Set theory is the branch of mathematical logic that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics. The language of set theory can be used in the definitions of nearly all mathematical objects.
The product of cardinals comes from the cartesian product.
|X| · |Y| = |X × Y|
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Anonymous2013-08-31 13:43
Infinity is often used not only to define a limit but as a value in the affinely extended real number system. Points labeled +\infty and -\infty can be added to the topological space of the real numbers, producing the two-point compactification of the real numbers.
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Anonymous2013-08-31 14:28
Cantor's work initially polarized the mathematicians of his day. While Karl Weierstrass and Dedekind supported Cantor, Leopold Kronecker, now seen as a founder of mathematical constructivism, did not.
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Anonymous2013-08-31 15:13
A large cardinal is a cardinal number with an extra property. Many such properties are studied, including inaccessible cardinals, measurable cardinals, and many more.
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Anonymous2013-08-31 15:59
This guarantees for any partition of a set X the existence of a subset C of X containing exactly one element from each part of the partition.
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Anonymous2013-08-31 16:44
The axiom of choice is not the only significant statement which is independent of ZF. For example, the generalized continuum hypothesis (GCH) is not only independent of ZF, but also independent of ZFC. However, ZF plus GCH implies AC, making GCH a strictly stronger claim than AC, even though they are both independent of ZF.
Every Tychonoff space has a Stone–Čech compactification.
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Anonymous2013-08-31 18:14
A binary operation ∘, called composition of morphisms, such that for any three objects a, b, and c, we have hom(b, c) × hom(a, b) → hom(a, c). The composition of f : a → b and g : b → c is written as g ∘ f or gf,[4] governed by two axioms:
One of Cantor's most important results was that the cardinality of the continuum \mathbf c is greater than that of the natural numbers {leph_0}; that is, there are more real numbers R than natural numbers N. Namely, Cantor showed that \mathbf{c} = 2^{leph_0} > {leph_0} (see Cantor's diagonal argument or Cantor's first uncountability proof).
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Anonymous2013-08-31 21:46
Union of the sets A and B, denoted A ∪ B, is the set of all objects that are a member of A, or B, or both. The union of {1, 2, 3} and {2, 3, 4} is the set {1, 2, 3, 4} .
From set theory's inception, some mathematicians have objected to it as a foundation for mathematics. The most common objection to set theory, one Kronecker voiced in set theory's earliest years, starts from the constructivist view that mathematics is loosely related to computation. If this view is granted, then the treatment of infinite sets, both in naive and in axiomatic set theory, introduces into mathematics methods and objects that are not computable even in principle.
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Anonymous2013-08-31 23:17
In the even simpler case of a collection of one set, a choice function just corresponds to an element, so this instance of the axiom of choice says that every nonempty set has an element; this holds trivially. The axiom of choice can be seen as asserting the generalization of this property, already evident for finite collections, to arbitrary collections
Zorn's lemma: Every non-empty partially ordered set in which every chain (i.e. totally ordered subset) has an upper bound contains at least one maximal element.
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Anonymous2013-09-01 0:47
There exists a model of ZF¬C in which there is a field with no algebraic closure.
endomorphism if a = b. end(a) denotes the class of endomorphisms of a.
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Anonymous2013-09-01 2:17
Monomorphism: f : X → Y is called a monomorphism if f ∘ g1 = f ∘ g2 implies g1 = g2 for all morphisms g1, g2 : Z → X. It is also called a mono or a monic
Algebra of continuous functions: a contravariant functor from the category of topological spaces (with continuous maps as morphisms) to the category of real associative algebras is given by assigning to every topological space X the algebra C(X) of all real-valued continuous functions on that space. Every continuous map f : X → Y induces an algebra homomorphism C(f) : C(Y) → C(X) by the rule C(f)(φ) = φ o f for every φ in C(Y).