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The Heine-Borel theorem in extended basic logic

Published online by Cambridge University Press:  12 March 2014

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

A demonstrably consistent theory of real numbers has been outlined by the writer in An extension of basic logic1 (hereafter referred to as EBL). This theory deals with non-negative real numbers, but it could be easily modified to deal with negative real numbers also. It was shown that the theory was adequate for proving a form of the fundamental theorem on least upper bounds and greatest lower bounds. More precisely, the following results were obtained in the terminology of EBL: If С is a class of U-reals and is completely represented in Κ′ and if some U-real is an upper bound of С, then there is a U-real which is a least upper bound of С. If D is a class of (U-reals and is completely represented in Κ′, then there is a U-real which is a greatest lower bound of D.

Type
Articles
Information
The Journal of Symbolic Logic , Volume 14 , Issue 1 , 16 May 1949 , pp. 9 - 15
Copyright
Copyright © Association for Symbolic Logic 1949

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References

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

Certain minor corrections are required in the original paper on basic logic (vol. 7, pp. 105–114 of this Journal). The corrections were communicated to the Editor by the referee of another of the writer's papers. The referee's remarks seem to be worth quoting in full:

“P. 111, lines 15–16. Delete “and only if” in line 15, and insert after line 16: “, and G bears T to ‘a’ only if ’(lga)’ is a member of F for all ‘g’ such that ‘g’ represents G in F.”

“P. 111, line 21. Read “R” for “‘r’”.

“P. 112, line 22 from below. Read “[=]” for “[Q]”.

“Without the change in 5.5, I was unable to prove 5.13 or to prove that ‘t2’ represents T2 in Κ (bottom of p. 112). (Note: A G which bears T2 to ‘b’ can be quite arbitrary, except that it must contain either both ‘a’ and ‘[a ⊃ b]’ or else ‘b’. Such a G might not be represented by any ‘g’.) This correction apparently does not affect Fitch's later papers, where a different treatment is used.”