Zermelo began to axiomatize set theory in 1905; in 1908, he published his results despite his failure to prove the consistency of his axiomatic system.
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Anonymous2013-08-31 13:11
κμ + ν = κμ·κν.
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Anonymous2013-08-31 13:57
Infinite-dimensional spaces are widely used in geometry and topology, particularly as classifying spaces, notably Eilenberg−MacLane spaces. Common examples are the infinite-dimensional complex projective space K(Z,2) and the infinite-dimensional real projective space K(Z/2Z,1).
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Anonymous2013-08-31 14:42
Power set of a set A is the set whose members are all possible subsets of A. For example, the power set of {1, 2} is { {}, {1}, {2}, {1,2} } .
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Anonymous2013-08-31 15:27
Principles such as the axiom of choice and the law of the excluded middle appear in a spectrum of different forms, some of which can be proven, others which correspond to the classical notions; this allows for a detailed discussion of the effect of these axioms on mathematics.
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Anonymous2013-08-31 16:13
This method cannot, however, be used to show that every countable family of nonempty sets has a choice function, as is asserted by the axiom of countable choice. If the method is applied to an infinite sequence (Xi : i∈ω) of nonempty sets, a function is obtained at each finite stage, but there is no stage at which a choice function for the entire family is constructed, and no "limiting" choice function can be constructed, in general, in ZF without the axiom of choice.
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Anonymous2013-08-31 16:57
Every unital ring other than the trivial ring contains a maximal ideal.
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Anonymous2013-08-31 17:42
In all models of ZF¬C, the generalized continuum hypothesis does not hold.
The morphism f has a right-inverse if there is a morphism g : Y → X such that f ∘ g = idY. The right inverse g is also called a section of f. Morphisms having a right inverse are always epimorphisms, but the converse is not always true in every category, as an epimorphism may fail to have a right inverse.
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Anonymous2013-08-31 19:44
Cantor applied his concept of one-to-one correspondence to infinite sets;[1] e.g. the set of natural numbers N = {0, 1, 2, 3, ...}. Thus, all sets having a one-to-one correspondence with N he called denumerable (countably infinite) sets and they all have the same cardinal number. This cardinal number is called ℵ0, aleph-null. He called the cardinal numbers of these infinite sets, transfinite cardinal numbers.
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Anonymous2013-08-31 20:30
Assuming the axiom of choice and, given an infinite cardinal κ and a finite cardinal μ greater than 0, the cardinal ν satisfying νμ = κ will be κ.
Sometimes infinite result of a physical quantity may mean that the theory being used to compute the result may be approaching the point where it fails. This may help to indicate the limitations of a theory.
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Anonymous2013-08-31 22:00
Sets alone. This includes the most common axiomatic set theory, Zermelo–Fraenkel set theory (ZFC), which includes the axiom of choice. Fragments of ZFC include:
The paradoxes of naive set theory can be explained in terms of the inconsistent assumption that "all classes are sets". With a rigorous foundation, these paradoxes instead suggest proofs that certain classes are proper.
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Anonymous2013-08-31 23:30
One might say, "Even though the usual ordering of the real numbers does not work, it may be possible to find a different ordering of the real numbers which is a well-ordering. Then our choice function can choose the least element of every set under our unusual ordering." The problem then becomes that of constructing a well-ordering, which turns out to require the axiom of choice for its existence; every set can be well-ordered if and only if the axiom of choice holds.
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.
In the category of smooth manifolds, morphisms are smooth functions and isomorphisms are called diffeomorphisms.
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Anonymous2013-09-01 3:17
A small category with a single object is the same thing as a monoid: the morphisms of a one-object category can be thought of as elements of the monoid, and composition in the category is thought of as the monoid operation. Functors between one-object categories correspond to monoid homomorphisms. So in a sense, functors between arbitrary categories are a kind of generalization of monoid homomorphisms to categories with more than one object.