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On normalization of proofs in set theory

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Hallnäs L.

Instytut Matematyczny Polskiej Akademii Nauk

CONTENTS Introduction..............................................................................................................................................5 I. Naive set theory.....................................................................................................................................6 1. The formal system................................................................................................................................6 2. Inversion and reduction properties of the rules of inference..............................................................12 3. Rules of contraction...........................................................................................................................17 4. Normalizability.....................................................................................................................................22 5. Counter-examples to normalizability in set theory...............................................................................24 6. C-normalizability.................................................................................................................................28 7. Concepts of normalizability and C-normalizability for naive set theory with intuitionistic logic.............33 II. Well-founded fragments of naive set theory........................................................................................33 1. Basic definitions, properties of C-normal deductions..........................................................................33 2. Axioms for set theory..........................................................................................................................37 3. The language SET..............................................................................................................................50 4. Semantical motivation for the rules of contraction..............................................................................54 III. C-normalizability of deductions in well-founded fragments of naive set theory...................................62 1. Well-foundedness predicates and well-foundedness objects.............................................................63 2. $ϕ_{T(x̅)}(α̅(t̅))$..................................................................................................................................66 3. The substitution property....................................................................................................................84 4. Every deduction in N satisfies some negation closed W-predicate.....................................................87 5. C-normalizability for well-founded fragments of naive set theory with intuitionistic logic.....................92 6. Notes..................................................................................................................................................94 References.............................................................................................................................................95

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1 Hallnäs L.

1 1988

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On normalization of proofs in set theory