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An investigation on the logical structure of mathematics (V).1 Contradictions of Russell's type
Published online by Cambridge University Press: 12 March 2014
Extract
If a contradiction is deduced by using only the defining formula of a set, it is called a contradiction of Russell's type, and the set is called inconsistent. A set is called consistent, if no contradiction can be deduced from the defining formula of the set.
The object of this note is to determine some conditions for some sets to be consistent or inconsistent.
In §1 notations and terms are defined. In §2 main results are stated. In §3, the proof in UL of a contradiction is simplified for a special kind of sets, so that a simplified definition of the proof of a contradiction is given for the kind of sets, and some properties of a proof in UL are stated concerning the simplified definition. This definition and the properties of a proof are the basis of the proof of the results given in §§5–10. Thus, this Part (V), except §3, is written independently of other Parts as far as possible. However, it depends essentially on Parts (I) and (II). In §4 some examples are given.
- Type
- Research Article
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- Copyright
- Copyright © Association for Symbolic Logic 1958
Footnotes
1
This is the continuation of the work with the same major title. This Part (V), except § 3 (cf. below), can be read independently of other parts.
References
2 Universal logic; cf. the beginning of the introduction to Part (II).
3 Parts (I) and (II) are forthcoming in the next volume of Hamburger Abhandlungen. For further continuation, see Nagoya Mathematical Journal.
4 Professor A. Church in 1955 kindly called the author's attention to the works of D. A. Bočvar and P. S. Novikov on the question of the paradoxes of mathematical logic and the theory of sets. Cf. the reviews in this Journal, vol. 11 p. 129, vol. 12 p. 27, vol. 13 p. 170, and also Church, A., A brief bibliography of formal logic, Proceedings of the American Academy of Arts and Sciences, vol. 80 (1952), p. 164Google Scholar. Because of linguistic difficulty, the author could not compare in detail the Russian papers of Bočvar and Novikov with this note. However, as far as he was informed through the reviews of the papers, the method applied and the results obtained in this note are different from theirs. Cf. also Novikov, P. S., On logical paradoxes (Russian), Doklady Akademii Nauk SSSR (n.s.), vol. 56 (1947) (reviewed in Mathematical Reviews, vol. 9 (1948), p. 1)Google Scholar.
5 For the constant p defined by (1) is used as a set in a proof of a contradiction. For the definition of sets in a proof, cf. Part (X).
6 Cf. Part (I), §11; Part (II), § 15.
7 Cf. Part (IV), § 5.
8 Cf. Part (I), § 11.
8 Cf. Part (II), § 17.
9 Cf. Part (IV), § 2, Theorem 1.
11 Cf. Part (IV), § 3.
12 Cf. Part (II), § 13, Theorems 4 and 6.
13 A proof constituent E is called a derivative of a proof formula F, if E is derived from F by successive association of a proof constituent to a formula. Cf. also Part (II), § 19.
14 The notation [σ] etc. attached to a horizontal line, as below, indicates the substitution σ etc. in the proof constituent carried by the horizontal line.
15 Cf. Part (II), § 12, Theorem 1.
16 Cf. the end of § 1, (iii). Part (IV).
17 A species or a formula is called bottom species or bottom formula of a proof P, if there is no proof constituent under it. A proof constituent E is called a. bottom constituent, if there is no proof constituent under any formula or species carried by E.
18 Notice that (24) and (25) are deduced under the assumption that there is no singular [A] in the proof P of pp. Since there is no singular [N] in P, this assumption is equivalent to the assumption that there is no singular P-constituent.
19 The constituent (16) must be used at least once in any proof of a contradiction with m = p.
20 Cf. Part (II), § 20.