Transfinite number

Lore & Background

Transfinite numbers were introduced by Georg Cantor in 1895, who coined the term to avoid the theological implications of the word 'infinite.' Cantor believed that 'truly infinite' is a perfect and divine quality and thus refused to attribute this term to mathematical constructs comprehensible by humans. The concepts include transfinite cardinals, which measure the size of infinite sets, and transfinite ordinals, which describe the order within infinite ordered sets. For finite numbers, ordinal and cardinal concepts are in one-to-one correspondence, but for transfinite numbers they diverge. The lowest transfinite ordinal is ω (omega), the order type of natural numbers under their usual ordering. The first transfinite cardinal is ℵ₀ (aleph-null), the cardinality of the natural numbers. If the axiom of choice holds, the next higher cardinal is ℵ₁; if not, there may be other cardinals incomparable with ℵ₁ and larger than ℵ₀. Either way, there are no cardinals between ℵ₀ and ℵ₁. The continuum hypothesis proposes that there are no intermediate cardinals between ℵ₀ and the cardinality of the continuum (the set of real numbers), or equivalently that ℵ₁ is the cardinality of the real numbers. In Zermelo–Fraenkel set theory, neither the continuum hypothesis nor its negation can be proved.

Reader's Guide

Transfinite numbers are significant because they extend the natural number system to infinite magnitudes, providing a rigorous foundation for set theory and the study of infinity. They allow mathematicians to compare the sizes of infinite sets and to order infinite sequences. Notable work on transfinite numbers was done by Wacław Sierpiński, described in his 1928 book Leçons sur les nombres transfinis, much expanded into Cardinal and Ordinal Numbers in 1958, with a second slightly revised edition in 1965. The concepts of transfinite cardinals and ordinals are now standard in mathematics, though the term 'transfinite' remains in use alongside 'infinite.' Some authors, including P. Suppes and J. Rubin, use the term 'transfinite cardinal' to refer to the cardinality of a Dedekind-infinite set in contexts where the axiom of countable choice is not assumed or not known to hold. Under this definition, a cardinal m is transfinite if and only if there is a Dedekind infinite set A with that cardinality, or equivalently if m + 1 = m, or ℵ₀ ≤ m, or there is a cardinal n such that ℵ₀ + n = m. Although transfinite ordinals and cardinals generalize only the natural numbers, other systems like hyperreal numbers and surreal numbers generalize the real numbers. The legacy of transfinite numbers lies in their role in resolving paradoxes of infinity and in enabling the development of modern set theory and mathematical logic.

Did You Know?

• The term 'transfinite' was coined in 1895 by Georg Cantor, who wished to avoid the theological implications of the word 'infinite.'
• The lowest transfinite ordinal is ω (omega), which is the order type of the natural numbers under their usual linear ordering.
• The first transfinite cardinal is ℵ₀ (aleph-null), the cardinality of the natural numbers.
• In Zermelo–Fraenkel set theory, neither the continuum hypothesis nor its negation can be proved.