The notion of Infinity within the Zermelo system and its relation to the Axiom of Countable Choice

TL;DR

This paper explores different ways to define an infinite set within Zermelo's axiomatic system, examining how these definitions relate to the Axiom of Countable Choice.

Contribution

It introduces alternative definitions of infinity in Zermelo's system and analyzes their implications for the Axiom of Countable Choice.

Findings

Different definitions of infinity are compared within Zermelo's system.

The relationship between these definitions and the Axiom of Countable Choice is clarified.

Implications for foundational set theory are discussed.

Abstract

In this article we consider alternative definitions-descriptions of a set being Infinite within the primitive Axiomatic System of Zermelo.