The Mystery of the Aleph

The Mystery of the Aleph: Mathematics, the Kabbalah, and the Search for Infinity by Amir D. Aczel tells the story of Georg Cantor, a mathematician who pursued the concept of Infinity and tried to define it. This book was an interesting guide to mathematics research in the late 1800s and early 1900s. An easy read, the story also gives incite to the complexity of infinity. Below are some excerpts that discuss some of the puzzles of infinity.

Nicholas likened the knowledge of God to a circle. He visualized human knowledge of God to a circle. He visualized human knowledge as a polygon inscribed within the circle. From these principles, Nicholas constructed a limit argument whereby as human knowledge increases, the polygon gains more and more sides, their number approaching infinity. But Nicholas concluded that no matter how much such knowledge grows, it could never reach God's knowledge in the same way that an inscribed polygon never actually becomes the circle - no matter how many sides it has.

Salviati sets up a one-to-one correspondence between the integers and all he squares of integers and says "We must conclude that there are as many squares as there are numbers." Thus an infinite set, the set of all whole numbers, is shown to be "equal in number" to the set of all squares of whole numbers, which is a proper subset of the set of whole numbers. How can this be possible?

He looked at a very simple mathematical function, y=2x. He let this function act on all the numbers in the domain space: all the numbers between zero and one. For each of these numbers, the function y=2x assigns a unique number in the range space: the space of all numbers between zero and two. For example, the number 0.5, which lives int he domain between 0 and 1 is now assigned a value in the range (0 to 2) given by: y=2x=2(0.5)=1. In the same way, every real number (real number means a rational number or an irrational number) between 0 and 1 is assigned a unique companion between 0 and 2. Therefore, Bolzano concluded, there are as many numbers between 0 and 1 as there are in the interval 0 to 2, which has twice the length of the 0 to 1 interval.