Cantor and the actual infinite

Set theory as we use it today was born between 1874 and 1897 from the work of Georg Cantor. His revolutionary idea was to treat the actual infinite — an infinite set regarded as a whole, already complete — rather than the merely potential infinite (a process that never ends) that mathematics had used ever since Aristotle.

Comparing infinities

Cantor proposed a very simple criterion for saying when two sets have “the same amount” of elements: there exists a one-to-one correspondence between them. With this yardstick he discovered surprising facts. The natural numbers $\mathbb{N}$, the integers $\mathbb{Z}$ and even the rationals $\mathbb{Q}$ all have the same cardinality: they are said to be countable. But the real numbers $\mathbb{R}$ are not. With his celebrated diagonal argument (1891) Cantor proved that no list $x_1, x_2, x_3, \ldots$ can contain all the reals of the interval $[0,1]$: by building a number that differs from the $n$-th one in its $n$-th decimal digit, one always obtains a real missing from the list. There are therefore infinities of different sizes, which Cantor denoted with the aleph numbers: $\aleph_0$ for the countable, a greater cardinality for the continuum (Boyer; Dunham).

Dedekind and the definition of an infinite set

A friend and correspondent of Cantor, Richard Dedekind gave in 1888 an elegant, paradoxical definition: a set is infinite if it can be put into one-to-one correspondence with a proper part of itself. For instance $n \mapsto 2n$ places $\mathbb{N}$ in bijection with the even numbers alone, which are “half” of them yet just as numerous — precisely the oddity that Galileo had already noticed. Dedekind also used his cuts to construct the real numbers rigorously starting from the rationals (Katz; Stillwell).

The paradoxes and the crisis of foundations

This freedom had a price. In 1901 Bertrand Russell discovered a paradox that shakes the “naïve” theory: consider the set $R$ of all sets that do not belong to themselves. If $R$ belongs to itself, then by definition it must not; if it does not belong to itself, then it must. An irreparable contradiction, made popular by the image of the barber who shaves all and only those who do not shave themselves. Mathematics’ answer was not to abandon sets but to found them on cautious axioms (Zermelo–Fraenkel theory). David Hilbert defended Cantor’s legacy with words that have remained famous: “No one shall drive us from the paradise which Cantor created for us” (Boyer; Dunham).

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Topics: Set theory Concepts: Set · Cardinality · Infinity People: Georg Cantor · Richard Dedekind · Bertrand Russell