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 13:08
Much like cute flowers, anime girls bloom during spring. That's a goddamn ton of cartoons we're getting.
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Anonymous2013-08-31 13:20
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.
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Anonymous2013-08-31 13:53
Think of useing a usd, as a mental tax wright-off using it as toilet paper. To most people its worth the same amount as before worn down threw a dishwasher.
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Anonymous2013-08-31 14:06
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 14:38
In general: decompose the logical units of your system into the smallest referentially transparent components possible, them implement them in modules. Global or local environments for sets of components (or inside components) might be mapped to monads. Use algebraic data types to describe core data structures. Share those definitions widely.
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Anonymous2013-08-31 14:51
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:
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Anonymous2013-08-31 15:24
お待ちしております、
ここに書いてね
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Anonymous2013-08-31 15:36
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 16:21
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.
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 20:38
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 22:08
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 23:38
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.
Every infinite game G_S in which S is a Borel subset of Baire space is determined.
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Anonymous2013-09-01 1:09
A group homomorphism between two groups "preserves the group structure" in a precise sense – it is a "process" taking one group to another, in a way that carries along information about the structure of the first group into the second group. The study of group homomorphisms then provides a tool for studying general properties of groups and consequences of the group axioms.
Many of the above concepts, especially equivalence of categories, adjoint functor pairs, and functor categories, can be situated into the context of higher-dimensional categories. Briefly, if we consider a morphism between two objects as a "process taking us from one object to another", then higher-dimensional categories allow us to profitably generalize this by considering "higher-dimensional processes".