The History of Infinity

In this chapter we will focus on infinity, from a social, psychological, practical, but mostly mathematical view point. Infinity can be viewed as the end of the road from learning to count. "How far can we count?.........," some ancient mathematician no doubt wondered, perhaps while peering into the endless space of some stary night. On the contrary, given a droplet of water, how far can it be decomposed by halving? Each of these involves at its basis an infinite process.

These issues were carried around as philosophical and mathematical baggage for millenia until Cantor gave a precise definition of infinity hardly more than a century ago.

Goals.

• How are infinity and the theory of sets connected?
• Is a theory of infinity possible without an accompanying theory of sets?
• What were the stimuli for the invention of infinity? Were they mathematical? Was some philosophical or religious objection overcome?
• The paradoxes of infinity have caused many difficulties in the early days of set theory. What are they and how were they resolved?
• Trace the role of Fourier series through the development of the modern theory of infinity.
• What is constructivism? Regarding infinity, what role did the constructivists, led say by Kronecker, play? What role do constructivists play today?
• Who is Kurt Godel? What was his impact on infinity and set theory and mathematics in general? How did his work impact Hilbert's program?