Arithmetic based on the church numerals in illative combinatory logic

Studia Logica 47 (2):129 - 143 (1988)
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Abstract

In the early thirties, Church developed predicate calculus within a system based on lambda calculus. Rosser and Kleene developed Arithmetic within this system, but using a Godelization technique showed the system to be inconsistent.Alternative systems to that of Church have been developed, but so far more complex definitions of the natural numbers have had to be used. The present paper based on a system of illative combinatory logic developed previously by the author, does allow the use of the Church numerals. Given a new definition of equality all the Peano-type axioms of Mendelson except one can be derived. A rather weak extra axiom allows the proof of the remaining Peano axiom. Note. The illative combinatory logic used in this paper is similar to the logic employed in computer languages such as ML.

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References found in this work

The Calculi of Lambda-Conversion.Alonzo Church - 1941 - Princeton, NJ, USA: Princeton University Press.
Combinatory Logic, Volume I.Haskell B. Curry, Robert Feys & William Craig - 1959 - Philosophical Review 68 (4):548-550.
Propositional and Predicate Calculuses Based on Combinatory Logic.M. W. Bunder - 1974 - Notre Dame Journal of Formal Logic 15 (1):25-34.

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