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absolute with respect aeOn Af-complete algebra of regular assume Axiom of Extensionality Axiom of Powers Axiom of Replacement Axiomatic Set Theory axioms of ZF B-valued structure Boolean valued Borel sets compatible complete Boolean algebra constant term contradiction Corollary defined Definition denotes dense element elementary substructure exists extensional finite intersection property function Furthermore G is P-generic Hausdorff space hence induction hypothesis isomorphic Ka(B Lemma limited formulas M-complete model of ZF open subsets Ord1 Ord2 ordinals partial order structure Proo proof of Theorem prove the following ramified language rank recursion regular open sets regular open subsets relative constructibility Remark satisfies the Axiom satisfies the c.c.c. satisfies the UCL set theory standard transitive model Suppose topological space u e K(B ultrafilter Va(B