A Structural Analysis of Infinity in Set Theory and Modern Algebra
Noah Betz

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 -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…
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsMathematical and Theoretical Analysis
