Georg Cantor (1845–1918) was a German mathematician who is credited with launching modern set theory. Cantor’s work has been of high importance to logic, especially with respect to the twentiety-century project of reducing mathematics to logic. In addition, Cantor’s conception of multiple infinities had some influence of philosophy. Cantor even received criticism from religious groups who believed that Cantor challenged the unique infinite nature of their god.

Transfinite Sets and Uncountability

Among Cantor’s most important achievements was the realization that there are degrees of infinity: that some infinite, or transfinite, sets are actually larger than others. Among these discoveries was that there are more real numbers (which include any numbers along a continuum) than there are natural numbers (whole, positive numbers), despite the fact that both groups contain infinitely-many numbers.

Cantor also illustrated that sets larger than the set of natural numbers are uncountable. That is, there is no way in which one can completely count every element of an uncountable set one by one and cover every member, even if given an infinite amount of time. By another definitition, a set is countable if and only if it can be mapped onto the set of natural numbers, or some subset of it.

Cardinality and Cardinal Numbers

Cantor also described the cardinal numbers, which measure the sizes of sets. Sets share the same cardinality if there is a bijection between them, that is, if two sets have a one-to-one correspondence between their members. Thus, the sets { 1, 2, 3 } and { a, b, c } are different sets, but share the same cardinality, the cardinal number 3.

For sets of finite size, Cantor uses the natural numbers, beginning with 0, to indicate their cardinality. For cardinalities of infinite sets, Cantor used the Hebrew letter aleph, ℵ, with a subscript number to indicate cardinality. Thus, the cardinal numbers are:

0, 1, 2, …, ℵ₀, ℵ₁, ℵ₂ …

The smallest cardinal number for transfinite sets is ℵ₀, read alpeh-naught. ℵ₀ is the cardinality of the set of natural numbers.

Later life

Cantor struggeld with depression throughout his later life, a likely result of bi-polar disorder. He was hospitalized in sanitoria multiple times until his death in 1918.

Selected Works

• “Über eine Eigenschaft des Ingebriffes aller reelen algebraischen Zahlen” (1915)
• Contributions to the Founding of the Theory of Transfinite Numbers (1915)