Normal and Non-Normal Classes Versus the Set-Theoretical and the Mereological Concept of Class
We shall concern ourselves here with the transition from the current concept of class to the distributive (set-theoretical) and the collective (mereological) concept of class. This transition is linked to the concepts of normal and non-normal class. Preliminary remarks on that issue have already been made in Sec. 8.
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- 1.This is a continuation of the preceding paper by the same authors. It also assumes a knowledge of that paper on the part of the reader.Google Scholar
- 2.Note that the tentative reconstructions of Russell’s antinomy, as carried out in Sec. 6, refer to the existential axiom (3.1). The symbolisms adopted there (k1 = K (KN1), k2 = K (KN2), q = K (KN),q* = K (non-KNN)) include implicit assumptions of the existence of the respective classes K (KN1),K (KN2), K (KN), K (non-KNN) in accordance with (3.1).Google Scholar
- 3.Note that the axiom of extensionality: x is KL ∧ y is KL∧El (x) = El (y) →x = y, follows easily from (2.2) and hence is a theorem for the current interpretation of the concepts of class and element.Google Scholar
- 4.We are using abbreviated symbolism here: for a and b we should write Id (a) and Id (b), respectively (cf. footnote3 to the first paper by the same authors).Google Scholar
- 5.This is the situation described by (7.8) in Sec. 7 we have: c is KN (C) ∧c is KNN (c).Google Scholar
- 6.S. Lesniewski, Podstawy ogólnej teorii mnogości (The Foundation of General Set Theory), I, Moscow 1916, and *O podstawach matematyki’ (On the Foundations of Mathematics), Przeglqd Filozoficzny, 30-34, Warsaw 1927–1934.Google Scholar