Infinity - A simple, but not too simple introduction

TL;DR

This paper provides an accessible, rigorous introduction to the mathematical properties of infinite sets, focusing on key concepts like cardinality, countability, and Cantor's groundbreaking work, suitable for readers without advanced mathematics.

Contribution

It offers a clear, beginner-friendly explanation of infinity concepts, emphasizing historical context and detailed reasoning, making complex ideas approachable for novices.

Findings

Explains potential and actual infinity concepts

Introduces Cantor's diagonal argument and Hilbert's hotel

Discusses the continuum hypothesis and its implications

Abstract

This text tries to give an elementary introduction to the mathematical properties of infinite sets. The aim is to keep the approach as simple as possible. Advanced knowledge of mathematics is not necessary for a proper understanding, and there is (almost) no use of formulas. At the same time, it is tried to keep the reasoning rigorous and transparent. Major arguments are explained in great detail, while it is explicitly stated if something is not further explained. However, also a fully guided trail can become difficult at times. Persistence is necessary to follow. No reader should be discouraged when he or she cannot understand everything on first readings - infinity is not a piece of cake. The reasoning presented here is mainly due to the famous German mathematician Georg Cantor (1845 - 1918). Despite the resistance from the mathematical community at that time, he opened the door to infinity almost on his own, dramatically changing the foundations of mathematics. The following notions and topics are treated: potential und actual infinity, cardinality and countability, Hilbert's hotel, Cantor's diagonal arguments and the highly astonishing answer to the question of the validity of Cantor's continuum hypothesis.