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On “Banach’s principle”. (Hungarian. English, Russian summaries) Zbl 0753.03020

Summary: M. Ajtai proved [Period. Math. Hung. 5, 343-352 (1974; Zbl 0307.02046)] that adding a Cohen-generic real to any model of ZFC, in the generic extension every linear operator between Banach spaces (all the spaces and the operators are definable by first-order formulae) is continuous.
When the formulae above satisfy certain absoluteness conditions, we can derive absolute theorems (see, e.g., Theorems 2.4 and 2.6). We generalize this and Ajtai’s other theorems in Theorems 4.15 and 4.21. Further, we give applications: derive absolute theorems in analysis using mathematical logical methods (Theorems 3.21 and 4.19). We also give an elementary introduction into the theory of first-order logics, constructive set theory (Section 1) and Cohen’s forcing method (Section 4).

MSC:

03E35 Consistency and independence results
03E75 Applications of set theory
47S30 Constructive operator theory
46B99 Normed linear spaces and Banach spaces; Banach lattices

Citations:

Zbl 0307.02046
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