Cantor's paradox

Cantor's paradox
The contradiction arising if we compare for size the set of all sets, and its own power set . By Cantor's theorem the power set must be bigger (contain more members). But it is itself a subset of the set of all sets, and so cannot be bigger. The paradox shows that the collection of all sets cannot itself be a set-theoretic object.

Philosophy dictionary. . 2011.

Игры ⚽ Поможем сделать НИР

Look at other dictionaries:

  • Cantor's paradox — In set theory, Cantor s paradox is the theorem that there is no greatest cardinal number, so that the collection of infinite sizes is itself infinite. Furthermore, it follows from this fact that this collection is not a set but a proper class; in …   Wikipedia

  • Cantor's diagonal argument — An illustration of Cantor s diagonal argument for the existence of uncountable sets. The sequence at the bottom cannot occur anywhere in the list of sequences above. Cantor s diagonal argument, also called the diagonalisation argument, the… …   Wikipedia

  • Paradox — For other uses, see Paradox (disambiguation). Further information: List of paradoxes A paradox is a seemingly true statement or group of statements that lead to a contradiction or a situation which seems to defy logic or intuition. Typically,… …   Wikipedia

  • Georg Cantor — Infobox Scientist name = Georg Ferdinand Ludwig Cantor image width=225px caption = birth date = birth date|1845|3|3 birth place = Saint Petersburg, Russia death date = death date and age|1918|1|6|1845|3|3 death place = Halle, Germany residence =… …   Wikipedia

  • Skolem's paradox — is the mathematical fact that every countable axiomatisation of set theory in first order logic, if consistent, has a model that is countable, even if it is possible to prove, from those same axioms, the existence of sets that are not countable.… …   Wikipedia

  • Russell's paradox — Part of the foundations of mathematics, Russell s paradox (also known as Russell s antinomy), discovered by Bertrand Russell in 1901, showed that the naive set theory of Frege leads to a contradiction.It might be assumed that, for any formal… …   Wikipedia

  • Hilbert's paradox of the Grand Hotel — is a mathematical paradox about infinite sets presented by German mathematician David Hilbert (1862–1943). The Paradox of the Grand Hotel Consider a hypothetical hotel with infinitely many rooms, all of which are occupied that is to say every… …   Wikipedia

  • Banach–Tarski paradox — The Banach–Tarski paradox is a theorem in set theoretic geometry which states that a solid ball in 3 dimensional space can be split into several non overlapping pieces, which can then be put back together in a different way to yield two identical …   Wikipedia

  • Olbers' paradox — in action In astrophysics and physical cosmology, Olbers paradox is the argument that the darkness of the night sky conflicts with the assumption of an infinite and eternal static universe. It is one of the pieces of evidence for a non static… …   Wikipedia

  • Barber paradox — This article is about a paradox of self reference. For an unrelated paradox in the theory of logical conditionals with a similar name, introduced by Lewis Carroll, see the Barbershop paradox. The Barber paradox is a puzzle derived from Russell s… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”