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An Investigation on the Logical Structure of Mathematics (VI)0): Consistent V-System T(V) (With Corrections to Part (XII))

Published online by Cambridge University Press:  22 January 2016

Sigekatu Kuroda
Sigekatu Kuroda*
Affiliation:
Mathematical Institute, Nagoya University
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The V-system T(V) is defined in §2 by using §1, and its consistency is proved in §3. The definition of T(V) is given in such a way that the consistency proof of T(V) in §3 shows a typical way to prove the consistency of some subsystems of UL. Otherwise we could define T(V) more simply by using truth values. After T(V)-sets are treated in §4, it is proved in §5 as a T(V)-theorem that T(V)-sets are all equal to V.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 14 , February 1959 , pp. 95 - 107
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1959

Footnotes

0)

Continuation of the author’s previous work with the same major title. This Part (VI) presupposes in particular the terminologies and the knowledge in Parts (I) and (II), forthcoming in Hamburger Abhandlungen, and §§ 75 9 of this Part (VI) can only be understandable after reading §§7, 10, 11 of Part (VII), appearing in this volume. The definition of concept and set is given in Part (X) forthcoming elsewhere. The references indicated by upper suffixes (I), (II),… refer to-the Part (I), (II),… respectively.

References

1) The formula (I) is neither of affirmative nor of negative type.

2) Roughly speaking, the sets(X) of a theory are those dependent variables which are allowed to substitute for bound variables in any proof of the theory.

3) The transfinite induction up to first e-number is used in the consistency proof of T1(N)which is given in Part (VIII). The author can not decide at present whethe Ñ≠V is T1Ñ-provable or unprovable.

4) In this §7 the proofs given in §§7, 10, 11 of Part (VII) are presupposed. The formula indicated by N* and V* are all found there.

5) It will be clear what is meant by the dual theory *T of T. The duality theorem C in §2, Part (V) (forthcoming in J. Symb. Logic), is a special case of the duality principle.

6) The T(V)-set 0E is defined by

7) Here the unspecialized definition(II) of Le, v is meant. As is said above, the specialized definition(II)) of Lr, v is the same as that of Russell’s set R.

8) T1(N) does not yet contain enough dependent variables as sets by which thè primitive recursive functions of natural numbers can be treated as sets. It is, however, very likely to be able to prove that (12) is a theorem under the assumption N≠V in such a minimal subsystem T/T1(N) of UL which has the simple types with the elements of N as basic type and which has the primitive recursive functions as sets.

9) Proc. Japan Acad. Vol. 34 (1958), pp. 400-403. This “corrections” was written in December, 1958.

10) Hamb. Abh. Vol. 22 (1958).

11) Still three errata in Part (XII): 1. page 401, in the proof figure, a line is to be inserted between the abovemost two formulas; 2. page 401, the last line of the foot-note, “y∈x” instead of “x∈”; 3. page 402, line 4 from bottom, insert a comma after “Part (II)”.