Pure Mathematics

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

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By G. B Keene

This textual content unites the logical and philosophical points of set thought in a fashion intelligible either to mathematicians with no education in formal good judgment and to logicians with no mathematical historical past. It combines an easy point of remedy with the top attainable measure of logical rigor and precision. 1961 variation.

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Proof Note: The application of the composition lemma to a class of pairs A and a class of pairs B, will be called the composition of A with B, or LEMMA 3 (sub-proof c(ii)): There exists the class of all sets of the form {{ab}{ba}}. Proof Lemma 4 (sub-proof c(ii)): There exists the class of all sets of the form: {{g{fk}}{{gf}k}}. Proof Sub-proof c(ii) Let C be a class of k-tuplets, such that or are applicable to its members. The result of applying or to the members of C is a class. ) Let H be the class admitted by Lemma 3.

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.

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 neither is nor is intended to be a completely rigorous formalization of the Bernays System. The definitions and the proofs of the major theorems are, for the most part, symbolizations of the definitions and proofs given by Bernays. Many of the other proofs, however, may well fall short of the elegance with which Bernays himself would have formally proved them.