In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality (size) of sets. The cardinality of a finite set is a natural number – the number of elements in the set. The transfinite cardinal numbers describe the sizes of infinite sets.
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Anonymous2013-08-31 13:14
Exponentiation is non-decreasing in both arguments:
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Anonymous2013-08-31 13:59
Leopold Kronecker was skeptical of the notion of infinity and how his fellow mathematicians were using it in 1870s and 1880s. This skepticism was developed in the philosophy of mathematics called finitism, an extreme form of the philosophical and mathematical schools of constructivism and intuitionism. Typical dumb goy.
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Anonymous2013-08-31 14:45
In modern set theory, it is common to restrict attention to the von Neumann universe of pure sets, and many systems of axiomatic set theory are designed to axiomatize the pure sets only.
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Anonymous2013-08-31 15:30
A class that is not a set (informally in Zermelo–Fraenkel) is called a proper class, and a class that is a set is sometimes called a small class. For instance, the class of all ordinal numbers, and the class of all sets, are proper classes in many formal systems.
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Anonymous2013-08-31 16:15
The difficulty appears when there is no natural choice of elements from each set. If we cannot make explicit choices, how do we know that our set exists? For example, suppose that X is the set of all non-empty subsets of the real numbers.
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Anonymous2013-08-31 17:00
Tychonoff's theorem stating that every product of compact topological spaces is compact.
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Anonymous2013-08-31 17:02
>>13
how to prove how to prove. Suppose A and B are compact how to show that A * B is compact
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Anonymous2013-08-31 17:03
A is compact iff every open cover admits a finite subcover
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Anonymous2013-08-31 17:03
want to show that every open cover of A * B admits a finite sub cover
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Anonymous2013-08-31 17:04
Let O be some open cover of A * B. we want to find a finite sub cover for O
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Anonymous2013-08-31 17:04
Each element of the open cover O is an open set of A * B
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Anonymous2013-08-31 17:06
The open sets of A * B are of the formed by unions of products of open sets in A and B
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Anonymous2013-08-31 17:06
can the unions be infinite? I don't remember. I think so.
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Anonymous2013-08-31 17:08
So each element o in O is an open set of A * B, and formed by an arbitrary union of base open sets, C*D where C and D are open in A and B respectively.
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Anonymous2013-08-31 17:09
Create a new open cover P, that is formed by the elements of O where each o is replaced by the arbitrary union used to form it from the base elements.
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Anonymous2013-08-31 17:10
So P is an open cover for A * B, where each element is of the form C*D where C and D are open in A and B respectively.
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Anonymous2013-08-31 17:10
So P is an open cover for A * B, where each element is of the form C*D where C and D are open in A and B respectively.
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Anonymous2013-08-31 17:11
Now consider this.....
\union C_i * D_i = \union C_i * \union D_i
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Anonymous2013-08-31 17:11
wait that's wrong
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Anonymous2013-08-31 17:14
i guess i'm not very good at this after all
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Anonymous2013-08-31 17:17
Consider the projection of the open cover P, down to A and B.
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Anonymous2013-08-31 17:18
let a be some arbitrary element of A, and b some arbitrary element of B
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Anonymous2013-08-31 17:18
Intersect P with a and b. Call them Pa and Pb
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Anonymous2013-08-31 17:20
Each Pa is an open cover of A. And each Pb is an open cover of B.
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Anonymous2013-08-31 17:20
Since A and B are compact, each Pa and Pb has a finite subcover
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Anonymous2013-08-31 17:21
Let the finite subcovers be Fa and Fb.
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Anonymous2013-08-31 17:22
Except, let Fa and Fb have the original elements of P, that were projected onto (a,_) and (_,b).
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Anonymous2013-08-31 17:45
"The Axiom of Choice is necessary to select a set from an infinite number of socks, but not an infinite number of shoes." — Bertrand Russell
His continuum hypothesis is the proposition that ???? is the same as ℵ1. This hypothesis has been found to be independent of the standard axioms of mathematical set theory; it can neither be proved nor disproved from the standard assumptions.
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Anonymous2013-08-31 20:32
The continuum hypothesis (CH) states that there are no cardinals strictly between ℵ0 and 2ℵ0. The latter cardinal number is also often denoted by ????; it is the cardinality of the continuum (the set of real numbers). In this case 2ℵ0 = ℵ1. The generalized continuum hypothesis (GCH) states that for every infinite set X, there are no cardinals strictly between | X | and 2| X |. The continuum hypothesis is independent of the usual axioms of set theory, the Zermelo-Fraenkel axioms together with the axiom of choice (ZFC).
Physicists however require that the end result be physically meaningful. In quantum field theory infinities arise which need to be interpreted in such a way as to lead to a physically meaningful result, a process called renormalization.
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Anonymous2013-08-31 22:02
General set theory, a small fragment of Zermelo set theory sufficient for the Peano axioms and finite sets;
ZF set theory does not formalize the notion of classes. They can instead be described in the metalanguage, as equivalence classes of logical formulas. For example, if \mathcal A is a structure interpreting ZF, then the metalanguage expression \{x\mid x=x \} is interpreted in \mathcal A by the collection of all the elements from the domain of \mathcal A; that is, all the sets in \mathcal A. So we can identify the "class of all sets" with the predicate x=x or any equivalent predicate.
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Anonymous2013-08-31 23:32
Similarly, although a subset of the real numbers that is not Lebesgue measurable can be proven to exist using the axiom of choice, it is consistent that no such set is definable.
Every continuous functor on a small-complete category which satisfies the appropriate solution set condition has a left-adjoint (the Freyd adjoint functor theorem).
The most accessible example of a category is the category of sets, where the objects are sets and the arrows are functions from one set to another. However, the objects of a category need not be sets, and the arrows need not be functions; any way of formalising a mathematical concept such that it meets the basic conditions on the behaviour of objects and arrows is a valid category, and all the results of category theory will apply to it.
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Anonymous2013-09-01 1:47
Hence, the challenge is to define special objects without referring to the internal structure of those objects. To define the empty set without referring to elements, or the product topology without referring to open sets, one can characterize these objects in terms of their relations to other objects, as given by the morphisms of the respective categories. Thus, the task is to find universal properties that uniquely determine the objects of interest.
In a functor category, the morphisms are natural transformations.
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Anonymous2013-09-01 3:19
Functors are often defined by universal properties; examples are the tensor product, the direct sum and direct product of groups or vector spaces, construction of free groups and modules, direct and inverse limits. The concepts of limit and colimit generalize several of the above.
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Anonymous2013-09-01 10:04
He graduated from Berlin's Luisenstädtisches Gymnasium in 1889. He then studied mathematics, physics and philosophy at the universities of Berlin, Halle and Freiburg. He finished his doctorate in 1894 at the University of Berlin, awarded for a dissertation on the calculus of variations (Untersuchungen zur Variationsrechnung). Zermelo remained at the University of Berlin, where he was appointed assistant to Planck, under whose guidance he began to study hydrodynamics. In 1897, Zermelo went to Göttingen, at that time the leading centre for mathematical research in the world, where he completed his habilitation thesis in 1899.
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Anonymous2013-09-01 10:50
κ0 = 1 (in particular 00 = 1), see empty function.