The intuition behind the formal definition of cardinal is the construction of a notion of the relative size or "bigness" of a set without reference to the kind of members which it has. For finite sets this is easy; one simply counts the number of elements a set has. In order to compare the sizes of larger sets, it is necessary to appeal to more subtle notions.
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Anonymous2013-08-31 8:00
Georg Cantor formalized many ideas related to infinity and infinite sets during the late 19th and early 20th centuries. In the theory he developed, there are infinite sets of different "sizes" (called cardinalities).
In 1584, the Italian philosopher and astronomer Giordano Bruno proposed an unbounded universe in On the Infinite Universe and Worlds: "Innumerable suns exist; innumerable earths revolve around these suns in a manner similar to the way the seven planets revolve around our sun. Living beings inhabit these worlds."
The above systems can be modified to allow urelements, objects that can be members of sets but that are not themselves sets and do not have any members.
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Anonymous2013-08-31 10:16
However, the set existence axioms of NBG are restricted so that they only quantify over sets, rather than over all classes. This causes NBG to be a conservative extension of ZF.
Another argument against the axiom of choice is that it implies the existence of counterintuitive objects. One example is the Banach–Tarski paradox which says that it is possible to decompose ("carve up") the 3-dimensional solid unit ball into finitely many pieces and, using only rotations and translations, reassemble the pieces into two solid balls each with the same volume as the original.
It was about Hikikomori as long as Misaki didn't show up.
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Anonymous2013-08-31 14:08
I remember reading and watching a documentary about Project Orion a few years ago, and it really amazed me how close they came to launching a manned rocket to Saturn. If anyone doesn't know what Orion is, see: http://en.wikipedia.org/wiki/Project_Orion_%28nuclear_propulsion%29
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Anonymous2013-08-31 14:53
Hey. I'm an american and i want to get news. I know CNN and FOX and all the crap that american's typicaly get is biased garbage. I dont trust BBC that much. What are some realiable new sources that i can look too.
The notion of cardinality, as now understood, was formulated by Georg Cantor, the originator of set theory, in 1874–1884. Cardinality can be used to compare an aspect of finite sets; e.g. the sets {1,2,3} and {4,5,6} are not equal, but have the same cardinality, namely three (this is established by the existence of a one-to-one correspondence between the two sets; e.g. {1->4, 2->5, 3->6}).
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Anonymous2013-08-31 20:28
If 2 ≤ κ and 1 ≤ μ and at least one of them is infinite, then:
One of the needs of any physical and scientific theory is to give usable formulas that correspond to or at least approximate reality. As an example if any object of infinite gravitational mass were to exist, any usage of the formula to calculate the gravitational force would lead to an infinite result, which would be of no benefit since the result would be always the same regardless of the position and the mass of the other object.
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Anonymous2013-08-31 21:58
This assumption gives rise to paradoxes, the simplest and best known of which are Russell's paradox and the Burali-Forti paradox. Axiomatic set theory was originally devised to rid set theory of such paradoxes.
Within set theory, many collections of sets turn out to be proper classes. Examples include the class of all sets, the class of all ordinal numbers, and the class of all cardinal numbers.
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Anonymous2013-08-31 23:28
Since X isn't measurable for any rotation-invariant countably additive finite measure on S, finding an algorithm to select a point in each orbit requires the axiom of choice. See non-measurable set for more details.
On the other hand, other foundational descriptions of category theory are considerably stronger, and an identical category-theoretic statement of choice may be stronger than the standard formulation, à la class theory, mentioned above.
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Anonymous2013-09-01 0:59
This quote comes from the famous April Fools' Day article in the computer recreations column of the Scientific American, April 1989.
If F and G are (covariant) functors between the categories C and D, then a natural transformation η from F to G associates to every object X in C a morphism ηX : F(X) → G(X) in D such that for every morphism f : X → Y in C, we have ηY ∘ F(f) = G(f) ∘ ηX; this means that the following diagram is commutative:
In the concrete categories studied in universal algebra (groups, rings, modules, etc.), morphisms are usually homomorphisms. Likewise, the notions of automorphism, endomorphism, epimorphism, homeomorphism, isomorphism, and monomorphism all find use in universal algebra.
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Anonymous2013-09-01 3:15
if f is an isomorphism in C, then F(f) is an isomorphism in D.
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Anonymous2013-09-01 10:09
In 1922, Adolf Fraenkel and Thoralf Skolem independently improved Zermelo's axiom system. The resulting 10 axiom system, now called Zermelo-Fraenkel axioms (ZF), is now the most commonly used system for axiomatic set theory.