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A demonstrably consistent mathematics—Part I

Published online by Cambridge University Press:  12 March 2014

Frederic B. Fitch
Frederic B. Fitch*
Affiliation:
Yale University
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Extract

This paper continues the exposition of a demonstrably consistent foundation for mathematics begun in An extension of basic logic (hereafter referred to as EBL) and in The Heine-Borel theorem in extended basic logic (hereafter referred to as HB). Still earlier papers related to these are A basic logic (referred to as BL) and Representations of calculi (referred to as RC). The main conventions and results of all the above papers will be presupposed in what follows. The numbering of sections and paragraphs will be a continuation of that of EBL and HB.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 15 , Issue 1 , March 1950 , pp. 17 - 24
Copyright
Copyright © Association for Symbolic Logic 1951

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References

1 This Journal, vol. 13 (1948), pp. 95–106.

2 Ibid., vol. 14 (1949), pp. 9–15.

3 Ibid., vol. 7 (1942), pp. 105–114.

4 Ibid., vol. 9 (1944), pp. 57–62.

5 This was pointed out by Kleene in his review of EBL, this Journal, vol. 14 (1949), pp. 68–69. Kleene also pointe out the redundancy of using the operation of proper ancestrally in characterizing recursively definite relatione, and he notes the equivalence of the notion of “recursively definite” with Gödel's notion of “arithmetical” and his own notion of “elementary”. The redundancy and equivalence were also called to my attention by John R. My hill. I am not sure just what Kleene means in line 7, p. 69 of his review in speaking of “any enumeration of the U-reals completely represented in K′.” If the complete representation refers to the class of all U-reals, the answer is that this class is not completely represented in K′ for reasons given in 10.4 below. Probably, however, Kleene means that the enumerating relation cannot be completely represented in K′ without making possible Cantor's diagonal argument. This is correct, and it leads to the conclusion that the class of U-reals can be enumerated only by enumerating relatione that are not completely represented in K′. In fact any class of U-expressions which is enumerable in the sense of being in a one-to-one correspondence with the class of all U-numerals by way a correlating relation completely represented in K′, is itself completely represented in K′ owing to the fact that the class of U-numerals is completely represented in K′. (These remarks have a close bearing on 12.8 below.) Kleene's surmise that the axiom of choice fails to hold in K′ is incorrect. The correlating relation represented completely in K′ by ’F5‘ (on p. 61 of RC) provides a rather strong axiom of choice, and enables us to order entities with respect to the magnitudes of the Gödel numbers of expressions denoting them. In reply to Kleene's last paragraph, it can be pointed out that the pattern of any given formalism appears in K in infinitely many instances, so that each one of infinitely many interpretations of the formalism can be assigned to one of the instances. This should suffice to take care of all the interpretations unless one wishes to invoke non-enumerable infinities and claim that a non-enumerable infinity of interpretations must be provided for. My standpoint rejects non-enumerable infinities.

6 This Journal, vol. 14 (1949), pp. 209–218.