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Name:
Anonymous2013-08-31 12:38
More formally, a non-zero number can be used for two purposes: to describe the size of a set, or to describe the position of an element in a sequence. For finite sets and sequences it is easy to see that these two notions coincide, since for every number describing a position in a sequence we can construct a set which has exactly the right size, e.g. 3 describes the position of 'c' in the sequence <'a','b','c','d',...>, and we can construct the set {a,b,c} which has 3 elements. However when dealing with infinite sets it is essential to distinguish between the two — the two notions are in fact different for infinite sets. Considering the position aspect leads to ordinal numbers, while the size aspect is generalized by the cardinal numbers described here.
In mathematics, "infinity" is often treated as if it were a number (i.e., it counts or measures things: "an infinite number of terms") but it is not the same sort of number as the real numbers. In number systems incorporating infinitesimals, the reciprocal of an infinitesimal is an infinite number, i.e., a number greater than any real number.
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Anonymous2013-08-31 14:09
This does not necessarily mean that physical infinities exist; it may mean simply that the theory is incapable of describing the situation properly. Two other examples occur in inverse-square force laws of the gravitational force equation of Newtonian gravity and Coulomb's law of electrostatics. At r=0 these equations evaluate to infinities.
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Anonymous2013-08-31 14:55
Sets and proper classes. These include Von Neumann–Bernays–Gödel set theory, which has the same strength as ZFC for theorems about sets alone, and Morse-Kelley set theory and Tarski–Grothendieck set theory, both of which are stronger than ZFC.
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Anonymous2013-08-31 15:40
Another approach is taken by the von Neumann–Bernays–Gödel axioms (NBG); classes are the basic objects in this theory, and a set is then defined to be a class that is an element of some other class.
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Anonymous2013-08-31 16:25
This has been used as an argument against the use of the axiom of choice.
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Anonymous2013-08-31 17:10
These axioms are sufficient for many proofs in elementary mathematical analysis, and are consistent with some principles, such as the Lebesgue measurability of all sets of reals, that are disprovable from the full axiom of choice.
Categories were first introduced by Samuel Eilenberg and Saunders Mac Lane in 1942–45, in connection with algebraic topology.
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Anonymous2013-08-31 18:40
It is a natural question to ask: under which conditions can two categories be considered to be "essentially the same", in the sense that theorems about one category can readily be transformed into theorems about the other category? The major tool one employs to describe such a situation is called equivalence of categories, which is given by appropriate functors between two categories. Categorical equivalence has found numerous applications in mathematics.
Functors were first considered in algebraic topology, where algebraic objects (like the fundamental group) are associated to topological spaces, and algebraic homomorphisms are associated to continuous maps. Nowadays, functors are used throughout modern mathematics to relate various categories. Thus, functors are generally applicable in areas within mathematics that category theory can make an abstraction of.
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Anonymous2013-08-31 19:45
Cantor also developed a large portion of the general theory of cardinal numbers; he proved that there is a smallest transfinite cardinal number (ℵ0, aleph-null) and that for every cardinal number, there is a next-larger cardinal
The logarithm of an infinite cardinal number κ is defined as the least cardinal number μ such that κ ≤ 2μ. Logarithms of infinite cardinals are useful in some fields of mathematics, for example in the study of cardinal invariants of topological spaces, though they lack some of the properties that logarithms of positive real numbers possess.
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Anonymous2013-08-31 21:16
This point of view does not mean that infinity cannot be used in physics. For convenience's sake, calculations, equations, theories and approximations often use infinite series, unbounded functions, etc., and may involve infinite quantities.
Zermelo set theory, which replaces the axiom schema of replacement with that of separation;
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Anonymous2013-08-31 22:46
For example, Russell's paradox suggests a proof that the class of all sets which do not contain themselves is proper, and the Burali-Forti paradox suggests that the class of all ordinal numbers is proper.
A proof requiring the axiom of choice may establish the existence of an object without explicitly defining the object in the language of set theory. For example, while the axiom of choice implies that there is a well-ordering of the real numbers, there are models of set theory with the axiom of choice in which no well-ordering of the reals is definable.
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Anonymous2013-09-01 0:16
If two small categories are weakly equivalent, then they are equivalent.
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Anonymous2013-09-01 0:17
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Anonymous2013-09-01 1:02
Many significant areas of mathematics can be formalised by category theory as categories. Category theory is an abstraction of mathematics itself that allows many intricate and subtle mathematical results in these fields to be stated, and proved, in a much simpler way than without the use of categories.
Each category is distinguished by properties that all its objects have in common, such as the empty set or the product of two topologies, yet in the definition of a category, objects are considered to be atomic, i.e., we do not know whether an object A is a set, a topology, or any other abstract concept.
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Anonymous2013-09-01 2:32
In the category of small categories, functors can be thought of as morphisms.
Let C and D be categories. The collection of all functors C→D form the objects of a category: the functor category. Morphisms in this category are natural transformations between functors.
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Anonymous2013-09-01 10:14
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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Anonymous2013-09-01 11:00
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.