By G. B Keene

ISBN-10: 0486155005

ISBN-13: 9780486155005

This textual content unites the logical and philosophical points of set idea in a way intelligible either to mathematicians with out education in formal common sense and to logicians with out a mathematical historical past. It combines an hassle-free point of remedy with the top attainable measure of logical rigor and precision. 1961 variation.

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**Extra resources for Abstract Sets and Finite Ordinals. An Introduction to the Study of Set Theory**

**Sample text**

If both are reading the same or related subjects, on the other hand, the classes are likely to overlap; and it is just conceivable, if both are attending the same course, that the two classes may be identical. But in each case the identity or otherwise of the two classes is determined by whether or not the conditions-for-membership of the one class is fulfilled by at least one book which fails to fulfil the conditions-for-membership of the other class. In general, two class-references are references to different classes only so long as there is something to which the predicate of the one reference applies, and to which the predicate of the other reference does not.

Note: In the above rule, c is an expression for a class or a set; if φ has no parameters then c is a new individual symbol; if φ has one or more parameters then c consists of a new function symbol with those parameters as its (only) variables. Furthermore, the quantifier (∃! x) is defined as follows: (∃! x)φ for: (∃x)(φ ⋅(z)(φ* ⊃) z= x)), where φ* is like φ except that z occurs wherever x occurs in φ; that is, “There exists a unique x such that φ”. 29) we can infer, as before: and since (introducing instead say G): entails: we have G = D and we can, having thus proved the uniqueness of D, introduce, in accordance with our new rule, a particular individual constant, say A ∩B, by means of the equivalence: Again, given: we can infer: from which it follows that: and since (introducing, say G) : entails G = D, we can, having again proved the uniqueness of D, introduce a particular individual constant, say , by means of the equivalence: Where, as in these examples, parameters occur, the new symbol is, of course, an individual constant (rather than a new function symbol) only if the parameters have their values fixed by reference to previous lines of the proof in which the new symbol is introduced.

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. P β C. p is of one of the forms: {{ab}c}, {a{bc}}, {{ab}{cd}}. Let p be of the form {a{bc}}. {a{bc}} β C ⋅ {{bc}{cb}} β H. {a{cb}} β {a{cb}} is the result of applying to {bc} in p. In case p is of the form {{ab}{cd}} the proof is analogous for the result of applying to {cd} in p. Let p be of the form {{ab}c}.

### Abstract Sets and Finite Ordinals. An Introduction to the Study of Set Theory by G. B Keene

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