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

HPA533

Generic Sets,In the material ahead we will be interested in standard transitive models . of . and in partial order structures P =

, ≤> for which P ? M. Although some of the results hold under more general conditions we will assume hereafter that this is the case i.e., M is a standard transitive model of ., P = is a partial order structure and P ? ..

ADOPT

Distributive Laws,In this section we wish to discuss several generalized distributive laws for Boolean algebras that will be of importance in the work to follow.

白楊

Relative Constructibility and Ramified Languages,Using a ramified language we shall give another definition of . a definition that has many applications since it only uses the concepts of ordinal number and transfinite induction. On the other hand, to carry out the actual induction steps may become rather complicated in particular cases where definitions by simultaneous recursion are involved.

漂亮才會(huì)豪華

The Independence of , = , and the ,Cohen’s technique of forcing was created for the specific purpose of proving the independence of several axioms of set theory from those of general set theory. In this section we will use Cohen’s method to prove the independence of . = . and the . from the axioms of ..

Antioxidant

Boolean-Valued Set Theory,The use of ramified language in Cohen-type independence proofs often requires proofs by induction which may become rather cumbersome in special cases. A different though essentially equivalent approach which avoids ramified language is provided by the theory of Boolean-valued models as developed by Scott and Solovay.

SOBER

Another Interpretation of V(B),The aim of this section is to prove that “M is a standard transitive model of .containing all the ordinals” and . = . [.]. hold in V. for suitable . and . (Theorems 14.21 and 14.24).

BORE

An Elementary Embedding of ,[,] in V,We have seen that in ., . = . [.]. Since . (.) expresses . ∈ . in the Boolean sense, we might expect some relationship between the Boolean-valued structures . and .. Again let . be a complete Boolean algebra and . → . be defined by .(.) = . for . as in §14. Furthermore, let . be the identity on ..

ALE

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嚴(yán)重傷害

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