Abstract Sets and Finite Ordinals. An Introduction to the by G. B Keene

By G. B Keene

This textual content unites the logical and philosophical elements of set concept in a fashion intelligible either to mathematicians with out education in formal common sense and to logicians with no mathematical history. It combines an undemanding point of remedy with the top attainable measure of logical rigor and precision. 1961 version.

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And” We use the sign ⋅ (read “dot”) between formulae, and it is defined by the following matrix: P Q PQ T T T F T F T F F F F F The matrix shows that if dot occurs between two formulae each of which represents a true proposition, the result is a formula representing a true proposition. In every other possible case the result is shown to be a formula representing a false proposition. e. e. 24. “Or” We use the sign v (read “vel”) between formulae, and it is defined by the following matrix: P Q P v Q T T T F T T T F T F F F The matrix shows that if vel occurs between two formulae each of which represents a false proposition, the result is a formula representing a false proposition.

As Bernays himself says: “it adopts the principal idea of von Neumann, that the elimination of the undefined notion of a property … which occurs in the original axiom system of Zermelo, can be accomplished in such a way as to make the resulting axiom system elementary …” (op. , Pt. I, p. 65). In the following pages a small part of the Bernays System has been expanded and set into a formal framework. This framework has been devised with the sole aim of combining, as far as possible, simplicity with formality.

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. This completes the sub-proof c(ii) and thereby the proof for case 3 (see above, page 53). As we have now considered all possible cases of non-primary standard expressions, this completes the proof of the class theorem.