A conjecture on numeral systems

TL;DR

This paper investigates the properties of numeral systems in lambda calculus, proving the independence of key functions needed for representing all total recursive functions and proposing a conjecture on the minimal functions required.

Contribution

It demonstrates the independence of Successor, Predecessor, and Zero Test functions in numeral systems and introduces a conjecture on the minimal set of unary functions for full recursive function representation.

Findings

Successor, Predecessor, and Zero Test functions are independent in numeral systems.

Any of these three functions alone cannot represent all total recursive functions.

A conjecture on the minimal number of unary functions needed for complete recursive function representation.

Abstract

A numeral system is an infinite sequence of different closed normal $\lambda$-terms intended to code the integers in $\lambda$-calculus. H. Barendregt has shown that if we can represent, for a numeral system, the functions : Successor, Predecessor, and Zero Test, then all total recursive functions can be represented. In this paper we prove the independancy of these particular three functions. We give at the end a conjecture on the number of unary functions necessary to represent all total recursive functions.