A Structural Analysis of Infinity in Set Theory and Modern Algebra
Noah Betz
TL;DR
This work presents a dual lens on infinity: a set-theoretic treatment via cardinals, ordinals, and CH, and an algebraic treatment via groups, rings, and modules. It reveals how infinity shapes structural results—from Schröder–Bernstein and well-orderings to infinite direct products, free modules, and Noetherian theory—while highlighting the foundational role of the axiom of choice and Zorn’s lemma in establishing key existence and classification results. Notable contributions include connecting infinite-dimensional phenomena in vector spaces and modules to classical theorems (e.g., the Fundamental Theorem of finitely generated Abelian groups, Hilbert’s Basis Theorem, First Isomorphism Theorem) and illustrating how infinite structures arise naturally in algebraic decompositions and module theory. The findings underscore a deep interplay between set-theoretic infinity and algebraic structure, clarifying how infinity pervades both foundational questions and concrete classifications across mathematics.
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 Schröder-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.