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 20:34
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."
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Anonymous2013-08-31 22:04
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.
>>15,17-18
not sure about you guys, but i'm never coming back to this shithole
see you at the DPT
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Anonymous2013-08-31 22:49
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.
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Anonymous2013-09-01 0:20
Other choice axioms weaker than axiom of choice include the Boolean prime ideal theorem and the axiom of uniformization. The former is equivalent in ZF to the existence of an ultrafilter containing each given filter, proved by Tarski in 1930.
Category theory has several faces known not just to specialists, but to other mathematicians. A term dating from the 1940s, "general abstract bullshite", refers to its high level of abstraction, compared to more classical branches of mathematics. Homological algebra is category theory in its aspect of organising and suggesting manipulations in abstract algebra.
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Anonymous2013-09-01 1:50
The definitions of categories and functors provide only the very basics of categorical algebra; additional important topics are listed below. Although there are strong interrelations between all of these topics, the given order can be considered as a guideline for further reading.
Let C and D be categories. A functor F from C to D is a mapping that
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Anonymous2013-09-01 3:21
If F and G are 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) between objects of D, called the component of η at X, such that for every morphism f : X → Y in C we have:
\eta_Y \circ F(f) = G(f) \circ \eta_X
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Anonymous2013-09-01 10:32
We can define arithmetic operations on cardinal numbers that generalize the ordinary operations for natural numbers. It can be shown that for finite cardinals these operations coincide with the usual operations for natural numbers. Furthermore, these operations share many properties with ordinary arithmetic.
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Anonymous2013-09-01 11:19
Leibniz, one of the co-inventors of infinitesimal calculus, speculated widely about infinite numbers and their use in mathematics. To Leibniz, both infinitesimals and infinite quantities were ideal entities, not of the same nature as appreciable quantities, but enjoying the same properties