Infinity in mathematics and learning calculus : potential infinity versus actual infinity.

Abstract

The discovery of different infinities in mathematics brought with it a discussion about their existence from very early times. Eleatic philosophy (5th century B.C.), through Zeno's paradoxes, tried to show philosopher-mathematicians that the conceptions held about infinity led to contradictions. Aristotle (384-322 B.C.) wanted to close the chapter by arguing that there is only one infinity in mathematics (the potential infinity) and that the real infinity had no place at all. An implication of this position can be seen in Euclid's Axiom 8 (325-265 B.C.): "The whole is greater than the infinite". Many attempts were made, but it was not until the work of Kant (1790) in philosophy and Bolzano (1817 and 1851) in mathematics (on the continuity of functions and on the paradoxes of infinity) that the problem of potential and real infinity could be better understood, passing from a contradictory status to a paradoxical one. Cantor (1883) proposed his theory of transfinite numbers and set theory, providing mathematics with a structure that integrates the different infinities.