Set theory is the branch of mathematical logic that studies sets, informally described as collections of objects, along with their sizes, structures and the axioms that can found the rest of mathematics upon them. It emerged in the 1870s through the work of Georg Cantor and Richard Dedekind. This description is adapted from Wikipedia contributors under CC BY-SA 4.0; changes were made. https://creativecommons.org/licenses/by-sa/4.0/
Facts
Central QuestionWhat are the possible sizes of infinite collections, and what axioms can consistently and completely found the rest of mathematics on the notion of a set? 1 Key DebateWhether the axiom of choice, which guarantees a selection function for any collection of nonempty sets without constructing one, should be accepted as a foundational axiom. Zermelo-Fraenkel set theory adopts it as standard (ZFC), but constructivist mathematicians reject the non-computable objects it entails, and Bertrand Russell's discovery that unrestricted set formation produces contradictions forced the field away from naive set theory entirely. 1 Classification
Pure or Applied Connections
Associated With
Founder of set theory, 1870s.
Source Set Theory (Wikipedia)
Includes
Source Wikipedia: Aronszajn tree
Source Axiom of Infinity (Wikipedia)
Source Beth Number (Wikipedia)
Source Borel determinacy theorem (Wikipedia)
Source Cantor Set (Wikipedia)
Source Cardinality of the Continuum (Wikipedia)
Source Constructible Universe (Wikipedia)
Source Easton's theorem, Wikipedia
Source Erdos-Rado theorem (Wikipedia)
Source Fodor's lemma, Wikipedia
Source Inaccessible cardinal (Wikipedia)
Source Measurable cardinal (Wikipedia)
Source Wikipedia: Ordinal number
Source Solovay model, Wikipedia
Source Wikipedia: Tree (set theory)
Source Thoralf Skolem (Wikipedia)
Source Wikipedia: Von Neumann universe
Source Woodin cardinal (Wikipedia)
Sources
1. Set Theory (Wikipedia)
WikipediaLead and origins sections
the branch of mathematical logic that studies sets, which can be informally described as collections of objects
Origins and central questions
Cantor is commonly considered the founder of set theory
Notable debates section
Let R be the set of all sets that are not members of themselves
Associated With: Georg Cantor, Origins section
Cantor is commonly considered the founder of set theory
View the Source Cantor Set (Wikipedia)
Wikipedia: Tree (set theory)
Woodin cardinal (Wikipedia)
Includes: Woodin Cardinal, Lead sentenceQuote, Includes: Woodin Cardinal, Lead sentence
In set theory, a Woodin cardinal (named for W.
Borel determinacy theorem (Wikipedia)
Includes: Borel Determinacy Theorem, Lead sentenceQuote, Includes: Borel Determinacy Theorem, Lead sentence
In descriptive set theory, the Borel determinacy theorem states that any Gale-Stewart game whose payoff set is a Borel set is dete
Beth Number (Wikipedia)
Includes: Beth number, Lead sentenceQuote, Includes: Beth number, Lead sentence
In mathematics, particularly in set theory, the beth numbers form a certain (unset) sequence of infinite cardinal numbers (also kn
Wikipedia: Ordinal number
Includes: Ordinal Number, Lead sentenceQuote, Includes: Ordinal Number, Lead sentence
In set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, nth, etc.) aimed to extend e
Wikipedia: Von Neumann universe
Includes: Von Neumann Universe, Lead sentenceQuote, Includes: Von Neumann Universe, Lead sentence
In set theory and related branches of mathematics, the von Neumann universe, or von Neumann hierarchy of sets, denoted by V, is th
Wikipedia: Aronszajn tree
Includes: Aronszajn Tree, Lead sentenceQuote, Includes: Aronszajn Tree, Lead sentence
In set theory, an Aronszajn tree is a tree of uncountable height with no uncountable branches and no uncountable levels.
Inaccessible cardinal (Wikipedia)
Includes: Inaccessible Cardinal, Lead sentenceQuote, Includes: Inaccessible Cardinal, Lead sentence
In set theory, a cardinal number is a strongly inaccessible cardinal if it is uncountable, regular, and a strong limit cardinal.
Measurable cardinal (Wikipedia)
Includes: Measurable Cardinal, Lead sentenceQuote, Includes: Measurable Cardinal, Lead sentence
In mathematics, specifically in set theory, a measurable cardinal is a certain kind of large cardinal number.
Erdos-Rado theorem (Wikipedia)
Fodor's lemma, Wikipedia
Includes: Fodor's Lemma, Lead sentenceQuote, Includes: Fodor's Lemma, Lead sentence
In mathematics, particularly in set theory, Fodor's lemma (or the pressing-down lemma) states: In modern parlance, the nonstationa
Solovay model, Wikipedia
Includes: Solovay's Theorem, Lead sentenceQuote, Includes: Solovay's Theorem, Lead sentence
In the mathematical field of set theory, the Solovay model is a model constructed by Robert M.
Easton's theorem, Wikipedia
Includes: Easton's Theorem, Lead sentenceQuote, Includes: Easton's Theorem, Lead sentence
In set theory, Easton's theorem is a result on the possible cardinal numbers of powersets.
Cardinality of the Continuum (Wikipedia)
Axiom of Infinity (Wikipedia)
Includes: Axiom of Infinity, Lead sentenceQuote, Includes: Axiom of Infinity, Lead sentence
In axiomatic set theory and the branches of mathematics and philosophy that use it, the axiom of infinity is one of the axioms of
Constructible Universe (Wikipedia)
Thoralf Skolem (Wikipedia)
Includes: Thoralf Skolem, Lead paragraph [in-branch]Quote, Includes: Thoralf Skolem, Lead paragraph [in-branch]
set theory
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