By G. B Keene

This textual content unites the logical and philosophical points of set thought in a way intelligible either to mathematicians with out education in formal good judgment and to logicians with no mathematical heritage. It combines an effortless point of therapy with the top attainable measure of logical rigor and precision. 1961 variation.

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**Sample text**

24. The Class Theorem (CT): If is a standard expression then Proof The proof will be in two parts, in the first of which we prove that the assertion (CT) holds for all possible cases of primary standard expressions. Proof: part I. Let be a primary expression. Then it has one of the forms: (i) v1 β C (iv) v1 ε v2 (vii) v2 = v1 (ii) v1 = r (V) v2 ε v1 (viii) vl ε r (iii) r = v1 (vi) v1 = v2 (ix) r ε v1 where the parameters are indicated by C and r (referring to classes and to sets, respectively).

Sub-proof (α) We have, as a class whose elements are the (j + k)-tuplets {ce}, (c being a normal j-tuplet corresponding to a system of values c1…, cj of u1…, uj, and e being a normal k-tuplet corresponding to a system of values e1, …, ek of v1, …, vk) such that: (ψ ⋅ φ) holds of c1, …, cj, e1, …, ek. This class is not, however, a class of normal (j+k)-tuplets. We have now to show that the required class can be constructed from it. In doing so, we allow at the same time for the case in which the two component expressions have at least one argument in common.

Let p be of the form {{ab}c}. {c{ab}} β C. {c{ba}} β {c{ba}} is the result of applying to {ab} in p. In case p is of the form {{ab}{cd}} the proof is analogous for the result of applying to {ab} in p. The result of applying to those members of C to which it is applicable is the class Let G be the class admitted by Lemma 4. The result of applying to those members of G to which it is applicable is the class The proof is analogous to the proof for the application of The result of applying to those members of C to which such a step is applicable, is a class.