Set theory

The modern study of set theory was initiated by Georg Cantor and Richard Dedekind in the 1870s. After the discovery of paradoxes in naive set theory, numerous axiom systems were proposed in the early twentieth century, of which the Zermelo–Fraenkel axioms, with the axiom of choice, are the best-known. Set theory is commonly employed as a foundational system for mathematics, particularly in the form of Zermelo–Fraenkel set theory with the axiom of choice. Beyond its foundational role, set theory is a branch of mathematics in its own right, with an active research community. Contemporary research into set theory includes a diverse collection of topics, ranging from the structure of the real number line to the study of the consistency of large cardinals. History Mathematical topics typically emerge and evolve through interactions among many researchers. Cantor's work initially polarized the mathematicians of his day. Basic concepts and notation Some ontology Sets alone.

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Well-order Every non-empty well-ordered set has a least element. Every element s of a well-ordered set, except a possible greatest element, has a unique successor (next element), namely the least element of the subset of all elements greater than s. There may be elements besides the least element which have no predecessor (see Natural numbers below for an example). In a well-ordered set S, every subset T which has an upper bound has a least upper bound, namely the least element of the subset of all upper bounds of T in S. If ≤ is a non-strict well-ordering, then < is a strict well-ordering. A relation is a strict well-ordering if and only if it is a well-founded strict total order.

Glossary of category theory This is a glossary of properties and concepts in category theory in mathematics. Categories Morphisms A morphism f in a category is called: Functors A functor F is said to be: K-theory Early history If X is a smooth variety, the two groups are the same. If it is a smooth affine variety, then all extensions of locally free sheaves split, so the group has an alternative definition. Already in 1955, Jean-Pierre Serre had used the analogy of vector bundles with projective modules to formulate Serre's conjecture, which states that every finitely generated projective module over a polynomial ring is free; this assertion is correct, but was not settled until 20 years later. (Swan's theorem is another aspect of this analogy.)

Ordinal number Representation of the ordinal numbers up to ωω. Each turn of the spiral represents one power of ω In set theory, an ordinal number, or just ordinal, is the order type of a well-ordered set. They are usually identified with hereditarily transitive sets. Category theory A category with objects X, Y, Z and morphisms f, g, g ∘ f, and three identity morphisms (not shown) 1X, 1Y and 1Z. Several terms used in category theory, including the term "morphism", differ from their uses within mathematics itself. In category theory, a "morphism" obeys a set of conditions specific to category theory itself. Thus, care must be taken to understand the context in which statements are made. An abstraction of other mathematical concepts The most accessible example of a category is the category of sets, where the objects are sets and the arrows are functions from one set to another.

Iwasawa theory Formulation Iwasawa worked with so-called -extensions: infinite extensions of a number field with Galois group 0 (number) 0 (zero; BrE: /ˈzɪərəʊ/ or AmE: /ˈziːroʊ/) is both a number[1] and the numerical digit used to represent that number in numerals. It fulfills a central role in mathematics as the additive identity of the integers, real numbers, and many other algebraic structures. As a digit, 0 is used as a placeholder in place value systems. In the English language, 0 may be called zero, nought or (US) naught /ˈnɔːt/, nil, or — in contexts where at least one adjacent digit distinguishes it from the letter "O" — oh or o /ˈoʊ/. Informal or slang terms for zero include zilch and zip.[2] Ought or aught /ˈɔːt/ has also been used historically.[3] (See Names for the number 0 in English.)

Natural transformation Definition If F and G are functors between the categories C and D, then a natural transformation η from F to G is a family of morphisms that satisfy two requirements. The natural transformation must associate to every object X in C a morphism ηX : F(X) → G(X) between objects of D. The morphism ηX is called the component of η at X.Components must be such that for every morphism f : X → Y in C we have: The last equation can conveniently be expressed by the commutative diagram

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