Titlebook: Axiomatic Set Theory; Gaisi Takeuti,Wilson M. Zaring Textbook 1973 Springer-Verlag New York Inc. 1973 forcing.proof.set theory

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Boolean-Valued Structures,ruth 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.

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Boolean-Valued Relative Constructibility,will denote the language of the first-order predicate calculus with predicate constants = and ?. In addition L is a first order language that is an extension of L.. In most applications L will have only finitely many constants but it may have infinitely many. M and M’ will be two B-valued structures

鎮(zhèn)痛劑

Forcing,hroughout this section, . denotes a standard transitive model of ., . ∈ . is a partial order structure, and . is the corresponding .-complete Boolean algebra of regular open sets of . in the relative sense of .. Further-more we have.Such that ., ., and . are related to each as described in §2. Thus

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Cardinals in V(B),ollary 14.23. However, since this translation requires the existence of elementary subsystems of . and thus cannot be carried out in . we shall try to give direct proofs in .. Corresponding to the fact that every cardinal in .[.], where 〈., .〉 is a setting for forcing and . is .-generic over ., is a

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Axiomatic Set Theory978-1-4684-8751-0Series ISSN 0072-5285 Series E-ISSN 2197-5612

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