### Degrees of Infinity

Many of us recall the sense of wonder we felt upon learning that there is no biggest number; for some of us, that wonder has never quite gone away. It is obvious that, given any counting number, one can be added to it to give a larger number. But the implication that there is no limit to this process is perplexing.

Georg Cantor (1845 – 1918) around 1870 (left) and in later life (right).

The concept of infinity has exercised the greatest minds throughout the history of human thought. It can lead us into a quagmire of paradox from which escape seems hopeless. In the late 19th century, the German mathematician Georg Cantor showed that there are different degrees of infinity — indeed an infinite number of them — and he brought to prominence several paradoxical results that had a profound impact on the subsequent development of the subject.

Set Theory

Cantor was the inventor of set theory, which is a fundamental foundation of modern mathematics. A set is any collection of objects, physical or mathematical, actual or ideal. A particular number, say 4, is associated with all the sets having four elements. For any two of these sets, we can find a 1-to-1 correspondence, or bijection, between the elements of one set and those of the other. The number 4 is called the cardinality of these sets. Generalizing this argument, Cantor treated any two sets as being of the same size, or cardinality, if there is a 1-to-1 correspondence between them.

Bijection between two sets of cardinality 4.

But suppose the sets are infinite. As a concrete example, take all the natural numbers, 1, 2, 3, … as one set, and all the even numbers 2, 4, 6, … as the other. By associating any number n in the first set with 2n in the second, we have a perfect 1-to-1 correspondence. By Cantor’s argument, the two sets are the same size. But this is paradoxical, for the set of natural numbers contains all the even numbers and also all the odd ones so, in an intuitive sense, it is larger. The same paradoxical result had been deduced by Galileo some 250 years earlier.

Cantor carried these ideas much further, showing in particular that the set of all the real numbers (or all the points on a line) have a degree of infinity, or cardinality, greater than the counting numbers. He did this using an ingenious approach called the diagonal argument. This raised an issue, called the continuum hypothesis: is there a degree of infinity between these two? This question can not be answered within standard set theory.

Infinities without limit

Cantor introduced the concept of a power set: for any set A, the power set P(A) is the collection of all the subsets of A. Cantor proved that the cardinality of P(A) is greater than that of A. For finite sets, this is obvious; for infinite ones, it was startling. The result is now known as Cantor’s Theorem, and he used his diagonal argument in proving it. He thus developed an entire hierarchy of transfinite cardinal numbers. The smallest of these is the cardinality of the natural numbers, called Aleph-zero:

Aleph-zero, the cardinality of the natural numbers and the smallest transfinite number.

Cantor’s theory caused quite a stir; some of his mathematical contemporaries expressed dismay at its counter-intuitive consequences. Henri Poincaré, the leading luminary of the day, described the theory as a “grave disease” of mathematics, while Leopold Kronecker denounced Cantor as a renegade and a “corrupter of youth”. This hostility may have contributed to the depression that Cantor suffered through his latter years. But David Hilbert championed Cantor’s ideas, famously predicting that “no on will drive us from the paradise that Cantor has created for us”.

Ternary Variations: Cantor’s surprising ternary set.

Surreal Numbers: Conway’s ingenious transfinite numbers