The classic example used is that of the infinite hotel paradox, also called Hilbert's paradox of the Grand Hotel. Suppose you are an innkeeper at a hotel with an infinite number of rooms. The hotel is full, and then a new guest arrives. It's possible to fit the extra guest in by asking the guest who was in room 1 to move to room 2, the guest in room 2 to move to room 3, and so on, leaving room 1 vacant. We can explicitly write a segment of this mapping:
1 ↔ 2
2 ↔ 3
3 ↔ 4
...
n ↔ n + 1
...
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Anonymous2013-08-31 8:04
However, the earliest attestable accounts of mathematical infinity come from Zeno of Elea (c. 490 BCE? – c. 430 BCE?), a pre-Socratic Greek philosopher of southern Italy and member of the Eleatic School founded by Parmenides. Aristotle called him the inventor of the dialectic. He is best known for his paradoxes, described by Bertrand Russell as "immeasurably subtle and profound".
As to date, analysis of the radiation patterns recorded by the WMAP spacecraft hints that the universe has a flat topology. This would be consistent with an infinite physical universe.
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Anonymous2013-08-31 9:35
Many mathematical concepts can be defined precisely using only set theoretic concepts. For example, mathematical structures as diverse as graphs, manifolds, rings, and vector spaces can all be defined as sets satisfying various (axiomatic) properties.
The axiom of choice was formulated in 1904 by Ernst Zermelo in order to formalize his proof of the well-ordering theorem.
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Anonymous2013-08-31 11:05
The restriction to ZF renders any claim that relies on either the axiom of choice or its negation unprovable. For example, the Banach–Tarski paradox is neither provable nor disprovable from ZF alone: it is impossible to construct the required decomposition of the unit ball in ZF, but also impossible to prove there is no such decomposition.
In informal use, a cardinal number is what is normally referred to as a counting number, provided that 0 is included: 0, 1, 2, .... They may be identified with the natural numbers beginning with 0. The counting numbers are exactly what can be defined formally as the finite cardinal numbers. Infinite cardinals only occur in higher-level mathematics and logic.
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Anonymous2013-08-31 20:32
Infinity (symbol: ∞) is an abstract concept describing something without any limit and is relevant in a number of fields, predominantly mathematics and physics. The English word infinity derives from Latin infinitas, which can be translated as "unboundedness", itself calqued from the Greek word apeiros, meaning "endless".
However, there are some theoretical circumstances where the end result is infinity. One example is the singularity in the description of black holes. Some solutions of the equations of the general theory of relativity allow for finite mass distributions of zero size, and thus infinite density. This is an example of what is called a mathematical singularity, or a point where a physical theory breaks down.
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Anonymous2013-08-31 22:03
Kripke-Platek set theory, which omits the axioms of infinity, powerset, and choice, and weakens the axiom schemata of separation and replacement.
Because classes do not have any formal status in the theory of ZF, the axioms of ZF do not immediately apply to classes. However, if an inaccessible cardinal κ is assumed, then the sets of smaller rank form a model of ZF (a Grothendieck universe), and its subsets can be thought of as "classes".
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Anonymous2013-08-31 23:33
The axiom of choice produces these intangibles (objects that are proven to exist, but which cannot be explicitly constructed), which may conflict with some philosophical principles. Because there is no canonical well-ordering of all sets, a construction that relies on a well-ordering may not produce a canonical result, even if a canonical result is desired (as is often the case in category theory)
There are several weaker statements that are not equivalent to the axiom of choice, but are closely related. One example is the axiom of dependent choice (DC). A still weaker example is the axiom of countable choice (ACω or CC), which states that a choice function exists for any countable set of nonempty sets.
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Anonymous2013-09-01 1:04
One of the simplest examples of a category is that of groupoid, defined as a category whose arrows or morphisms are all invertible. The groupoid concept is important in topology. Categories now appear in most branches of mathematics, some areas of theoretical computer science where they correspond to types, and mathematical physics where they can be used to describe vector spaces.
Indeed, it turns out that numerous important constructions can be described in a purely categorical way. The central concept which is needed for this purpose is called categorical limit, and can be dualized to yield the notion of a colimit.
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Anonymous2013-09-01 2:34
A functor is a type of mapping between categories, which is applied in category theory. Functors can be thought of as homomorphisms between categories. In the category of small categories, functors can be thought of more generally as morphisms.
Universal constructions often give rise to pairs of adjoint functors.
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Anonymous2013-09-01 9:58
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"