Axiomatic set theory

Set theory is a branch of mathematics created principally by the German mathematician Georg Cantor at the end of the 19th century. Initially controversial, set theory has come to play the role of a foundational theory in modern mathematics, in the sense of a theory invoked to justify assumptions made in mathematics concerning the existence of mathematical objects (such as numbers or functions) and their properties.

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 manifold applications in logic and mathematics.

Related Topics:
Cardinality - Rational number - Countably infinite - Proper subset - Uncountable - Diagonal construction

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Cantor went right ahead and 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.

Related Topics:
Ordinal - Cardinal number - Finitist - Leopold Kronecker

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Cantor's development of set theory was still "naïve" 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. However, the last of these leads directly to 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.)

Related Topics:
Axiomatization - Axiom of extensionality - Axiom of infinity - Axiom schema of (unrestricted) comprehension - Russell's paradox - Contradiction - Classical logic - Intuitionism - Intuitionistic logic - Brazilian logic

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In order to avoid this and similar paradoxes, Ernst Zermelo put forth a system of axioms for set theory in 1908. He included in this system the axiom of choice, a truly 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.

Related Topics:
Ernst Zermelo - Axiom - 1908 - Axiom of choice - Well-ordering theorem - Adolf Fraenkel - Thoralf Skolem

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