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Cantor and the sizes of infinity

Georg CantorLeopold KroneckerDavid Hilbert

This page delivers no verdict; it presents claims and evidence with their sources. No theory in science is absolute. Method

For everyone

Cantor showed that infinities can be larger than one another: the natural numbers and the real numbers are both infinite, but the reals are "more". Some mathematicians such as Kronecker opposed the idea fiercely; Hilbert said that "no one shall expel us from the paradise that Cantor has created". Cantor also related his thinking on infinity to theological questions.

For specialists

In 1874 Cantor proved the real numbers uncountable; the 1891 diagonal argument is a more general proof. Every set's power set is strictly larger than the set (Cantor's theorem). The continuum hypothesis says there is no infinity of intermediate size between the naturals and the reals; Gödel (1940) and Cohen (1963) showed it independent of ZFC, neither provable nor refutable from those axioms.

Claims

Evidence

Supporting

Mathematical proof

The diagonal argument shows that a real number missing from any list of reals can always be constructed.[1]

Challenging

Philosophical / theological argument

Constructivist and finitist mathematicians argued that completed (actual) infinite sets are not legitimate mathematical objects.[2]

Connections

Parent

Mathematics

Related

The argument from origination (kalām cosmological argument)Gödel's incompleteness theoremsFrege and the birth of modern logic

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Sources

  1. ArticleCantor, G. (1874). Ueber eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen. Journal für die reine und angewandte Mathematik 77, 258–262DOI 10.1515/crll.1874.77.258archive
  2. BookDauben, J. W. (1979). Georg Cantor: His Mathematics and Philosophy of the Infinite. Harvard University Press

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Cite this page

APA

Athena contributors. (2026). Cantor and the sizes of infinity. Athena: The tree of human knowledge. Retrieved November 1, 2026, from https://latifkedi.github.io/athena/en/d/cantor/

BibTeX
@misc{athena_cantor,
  title        = {Cantor and the sizes of infinity},
  author       = {{Athena contributors}},
  year         = {2026},
  howpublished = {\url{https://latifkedi.github.io/athena/en/d/cantor/}},
  note         = {Athena: The tree of human knowledge. Accessed: 2026-11-01}
}

Added: September 27, 2026 · Last changed: September 27, 2026 · Page history

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