 87.

75 See p. 67.

76 Cf. p. 70.

77 Cf. Part III, p. 86.

78 Cf. Part V, p. 89.

79 Cf. Part III, pp. 68(bottom)-70, and p. 86.

80 Cf. p. 74.

81 See Part II, p. 12.

82 Cf. Part II, p. 4. It is noted here as consequence (6) of our axioms. The proof here given is by IV and Vb; a better known proof is by IV and Va, c.

83 Cf. Part I, p. 65, footnote 3.