The problem was when they left out large chunks of the story and the tiny things that made it enjoyable, the people who watched it without reading the VN were probably confused as fuck since they had to fit everything into 26 episodes.
They even left out a lot of red text. Also they barely even used the sound track, except for suspicion
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Anonymous2013-08-31 14:36
From this definition, it is clear that a set is a subset of itself; for cases where one wishes to rule out this, the term proper subset is defined. A is called a proper subset of B if and only if A is a subset of B, but B is not a subset of A.
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Anonymous2013-08-31 15:12
So what are the chances that these two are the same person, or just have a funny hat thing going on?
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Anonymous2013-08-31 15:21
Set-theoretic topology studies questions of general topology that are set-theoretic in nature or that require advanced methods of set theory for their solution. Many of these theorems are independent of ZFC, requiring stronger axioms for their proof.
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Anonymous2013-08-31 16:06
There is a set A such that for all functions f (on the set of non-empty subsets of A), there is a B such that f(B) does not lie in B.
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Anonymous2013-08-31 16:51
König's theorem: Colloquially, the sum of a sequence of cardinals is strictly less than the product of a sequence of larger cardinals. (The reason for the term "colloquially", is that the sum or product of a "sequence" of cardinals cannot be defined without some aspect of the axiom of choice.)
Note that the domain and codomain are in fact part of the information determining a morphism. For example, in the category of sets, where morphisms are functions, two functions may be identical as sets of ordered pairs (may have the same range), while having different codomains. The two functions are distinct from the viewpoint of category theory. Thus many authors require that the hom-classes hom(X, Y) be disjoint.
A set X is Dedekind-infinite if there exists a proper subset Y of X with |X| = |Y|, and Dedekind-finite if such a subset doesn't exist. The finite cardinals are just the natural numbers, i.e., a set X is finite if and only if |X| = |n| = n for some natural number n. Any other set is infinite. Assuming the axiom of choice, it can be proved that the Dedekind notions correspond to the standard ones. It can also be proved that the cardinal ℵ0 (aleph null or aleph-0, where aleph is the first letter in the Hebrew alphabet, represented ℵ) of the set of natural numbers is the smallest infinite cardinal, i.e. that any infinite set has a subset of cardinality ℵ0. The next larger cardinal is denoted by ℵ1 and so on. For every ordinal α there is a cardinal number ℵα, and this list exhausts all infinite cardinal numbers.
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Anonymous2013-08-31 20:44
It was introduced in 1655 by John Wallis, and, since its introduction, has also been used outside mathematics in modern mysticism and literary symbology.
Perspective artwork utilizes the concept of imaginary vanishing points, or points at infinity, located at an infinite distance from the observer. This allows artists to create paintings that realistically render space, distances, and forms
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Anonymous2013-08-31 22:14
Descriptive set theory is the study of subsets of the real line and, more generally, subsets of Polish spaces. It begins with the study of pointclasses in the Borel hierarchy and extends to the study of more complex hierarchies such as the projective hierarchy and the Wadge hierarchy.
For any set X of nonempty sets, there exists a choice function f defined on X.
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Anonymous2013-08-31 23:45
A cause for this difference is that the axiom of choice in type theory does not have the extensionality properties that the axiom of choice in constructive set theory does.
Stone's representation theorem for Boolean algebras needs the Boolean prime ideal theorem.
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Anonymous2013-09-01 1:15
Difficult topological questions can be translated into algebraic questions which are often easier to solve. Basic constructions, such as the fundamental group or fundamental groupoid of a topological space, can be expressed as fundamental functors to the category of groupoids in this way, and the concept is pervasive in algebra and its applications.
The subsequent development of category theory was powered first by the computational needs of homological algebra, and later by the axiomatic needs of algebraic geometry, the field most resistant to being grounded in either axiomatic set theory or the Russell-Whitehead view of united foundations. General category theory, an extension of universal algebra having many new features allowing for semantic flexibility and higher-order logic, came later; it is now applied throughout mathematics.