A set theory ZFI′ which does not employ the Law of the Excluded Middle φ ∀ ⊥ φ, for all φ, retians the stock of expressive capacities of the classical set theory ZF, on the one hand, and has many of the features of an effective theory on the other. In the article, a broad class of formulas σ is constructed for which ZF ⊥ σ implies ZFI′ ⊥ σ. This result provides a generalization of Friedman's theorem on AE-arithmetic formulas. Besides, we prove transfer theorems of classical logic for the case of rings; in particular, Hilbert's theorem on zeros and Artin's theorem on ordered fields are extended to the case of regular f-rings, and we bring in appropriate upper bounds for them.
KeywordsDivision Ring Atomic Formula Heyting Algebra Central Idempotent Complete Boolean Algebra
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- 1.H. Friedman, “Classically and intuitionistically provable recursive functions,” inHigher Set Theory, Lect. Notes Math.,669, 21–27 (1978).Google Scholar
- 2.V. A. Lyubetsky, “An approach to modelling intelligent systems,”Data Transfer Problems,29, No. 3, 107–109 (1993).Google Scholar
- 3.V. A. Lyubetsky, “Heyting-valued analysis: P. S. Novikov's hypothesis,”Cont. Math.,131, Part 3, 565–582 (1992).Google Scholar
- 4.V. A. Lyubetsky, “Evaluations and sheaves. Some problems of nonstandard analysis,”Usp. Mat. Nauk,44, No. 4, 99–153 (1989).Google Scholar
- 5.V. A. Lyubetsky, “On some applications of Heyting-valued analysis. II,” inLect. Notes Comp. Science,417, 122–145 (1988).Google Scholar
- 6.M. P. Fourman and D. S. Scott, “Sheaves and logic”, inApplications of Sheaves, Lect. Notes Math.,753, 302–401 (1979).Google Scholar
- 7.V. A. Lyubetsky, Evaluations and Sheaves: Transfer Theorems, Doctoral Dissertation, Institute of Data Transfer Problems, Ross. Akad. Nauk, Moscow (1991).Google Scholar
- 8.V. A. Lyubetsky, Evaluations and Sheaves: Transfer Theorems, Abstr. Doct. Dissertation, Institute of Data Transfer Problems, Ross. Acad. Nauk, Moscow (1991).Google Scholar