Cardinality is studied for its own sake as part of set theory. It is also a tool used in branches of mathematics including combinatorics, abstract algebra, and mathematical analysis. In category theory, the cardinal numbers form a skeleton of the category of sets.
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Anonymous2013-08-31 7:53
All the remaining propositions in this section assume the axiom of choice:
If κ and μ are both finite and greater than 1, and ν is infinite, then κν = μν.
If κ is infinite and μ is finite and non-zero, then κμ = κ.
The practice of refusing infinite values for measurable quantities does not come from a priori or ideological motivations, but rather from more methodological and pragmatic motivations.
Elementary set theory can be studied informally and intuitively, and so can be taught in primary schools using Venn diagrams. The intuitive approach tacitly assumes that a set may be formed from the class of all objects satisfying any particular defining condition.
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Anonymous2013-08-31 10:09
The surreal numbers are a proper class of objects that have the properties of a field.
Hence S breaks up into uncountably many orbits under G. Using the axiom of choice, we could pick a single point from each orbit, obtaining an uncountable subset X of S with the property that all of its translates by G are disjoint from X. The set of those translates partitions the circle into a countable collection of disjoint sets, which are all pairwise congruent.
Ernst Friedrich Ferdinand Zermelo (German: [ʦɛrˈmeːlo]; 1871–1953) was a German logician and mathematician, whose work has major implications for the foundations of mathematics.
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Anonymous2013-08-31 13:06
Assuming the axiom of choice and, given an infinite cardinal π and a non-zero cardinal μ, there exists a cardinal κ such that μ · κ = π if and only if μ ≤ π. It will be unique (and equal to π) if and only if μ < π.
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Anonymous2013-08-31 13:22
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Anonymous2013-08-31 13:51
If a set is too large to be put in one to one correspondence with the positive integers, it is called uncountable. Cantor's views prevailed and modern mathematics accepts actual infinity. Certain extended number systems, such as the hyperreal numbers, incorporate the ordinary (finite) numbers and infinite numbers of different sizes.
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Anonymous2013-08-31 14:07
So I'm getting a urine test at the lab in the hospital. Chick was processing me, but the chick next to her smiled at me, she was staring too. I smiled back and kind of giggled because I was happy. Then she started mumbling, I was like "hmmm?" but I finally picked up on what she was saying and she said "that’s the dr that falls in love with her". I was so confused. I turned back and there was a TV behind me. She smiled at the TV and was talking about it. I was invisible. I felt so embarrassed.
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Anonymous2013-08-31 14:37
Just as arithmetic features binary operations on numbers, set theory features binary operations on sets. The:
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Anonymous2013-08-31 14:52
I had a Russian BF once and he was hung like a dang elephant! I love anal, but with him it was a real challenge getting it done without a good amount of lube
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Anonymous2013-08-31 15:22
A famous problem is the normal Moore space question, a question in general topology that was the subject of intense research. The answer to the normal Moore space question was eventually proved to be independent of ZFC.
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Anonymous2013-08-31 16:07
The statement of the axiom of choice does not specify whether the collection of nonempty sets is finite or infinite, and thus implies that every finite collection of nonempty sets has a choice function. However, that particular case is a theorem of Zermelo–Fraenkel set theory without the axiom of choice (ZF); it is easily proved by mathematical induction.
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Anonymous2013-08-31 16:52
Every surjective function has a right inverse.
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Anonymous2013-08-31 17:37
There exists a model of ZF¬C in which real numbers are a countable union of countable sets.[16]
isomorphism if there exists a morphism g : b → a such that f ∘ g = 1b and g ∘ f = 1a.
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Anonymous2013-08-31 19:08
In practice, this is not a problem because if this disjointness does not hold, it can be assured by appending the domain and codomain to the morphisms, (say, as the second and third components of an ordered triple).
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"
Arithmetic operations similar to those given below for the extended real numbers can also be defined, though there is no distinction in the signs (therefore one exception is that infinity cannot be added to itself). On the other hand, this kind of infinity enables division by zero, namely z/0 = \infty for any nonzero complex number z. In this context it is often useful to consider meromorphic functions as maps into the Riemann sphere taking the value of \infty at the poles. The domain of a complex-valued function may be extended to include the point at infinity as well. One important example of such functions is the group of Möbius transformations.
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Anonymous2013-08-31 21:40
In 1899 Cantor had himself posed the question "What is the cardinal number of the set of all sets?", and obtained a related paradox. Russell used his paradox as a theme in his 1903 review of continental mathematics in his The Principles of Mathematics.
The axiom of determinacy (AD) is an important object of study; although incompatible with the axiom of choice, AD implies that all subsets of the real line are well behaved (in particular, measurable and with the perfect set property). AD can be used to prove that the Wadge degrees have an elegant structure.
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Anonymous2013-08-31 23:10
Authors who use this formulation often speak of the choice function on A, but be advised that this is a slightly different notion of choice function. Its domain is the powerset of A (with the empty set removed), and so makes sense for any set A, whereas with the definition used elsewhere in this article, the domain of a choice function on a collection of sets is that collection, and so only makes sense for sets of sets. With this alternate notion of choice function, the axiom of choice can be compactly stated as
There are important statements that, assuming the axioms of ZF but neither AC nor ¬AC, are equivalent to the axiom of choice. The most important among them are Zorn's lemma and the well-ordering theorem. In fact, Zermelo initially introduced the axiom of choice in order to formalize his proof of the well-ordering theorem.
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Anonymous2013-09-01 0:41
Note that strengthened negations may be compatible with weakened forms of AC. For example, ZF + DC + BP is consistent, if ZF is.
From the axioms, it can be proved that there is exactly one identity morphism for every object. Some authors deviate from the definition just given by identifying each object with its identity morphism.
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Anonymous2013-09-01 2:11
For every three objects X, Y, and Z, there exists a binary operation hom(X, Y) × hom(Y, Z) → hom(X, Z) called composition. The composite of f : X → Y and g : Y → Z is written g ∘ f or gf. The composition of morphisms is often represented by a commutative diagram. For example,
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Anonymous2013-09-01 2:21
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Anonymous2013-09-01 2:57
Diagonal functor: The diagonal functor is defined as the functor from D to the functor category DC which sends each object in D to the constant functor at that object.
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Anonymous2013-09-01 10:29
Since cardinality is such a common concept in mathematics, a variety of names are in use. Sameness of cardinality is sometimes referred to as equipotence, equipollence, or equinumerosity. It is thus said that two sets with the same cardinality are, respectively, equipotent, equipollent, or equinumerous.
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Anonymous2013-09-01 11:15
* Enumerable: lowest, intermediate, and highest
* Innumerable: nearly innumerable, truly innumerable, and innumerably innumerable