By G. B Keene

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

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

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21. Authorized Predicates The following notation will be used: k β A for: the relationship of being an element of between a set k and a class A, in that order. for: the well-formed formula φ with some or all of ν1,…,νk among its free set variables and γ1,…,γn as parameters. By the last of the above definitions, the notation ^ signifies that the expression in the square brackets following it, is a class condition on the set variables indicated, in the sense given by the definiens. That is, the formula φ in which ν1,…,νk are free variables for sets and γ1…,γn are the parameters, determines a class each member of which is a normal k-tuplet corresponding to some system of values of the variables ν1,…,νk.

P can be transformed into a k-tuplet {cbk}, c being a (k—1)-tuplet having b1,…, bk–1as its members. (Lemma 2 (above)) {cbk} is the required normal k-tuplet in case k = 2. k > 2. ) c can be transformed into a (k-1)-tuplet {dbk–1}, d being a (k—2)-tuplet having b1,…, bk–2as its members. (Lemma 2 (above)) p can be transformed into the k-tuplet {{dbk–1}bk}, by the application of and to {cbk}. p becomes {d{bk–1bk}}, by application of which is the required normal k-tuplet in case k = 3. k > 3. ) d can be transformed into a (k—2)-tuplet {ebk–2, e being a (k—3)-tuplet having b1, …,bk–3as its members.

In sub-proof c(ii) we shall prove that the result of applying any of these steps to the members of a class of k-tuplets (to whose members such a step is applicable) is a class. For this purpose we require four lemmas. 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.

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