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Mutual Contradiction of Two Self-Consistent Abstractions

Published online by Cambridge University Press:  22 January 2016

Katuzi Ono
Katuzi Ono*
Affiliation:
Mathematical Institute, Nagoya University
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A vast class of abstractions are proved self-contradictory by Russell-type paradoxes in the sense that the negation of any one of them can be proved tautologically. On the other hand, there are a vast class of abstractions, each being self-consistent. A simple criterion for abstractions to be self-consistent (a sufficient condition) can be given. However, even a fairly restricted class of abstractions, each satisfying the criterion to be self-consistent, may contradict to each other.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 28 , October 1966 , pp. 59 - 61
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1966

References

1) In a joint work with my younger colleague M. OHTA, I have given a sufficient condition for abstractions to be self-contradictory. Known Russell-type paradoxes satisfy the condition. The work will be published in the near future.

2) QUINE and HINTIKKA gave an example of a pair of mutually contradictory propositions, the one being a deformation of abstraction and the other being the natural assumption that there are at least two distinct objects. See QUINE, W. V., ‘On Frege’s way out’, Mind, N. S. 64 (1955), 145-159, and HINTIKKA, K. J. J., ‘Vicious circle principle and the paradoxes’, J. Symb. Log., 22 (1957), 245-249.

3) I have suggested to study theoretical systems starting exclusively from .abstractions for limited sets of relations (‘On a restricted abstraction principle’, spoken at the 1965 Annual Meeting of The Mathematical Society of Japan held at Waseda University on May 21, 1965.). Especially, I have suggested to study 0-abstractions for the pair set 0 consisting of two relations ⊆ and defined by

because I could show self-consistency of every So-abstraction and I could also develop a set theory starting exclusively from 0-abstractions.

The result of the present paper has been a byproduct of my unsuccessful struggle to find out a suitable system of relations that makes every -abstraction self-consistent and enables to develop a set theory safely starting exclusively from -abstractions. The example system given in the present paper may be one of the simplest systems which make each -abstraction self-consistent and also make a certain set of -abstractions mutually contradictory. Recently, the system 0 above mentioned has been also proved to belong to the same category by Y. INOUE, one of my younger colleagues.