By G. B Keene
This textual content unites the logical and philosophical facets of set thought in a fashion intelligible either to mathematicians with no education in formal good judgment and to logicians with out a mathematical heritage. It combines an uncomplicated point of therapy with the top attainable measure of logical rigor and precision. 1961 version.
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Extra resources for Abstract Sets and Finite Ordinals. An Introduction to the Study of Set Theory
I. Title. Y. 11501 Table of Contents Title Page Dedication Copyright Page PREFACE PART I - THE ELEMENTS OF SET THEORY SECTION 1 - THE BASIC LOGICAL CONCEPTS SECTION 2 - OPERATIONS ON CLASSES PART II - THE BERNAYS THEORY OF FINITE CLASSES AND FINITE SETS FOREWORD TO PART TWO INTRODUCTION SECTION 1 - THE BASIS OF THE SYSTEM SECTION - DEVELOPMENT OF THE SYSTEM: STAGE I SECTION 3 - DEVELOPMENT OF THE SYSTEM: STAGE II SECTION 4 - THE THEORY OF FINITE ORDINALS SECTION 5 - THE THEORY OF FINITE CLASSES AND FINITE SETS REFERENCES INDEX PREFACE IN this book an attempt is made to present, in a reasonably simple way, the outlines of a relatively complex subject.
Yet we may not infer from the fact that Mr X is a member of the British Nation and the fact that the British Nation is a member of the United Nations, that Mr X is a member of the United Nations. Thus, in general, a part of a part of a whole is always said to be a part of that whole, whereas a member of a member of a class is not always (although in certain cases it may be) said to be a member of that class. To this extent at least, then, the words “class” and “whole” have distinct uses and are therefore not synonymous.
LEMMA 1 (sub-proof c(ii)): The result of applying or to the members of a class of k-tuplets (to whose members such steps are applicable) is a class. Proof 1 Let B be a class of k-tuplets. ) 2 B is the required class, for (AxIII(c2), p. 45) 3 B or Bis the required class, for (AxIII(c3), p. 45 and T13, p. 50) It remains to be shown that the required class exists in case the steps applied are or The lemma needed for this purpose is also essential to some of the proofs in the later part of the system.
Abstract Sets and Finite Ordinals. An Introduction to the Study of Set Theory by G. B Keene