Elementary recursive algorithms

TL;DR

This paper explores the foundational aspects of elementary recursive algorithms, discussing their definitions and providing simple proofs of key mathematical results related to these algorithms.

Contribution

It offers a brief discussion on the foundational questions of algorithms and presents elementary proofs of important properties of elementary recursive algorithms.

Findings

Provides simple proofs of two basic mathematical results

Clarifies the definition and scope of elementary recursive algorithms

Highlights the foundational significance of recursive programs

Abstract

They run our lives, if you believe the hype in the news, but there is no precise definition of "algorithms" which is generally accepted by the mathematicians, logicians and computer scientists who create and study them. My main aims here are (first) to discuss briefly and point to the few publications that try to deal with this foundational question and (primarily) to outline in Sections 4 and 5 simple proofs of two basic mathematical results about the "elementary recursive algorithms" from specified primitives expressed by recursive (McCarthy) programs.