# 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 points of set concept in a fashion intelligible either to mathematicians with no education in formal good judgment and to logicians and not using a mathematical history. It combines an uncomplicated point of therapy with the top attainable measure of logical rigor and precision. 1961 variation.

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The Venn diagrams) do not provide for a clear and unambiguous visible means of distinction between class-inclusion and class-membership. A thorough grasp of this distinction is essential to an understanding of everything that follows. It can be illustrated by considering the inference already mentioned as invalid, from (i) “He is a member of the British Nation” and (ii) “The British Nation is a member of the United Nations”, to: “He is a member of the United Nations”. For we may contrast with this the inference from, for instance, (i) “The class of Greeks is included in the class of men”, and (ii) “The class of men is included in the class of mortals”, to: “The class of Greeks is included in the class of mortals”.

Figure 15 (C) =df the class defined by: [x = C] (CD) =df the class defined by: [x = C v x = D] Ordered Pair An ordered pair is a pair of classes in a given order and such that it differs from a pair having the same members in a different order. We shall expand this statement more fully later (cf. 11) confining ourselves here to an intuitive grasp of the concept. ) mem1A The class mem1A is a class whose members are those ordered pairs whose first member is a member of A. It is therefore the class defined by: “… is an ordered pair whose first member is a member of A”.

G. the Venn diagrams) do not provide for a clear and unambiguous visible means of distinction between class-inclusion and class-membership. A thorough grasp of this distinction is essential to an understanding of everything that follows. It can be illustrated by considering the inference already mentioned as invalid, from (i) “He is a member of the British Nation” and (ii) “The British Nation is a member of the United Nations”, to: “He is a member of the United Nations”. For we may contrast with this the inference from, for instance, (i) “The class of Greeks is included in the class of men”, and (ii) “The class of men is included in the class of mortals”, to: “The class of Greeks is included in the class of mortals”.