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- Dedekind-infinite_set abstract "In mathematics, a set A is Dedekind-infinite (named after the German mathematician Richard Dedekind) if some proper subset B of A is equinumerous to A. Explicitly, this means that there is a bijective function from A onto some proper subset B of A. A set is Dedekind-finite if it is not Dedekind-infinite.Proposed by Richard Dedekind in 1888, Dedekind-infiniteness was the first definition of "infinite" that did not rely on the definition of the natural numbers. Until the foundational crisis of mathematics showed the need for a more careful treatment of set theory most mathematicians assumed that a set is infinite if and only if it is Dedekind-infinite. In the early twentieth century Zermelo–Fraenkel set theory (ZF), today the most commonly used form of axiomatic set theory, was proposed as an axiomatic system to formulate a theory of sets without the paradoxes of naive set theory such as Russell's paradox. Using the axioms of ZF set theory with the originally highly controversial axiom of choice included (ZFC) one can show that a set is Dedekind-finite if and only if it is finite in the sense of having a finite number of elements. However, there exists a model of ZF in which there exists an infinite, Dedekind-finite set, showing that the axioms of ZF are not strong enough to prove that every set that is Dedekind-finite has a finite number of elements.There are other definitions of finiteness and infiniteness of sets that do not depend on the axiom of choice.A vaguely related notion is that of a Dedekind-finite ring. A ring is said to be a Dedekind-finite ring if ab=1 implies ba=1 for any two ring elements a and b. These rings have also been called directly finite rings.".
- Dedekind-infinite_set wikiPageID "1074742".
- Dedekind-infinite_set wikiPageLength "11036".
- Dedekind-infinite_set wikiPageOutDegree "49".
- Dedekind-infinite_set wikiPageRevisionID "625002722".
- Dedekind-infinite_set wikiPageWikiLink Alfred_North_Whitehead.
- Dedekind-infinite_set wikiPageWikiLink Axiom_of_choice.
- Dedekind-infinite_set wikiPageWikiLink Axiom_of_countable_choice.
- Dedekind-infinite_set wikiPageWikiLink Axiomatic_set_theory.
- Dedekind-infinite_set wikiPageWikiLink Axiomatic_system.
- Dedekind-infinite_set wikiPageWikiLink Bernard_Bolzano.
- Dedekind-infinite_set wikiPageWikiLink Bertrand_Russell.
- Dedekind-infinite_set wikiPageWikiLink Bijection.
- Dedekind-infinite_set wikiPageWikiLink Bijective_function.
- Dedekind-infinite_set wikiPageWikiLink Category:Basic_concepts_in_infinite_set_theory.
- Dedekind-infinite_set wikiPageWikiLink Category:Cardinal_numbers.
- Dedekind-infinite_set wikiPageWikiLink Charles_University_in_Prague.
- Dedekind-infinite_set wikiPageWikiLink Countable_set.
- Dedekind-infinite_set wikiPageWikiLink Equinumerosity.
- Dedekind-infinite_set wikiPageWikiLink Equinumerous.
- Dedekind-infinite_set wikiPageWikiLink Ernst_Zermelo.
- Dedekind-infinite_set wikiPageWikiLink Finite_set.
- Dedekind-infinite_set wikiPageWikiLink Foundational_crisis_of_mathematics.
- Dedekind-infinite_set wikiPageWikiLink Foundations_of_mathematics.
- Dedekind-infinite_set wikiPageWikiLink Function_(mathematics).
- Dedekind-infinite_set wikiPageWikiLink Graduate_Texts_in_Mathematics.
- Dedekind-infinite_set wikiPageWikiLink Graduate_texts_in_mathematics.
- Dedekind-infinite_set wikiPageWikiLink If_and_only_if.
- Dedekind-infinite_set wikiPageWikiLink Iff.
- Dedekind-infinite_set wikiPageWikiLink Image_(mathematics).
- Dedekind-infinite_set wikiPageWikiLink Infinite_set.
- Dedekind-infinite_set wikiPageWikiLink Injective_function.
- Dedekind-infinite_set wikiPageWikiLink Mathematician.
- Dedekind-infinite_set wikiPageWikiLink Mathematics.
- Dedekind-infinite_set wikiPageWikiLink Naive_set_theory.
- Dedekind-infinite_set wikiPageWikiLink Natural_number.
- Dedekind-infinite_set wikiPageWikiLink One-to-one.
- Dedekind-infinite_set wikiPageWikiLink Ordinal_number.
- Dedekind-infinite_set wikiPageWikiLink Richard_Dedekind.
- Dedekind-infinite_set wikiPageWikiLink Russells_paradox.
- Dedekind-infinite_set wikiPageWikiLink Sequence.
- Dedekind-infinite_set wikiPageWikiLink Sequences.
- Dedekind-infinite_set wikiPageWikiLink Set_theory.
- Dedekind-infinite_set wikiPageWikiLink Strict.
- Dedekind-infinite_set wikiPageWikiLink Strictly.
- Dedekind-infinite_set wikiPageWikiLink Subset.
- Dedekind-infinite_set wikiPageWikiLink Surjective_function.
- Dedekind-infinite_set wikiPageWikiLink Theory_of_sets.
- Dedekind-infinite_set wikiPageWikiLink Thomas_Jech.
- Dedekind-infinite_set wikiPageWikiLink Von_Neumann_regular_ring.
- Dedekind-infinite_set wikiPageWikiLink Well-ordering_theorem.
- Dedekind-infinite_set wikiPageWikiLink Zermelo–Fraenkel_set_theory.
- Dedekind-infinite_set wikiPageWikiLinkText "Dedekind-finite".
- Dedekind-infinite_set wikiPageWikiLinkText "Dedekind-infinite set".
- Dedekind-infinite_set wikiPageWikiLinkText "Dedekind-infinite".
- Dedekind-infinite_set wikiPageWikiLinkText "Dedekind-infinite_set".
- Dedekind-infinite_set wikiPageWikiLinkText "infinite set".
- Dedekind-infinite_set hasPhotoCollection Dedekind-infinite_set.
- Dedekind-infinite_set wikiPageUsesTemplate Template:Reflist.
- Dedekind-infinite_set subject Category:Basic_concepts_in_infinite_set_theory.
- Dedekind-infinite_set subject Category:Cardinal_numbers.
- Dedekind-infinite_set hypernym Dedekind-infinite.
- Dedekind-infinite_set type Concept.
- Dedekind-infinite_set comment "In mathematics, a set A is Dedekind-infinite (named after the German mathematician Richard Dedekind) if some proper subset B of A is equinumerous to A. Explicitly, this means that there is a bijective function from A onto some proper subset B of A. A set is Dedekind-finite if it is not Dedekind-infinite.Proposed by Richard Dedekind in 1888, Dedekind-infiniteness was the first definition of "infinite" that did not rely on the definition of the natural numbers.".
- Dedekind-infinite_set label "Dedekind-infinite set".
- Dedekind-infinite_set sameAs デデキント無限.
- Dedekind-infinite_set sameAs Dedekind-oneindige_verzameling.
- Dedekind-infinite_set sameAs Dedekind-infinito.
- Dedekind-infinite_set sameAs m.043qjn.
- Dedekind-infinite_set sameAs Q5249754.
- Dedekind-infinite_set sameAs Q5249754.
- Dedekind-infinite_set wasDerivedFrom Dedekind-infinite_set?oldid=625002722.
- Dedekind-infinite_set isPrimaryTopicOf Dedekind-infinite_set.