A Structural Analysis of Infinity in Set Theory and Modern Algebra

TL;DR

This paper explores the concept of infinity through set theory and algebra, analyzing foundational results and their interconnections to deepen understanding of infinite structures in mathematics.

Contribution

It provides a comprehensive, self-contained analysis of infinity from set-theoretic and algebraic perspectives, highlighting their interplay and foundational results.

Findings

Analysis of the continuum hypothesis and infinite cardinals

Connections between set theory and algebraic structures

Insights into the properties of infinite algebraic structures

Abstract

We present a self-contained analysis of infinity from two mathematical perspectives: set theory and algebra. We begin with cardinal and ordinal numbers, examining deep questions such as the continuum hypothesis, along with foundational results such as the Schr\"oder-Bernstein theorem, multiple proofs of the well-ordering of cardinals, and various properties of infinite cardinals and ordinals. Transitioning to algebra, we analyze the interplay between finite and infinite algebraic structures, including groups, rings, and $R$-modules. Major results, such as the fundamental theorem of finitely generated abelian groups, Krull's Theorem, Hilbert's basis theorem, and the equivalence of free and projective modules over principal ideal domains, highlight the connections and differences between finite and infinite structures, as well as demonstrating the relationship between set-theoretic and algebraic treatments of infinity. Through this approach, we provide insights into how key results about infinity interact with and inform one another across set-theoretic and algebraic mathematics.