We can then extend this to an equality-style relation. Two sets X and Y are said to have the same cardinality if there exists a bijection between X and Y. By the Schroeder-Bernstein theorem, this is equivalent to there being both a one-to-one mapping from X to Y and a one-to-one mapping from Y to X. We then write |X| = |Y|. The cardinal number of X itself is often defined as the least ordinal a with |a| = |X|. This is called the von Neumann cardinal assignment; for this definition to make sense, it must be proved that every set has the same cardinality as some ordinal; this statement is the well-ordering principle. It is however possible to discuss the relative cardinality of sets without explicitly assigning names to objects.
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Anonymous2013-08-31 13:27
The earliest recorded idea of infinity comes from Anaximander, a pre-Socratic Greek philosopher who lived in Miletus. He used the word apeiron which means infinite or limitless.
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Anonymous2013-08-31 14:13
If, on the other hand, the universe were not curved like a sphere but had a flat topology, it could be both unbounded and infinite. The curvature of the universe can be measured through multipole moments in the spectrum of the cosmic background radiation.
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Anonymous2013-08-31 14:58
An enrichment of ZFC called Internal Set Theory was proposed by Edward Nelson in 1977.
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Anonymous2013-08-31 15:44
In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that "the product of a collection of non-empty sets is non-empty". More explicitly, it states that for every indexed family (S_i)_{i \in I} of nonempty sets there exists an indexed family (x_i)_{i \in I} of elements such that x_i \in S_i for every i \in I.
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Anonymous2013-08-31 16:29
It is possible to prove many theorems using neither the axiom of choice nor its negation; such statements will be true in any model of Zermelo–Fraenkel set theory (ZF), regardless of the truth or falsity of the axiom of choice in that particular model.
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Anonymous2013-08-31 16:48
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If the set A is infinite, then there exists an injection from the natural numbers N to A (see Dedekind infinite).
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Anonymous2013-08-31 17:59
Instead of focusing merely on the individual objects (e.g., groups) possessing a given structure, category theory emphasizes the morphisms – the structure-preserving mappings – between these objects; by studying these morphisms, we are able to learn more about the structure of the objects. In the case of groups, the morphisms are the group homomorphisms.
>>18
I'VE TOLD YOU MANY FUCKING TIMES TO STOP TRYING TO FIT IT, BUT YOU WON'T LISTED.
ENJOY BEING LAUGHED AT BY THE LOCALS.
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Anonymous2013-08-31 18:44
Adjoint functors: A functor can be left (or right) adjoint to another functor that maps in the opposite direction. Such a pair of adjoint functors typically arises from a construction defined by a universal property; this can be seen as a more abstract and powerful view on universal properties.
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Anonymous2013-08-31 19:28
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Cardinality is defined in terms of bijective functions. Two sets have the same cardinality if and only if there is a bijection between them. In the case of finite sets, this agrees with the intuitive notion of size. In the case of infinite sets, the behavior is more complex. A fundamental theorem due to Georg Cantor shows that it is possible for infinite sets to have different cardinalities, and in particular the cardinality of the set of real numbers is greater than the cardinality of the set of natural numbers. It is also possible for a proper subset of an infinite set to have the same cardinality as the original set, something that cannot happen with proper subsets of finite sets.