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. Cantorian set theory eventually became widespread, due to the utility of Cantorian concepts, such as one-to-one correspondence among sets, his proof that there are more real numbers than integers, and the "infinity of infinities" ("Cantor's paradise") resulting from the power set operation. This utility of set theory led to the article "Mengenlehre" contributed in 1898 by Arthur Schoenflies to Klein's encyclopedia.
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Anonymous2013-08-31 7:39
Multiplication is non-decreasing in both arguments: κ ≤ μ → (κ·ν ≤ μ·ν and ν·κ ≤ ν·μ).
Dedekind's approach was essentially to adopt the idea of one-to-one correspondence as a standard for comparing the size of sets, and to reject the view of Galileo (which derived from Euclid) that the whole cannot be the same size as the part. An infinite set can simply be defined as one having the same size as at least one of its proper parts; this notion of infinity is called Dedekind infinite. The diagram gives an example: viewing lines as infinite sets of points, the left half of the lower blue line can be mapped in a one-to-one manner (green correspondences) to the higher blue line, and, in turn, to the whole lower blue line (red correspondences); therefore the whole lower blue line and its left half have the same cardinality, i.e. "size".
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Anonymous2013-08-31 9:09
Set theory begins with a fundamental binary relation between an object o and a set A. If o is a member (or element) of A, write o ∈ A. Since sets are objects, the membership relation can relate sets as well.
A cardinal invariant is a property of the real line measured by a cardinal number. For example, a well-studied invariant is the smallest cardinality of a collection of meagre sets of reals whose union is the entire real line.
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Anonymous2013-08-31 10:39
For any set A there is a function f such that for any non-empty subset B of A, f(B) lies in B.
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Anonymous2013-08-31 10:49
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Anonymous2013-08-31 11:24
Trichotomy: If two sets are given, then either they have the same cardinality, or one has a smaller cardinality than the other.
Kimi to Kanojo to Kanojo no Koi was poorly written and had little going for it other than the meta gimmick. Only play it if you're really interested in having someone preach at you about how horrible a person you are for playing eroge
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Anonymous2013-08-31 13:40
Infinity is also used to describe infinite series:
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Anonymous2013-08-31 14:02
KITES, MUTHAFUCKA! DO YOU FLY THEM!?
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Anonymous2013-08-31 14:26
Mathematical topics typically emerge and evolve through interactions among many researchers. Set theory, however, was founded by a single paper in 1874 by Georg Cantor: "On a Characteristic Property of All Real Algebraic Numbers".
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Anonymous2013-08-31 14:47
I was offered a position at a Japanese chemical company, starting stateside, and moving to Japan in a few years.
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Anonymous2013-08-31 15:11
An inner model of Zermelo–Fraenkel set theory (ZF) is a transitive class that includes all the ordinals and satisfies all the axioms of ZF. The canonical example is the constructible universe L developed by Gödel. One reason that the study of inner models is of interest is that it can be used to prove consistency results. For example, it can be shown that regardless of whether a model V of ZF satisfies the continuum hypothesis or the axiom of choice, the inner model L constructed inside the original model will satisfy both the generalized continuum hypothesis and the axiom of choice. Thus the assumption that ZF is consistent (has at least one model) implies that ZF together with these two principles is consistent.
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Anonymous2013-08-31 15:56
There are many other equivalent statements of the axiom of choice. These are equivalent in the sense that, in the presence of other basic axioms of set theory, they imply the axiom of choice and are implied by it.
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Anonymous2013-08-31 16:41
One argument given in favor of using the axiom of choice is that it is convenient to use it because it allows one to prove some simplifying propositions that otherwise could not be proved. Many theorems which are provable using choice are of an elegant general character: every ideal in a ring is contained in a maximal ideal, every vector space has a basis, and every product of compact spaces is compact. Without the axiom of choice, these theorems may not hold for mathematical objects of large cardinality.
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Anonymous2013-08-31 17:27
The Baire category theorem about complete metric spaces, and its consequences, such as the open mapping theorem and the closed graph theorem.
A class hom(C), whose elements are called morphisms or maps or arrows. Each morphism f has a source object a and target object b.
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Anonymous2013-08-31 18:57
In many fields of mathematics, morphism refers to a structure-preserving mapping from one mathematical structure to another. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms are functions; in linear algebra, linear transformations; in group theory, group homomorphisms; in topology, continuous functions, and so on.