Vol. 57, No. 2, 1975

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ISSN: 0030-8730
Provable equality in primitive recursive arithmetic with and without induction

Harvey Friedman

Vol. 57 (1975), No. 2, 379–392
Abstract

The set of identities provable in primitive recursive arithmetic without induction (PRE), with or without standard quantifier free successor axioms, is recursive. A finite number of identities can be added to PRE such that the set of identities provable become complete r.e. (with or without successor axioms). If the successor axiom y0 gx(S(x) = y) is added to PRE, then the set of identities provable become complete r.e. (with or without 10). If PRE is augmented by definition by cases, then the set of identities provable become complete r.e. (with or without successor axioms). Equivalents of primitive recursive arithmetic (PRA) are given, involving the rule of induction restricted to identities.

Mathematical Subject Classification
Primary: 02D99
Milestones
Received: 8 August 1974
Published: 1 April 1975
Authors
Harvey Friedman