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Axiomatic set theory - The origins of rigorous set theory | | Axiomatic set theory - The origins of rigorous set theory: Encyclopedia II - Axiomatic set theory - The origins of rigorous set theory | | The important idea of Cantor's, which got set theory going as a new field of study, was to define two sets A and B to have the same number of members (the same cardinality) when there is a way of pairing off members of A exhaustively with members of B. Then the set N of natural numbers has the same cardinality as the set Q of rational numbers (they are both said to be countably infinite), even though N is a proper subset of Q. On the other hand, the set R of real numbers d ...
See also:Axiomatic set theory, Axiomatic set theory - The origins of rigorous set theory, Axiomatic set theory - Axioms for set theory, Axiomatic set theory - Independence in ZFC, Axiomatic set theory - Set theory ZFC foundations for mathematics, Axiomatic set theory - Well-foundedness and hypersets, Axiomatic set theory - Objections to set theory | | | Axiomatic set theory, Axiomatic set theory - Axioms for set theory, Axiomatic set theory - Independence in ZFC, Axiomatic set theory - Objections to set theory, Axiomatic set theory - Set theory ZFC foundations for mathematics, Axiomatic set theory - The origins of rigorous set theory, Axiomatic set theory - Well-foundedness and hypersets, Alternative set theory, List of set theory topics, Zermelo-Fraenkel set theory, Simple theorems in the algebra of sets, Naive set theory, Cantor–Bernstein–Schroeder theorem, Zorn's lemma, Cantor's theorem, Cantor's diagonal argument, Model theory, Internal set theory, Kripke-Platek set theory with urelements | | |
| | | Axiomatic set theory: Encyclopedia II - Axiomatic set theory - The origins of rigorous set theory
Axiomatic set theory - The origins of rigorous set theory
The important idea of Cantor's, which got set theory going as a new field of study, was to define two sets A and B to have the same number of members (the same cardinality) when there is a way of pairing off members of A exhaustively with members of B. Then the set N of natural numbers has the same cardinality as the set Q of rational numbers (they are both said to be countably infinite), even though N is a proper subset of Q. On the other hand, the set R of real numbers does not have the same cardinality as N or Q, but a larger one (it is said to be uncountable). Cantor gave two proofs that R is not countable, and the second of these, using what is known as the diagonal construction, has been extraordinarily influential and has had many applications in logic and mathematics.
Cantor constructed infinite hierarchies of infinite sets, the ordinal and cardinal numbers. This was controversial in his day, with the opposition led by the finitist Leopold Kronecker, but there is no significant disagreement among mathematicians today that Cantor had the right idea.
Cantor's development of set theory was still "naive" in the sense that he did not have a precise axiomatization in mind. In retrospect, we can say that Cantor was tacitly using the axiom of extensionality, the axiom of infinity, and the axiom schema of (unrestricted) comprehension. Some do not agree that Cantor actually made the last assumption: Frege certainly did, and it was Frege's theory that Russell was actually addressing when he formulated Russell's paradox by constructing the set S := {A : A is not in A} of all sets that do not belong to themselves. (If S belongs to itself, then it does not, giving a contradiction, so S must not belong to itself. But then S would belong to itself, giving a final and absolute contradiction.) Therefore, set theorists were forced to abandon either classical logic or unrestricted comprehension, and the latter was far more reasonable to most. (Although intuitionism had a significant following, the paradox still goes through with intuitionistic logic. There is no paradox in Brazilian logic, but that was almost completely unknown at the time.)
In order to avoid this and similar paradoxes, Ernst Zermelo, working in Germany, put forth a system of axioms for set theory in 1908. He included in this system the axiom of choice, a controversial axiom that he needed to prove the well-ordering theorem. This system was later refined by Adolf Fraenkel and Thoralf Skolem, giving the axioms used today.
Other related archives"naive" or "intuitive" set theory, 1908, 19th century, Adolf Fraenkel, Alternative set theory, Axiom of Constructibility (V=L), Axiom of choice, Axiom of empty set, Axiom of extensionality, Axiom of infinity, Axiom of pairing, Axiom of power set, Axiom of regularity, Axiom of replacement, Axiom of separation, Axiom of union, Brazilian logic, Cantor's diagonal argument, Cantor's theorem, Cantor–Bernstein–Schroeder theorem, Continuum hypothesis, Dana Scott, Diamond principle, Ernst Zermelo, Errett Bishop, Frege, Georg Cantor, Germany, Godel's 2nd incompleteness theorem, Henri Poincaré, Internal set theory, Kripke-Platek set theory, Kripke-Platek set theory with urelements, Leopold Kronecker, Liar Paradox, List of set theory topics, Martin's axiom, Model theory, Morse-Kelley set theory, NF, Naive set theory, New Foundations, Peano Axioms, Quine, Russell, Russell's paradox, Simple theorems in the algebra of sets, Suslin hypothesis, Thoralf Skolem, Topos theory, Von Neumann-Bernays-Gödel set theory, Zermelo-Fraenkel set theory, Zorn's lemma, axiom of choice, axiom of extensionality, axiom of infinity, axiom of regularity, axiom schema of (unrestricted) comprehension, axiomatic, axiomatization, axioms, cardinal numbers, cardinality, classical logic, complex numbers, computable, constructible universe, constructivism, constructivist, continuous, contradiction, countably infinite, diagonal construction, discrete, disjoint sets, finitist, folklore, forcing, foundational theory, function, functions, fuzzy set theory, hyperset, iff, independent, inductively, inner models, integers, intuitionism, intuitionistic logic, large cardinals, list of statements undecidable in ZFC, logicians, mapping, mathematical rigor, mathematician, mathematics, naive set theory, natural numbers, non-well-founded sets, number, objected to using set theory as a foundation for mathematics, order, ordered lists, ordered pair, ordinal, paradoxes, positive set theory, power set, process algebra, proper subset, proposition, rational, rational numbers, real, real numbers, relation, relations, set, subset, uncountable, union, von Neumann, well-ordering theorem
 Adapted from the Wikipedia article "The origins of rigorous set theory", under the G.N U Free Docmentation License. Please also see http://en.wikipedia.org/wiki |
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