# Axiomatic Set Theory by Gaisi Takeuti, Wilson M. Zaring (auth.)

By Gaisi Takeuti, Wilson M. Zaring (auth.)

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Extra resources for Axiomatic Set Theory

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18 that G = G'. 20. If B is a natural Boolean algebra, if F is a proper Mcomplete ultrafilter for BM, and if G = {p I [p] - 0 E F} P = {b E IBI nM I b = b- o 1\ bn G =j: O} then F = P. Proof bE P = b- o ~ b ~ (3p E G)[p (3p)[[p]-O ~ 1\ bn G E b] E F 1\ =j: 0 [p]-O ::; bJ ~bEF. Thus F ' s; F. 16, P is a proper ultrafilter. 13, F = P. 34 3. 1. 1. A Boolean algebra B is a a-algebra iff (VA ~ IBD[1 = W~ aEA '2 aE IBI /\ 0 aE IBI1· aeA 2. A Boolean ideal I is a a-ideal iff (VA I)[1 = w~ '2 aEI-I· ~ aeA 3.

Throughout this section B = denotes a complete Boolean algebra. 1. i < ß· Then a B-valued interpretation of 2 is a pair is a mapping defined on the set of constants of the language 2" satisfying the following, 1. 4>(ct) E A, i < a. 2. 4>(R j ): An, -+ B, for j < ß where nj is the number of arguments of Rj • Remark. In order to define a truth value for closed formulas of 2 under a given B-valued interpretation we first extend 2! to a new language 2* ~ 2"(C(A» by introducing new individual constants Ca for each a E A.

If P is fine, then for each PEP [p]-O = [p]. Proo/. We have only to show [p]-O c:; [p]. , q 1, p. , Comp (r, p)]. Thisimplies [q] '/:. [p]-. , if q E [p]-O then q E [p]. Remark. 58 Many P's used in later sections are fine. 6. Boolean-Valued Structures The notion of a Boolean-valued structure is obtained from the definition of an ordinary 2-valued structure by replacing the Boolean algebra 2 of two truth values "truth" and "falsehood" by any complete Boolean algebra B. While some of the basic definitions and theorems can be generalized to the B-valued case almost mechanically the intuitive ideas behind these general notions are more difficult to perceive.