Georg Cantor's Infinity Paper: Historical Overview

Christopher Amick

MAT 135

Infinity Paper: Historical Overview

The Greeks and early Indians are the two ancient cultures that we recognize today as first recording the concept of infinity. Of the two, the first ever record of infinity comes from a pre-Socratic Greek philosopher named Anaximander. The word he chose was apeiron which means infinite or limitless. However, before this the earliest account of mathematical infinity came from Zeno of Elea (born c. 490 BCE). He was a pre-Socratic Greek philosopher as well and was from southern Italy. Zeno is best known for his paradoxes, and was called the inventor of the dialectic by Aristotle himself. Interestly the Greeks usually liked to distinguish the potential infinity from the actual infinity. For example instead of stating that there were an infinite number of primes they’d say that there are more prime numbers than contained in any given collection of prime numbers. Later on European mathematicians worked with irrational numbers, but there was a bit of confusion with infinity itself. St. Augustine had the view that God was infinite and had infinite thoughts, this was called the Platonic view. St Thomas Aquinas allowed the unlimitness of God, however he denied God made unlimited things. Later on in 1600 Galileo claimed the inclusion

Cantor’s work between 1874 and 1884 were the years of the origins of Set Theory. Before this the theory of a set hadn’t really come into play yet. Before Cantor, there were only finite sets, which were easily understood, and the infinite which was viewed as a philosophical and not a mathematical subject. Cantor proved that there are many different sizes of infinite sets and in doing so showed that Set Theory was something that needed to be seriously studied further. Later on Set Theory went on to become one of the founding theories in mathematics. Today the basic concepts of Set Theory are used throughout mathematics