One of Cantor's most important results was that the cardinality of the continuum \mathbf c is greater than that of the natural numbers {leph_0}; that is, there are more real numbers R than natural numbers N. Namely, Cantor showed that \mathbf{c} = 2^{leph_0} > {leph_0} (see Cantor's diagonal argument or Cantor's first uncountability proof).
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Anonymous2013-08-31 9:13
Union of the sets A and B, denoted A ∪ B, is the set of all objects that are a member of A, or B, or both. The union of {1, 2, 3} and {2, 3, 4} is the set {1, 2, 3, 4} .
From set theory's inception, some mathematicians have objected to it as a foundation for mathematics. The most common objection to set theory, one Kronecker voiced in set theory's earliest years, starts from the constructivist view that mathematics is loosely related to computation. If this view is granted, then the treatment of infinite sets, both in naive and in axiomatic set theory, introduces into mathematics methods and objects that are not computable even in principle.
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Anonymous2013-08-31 10:43
In the even simpler case of a collection of one set, a choice function just corresponds to an element, so this instance of the axiom of choice says that every nonempty set has an element; this holds trivially. The axiom of choice can be seen as asserting the generalization of this property, already evident for finite collections, to arbitrary collections
Zorn's lemma: Every non-empty partially ordered set in which every chain (i.e. totally ordered subset) has an upper bound contains at least one maximal element.
Since cardinality is such a common concept in mathematics, a variety of names are in use. Sameness of cardinality is sometimes referred to as equipotence, equipollence, or equinumerosity. It is thus said that two sets with the same cardinality are, respectively, equipotent, equipollent, or equinumerous.
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Anonymous2013-08-31 13:07
Would you guys recommend the old Kamen Rider shows (i.e., unhandsome Riders, I suppose), or old Super Sentai series? I thought about just watching some of the newer ones, but then I figured I might as well also watch something a little older.
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Anonymous2013-08-31 13:32
* Enumerable: lowest, intermediate, and highest
* Innumerable: nearly innumerable, truly innumerable, and innumerably innumerable
That's right; I'm BREAKING the NYPA trend and i'll be your PA! Of course, i'd appreciate others to participate in the thread as well.
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Anonymous2013-08-31 14:17
Some programming languages, such as Java[13] and J,[14] allow the programmer an explicit access to the positive and negative infinity values as language constants. These can be used as greatest and least elements, as they compare (respectively) greater than or less than all other values. They are useful as sentinel values in algorithms involving sorting, searching, or windowing.
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Anonymous2013-08-31 14:38
You can refactor in Haskell a lot. The types ensure your large scale changes will be safe, if you're using types wisely. This will help your codebase scale. Make sure that your refactorings will cause type errors until complete.
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Anonymous2013-08-31 15:03
Set theory as a foundation for mathematical analysis, topology, abstract algebra, and discrete mathematics is likewise uncontroversial; mathematicians accept that (in principle) theorems in these areas can be derived from the relevant definitions and the axioms of set theory.
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Anonymous2013-08-31 15:23
ちなみに
誤：海鮮問屋
正：廻船問屋
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Anonymous2013-08-31 15:48
One motivation for this use is that a number of generally accepted mathematical results, such as Tychonoff's theorem, require the axiom of choice for their proofs. Contemporary set theorists also study axioms that are not compatible with the axiom of choice, such as the axiom of determinacy.
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Anonymous2013-08-31 16:33
In Martin-Löf type theory and higher-order Heyting arithmetic, the appropriate statement of the axiom of choice is (depending on approach) included as an axiom or provable as a theorem.
The Lebesgue measure of a countable disjoint union of measurable sets is equal to the sum of the measures of the individual sets.
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Anonymous2013-08-31 18:03
Diagram chasing is a visual method of arguing with abstract "arrows" joined in diagrams. Functors are represented by arrows between categories, subject to specific defining commutativity conditions. Functors can define (construct) categorical diagrams and sequences (viz. Mitchell, 1965). A functor associates to every object of one category an object of another category, and to every morphism in the first category a morphism in the second.
Higher-dimensional categories are part of the broader mathematical field of higher-dimensional algebra, a concept introduced by Ronald Brown. For a conversational introduction to these ideas, see John Baez, 'A Tale of n-categories' (1996).
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Anonymous2013-08-31 19:39
Cardinality is defined in terms of bijective functions. Two sets have the same cardinality if and only if there is a bijection between them. In the case of finite sets, this agrees with the intuitive notion of size. In the case of infinite sets, the behavior is more complex. A fundamental theorem due to Georg Cantor shows that it is possible for infinite sets to have different cardinalities, and in particular the cardinality of the set of real numbers is greater than the cardinality of the set of natural numbers. It is also possible for a proper subset of an infinite set to have the same cardinality as the original set, something that cannot happen with proper subsets of finite sets.
In physics, approximations of real numbers are used for continuous measurements and natural numbers are used for discrete measurements (i.e. counting).
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Anonymous2013-08-31 21:54
There are many technical advantages to this restriction, and little generality is lost, because essentially all mathematical concepts can be modeled by pure sets. Sets in the von Neumann universe are organized into a cumulative hierarchy, based on how deeply their members, members of members, etc. are nested.
Outside set theory, the word "class" is sometimes used synonymously with "set". This usage dates from a historical period where classes and sets were not distinguished as they are in modern set-theoretic terminology.
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Anonymous2013-08-31 23:25
First we might try to proceed as if X were finite. If we try to choose an element from each set, then, because X is infinite, our choice procedure will never come to an end, and consequently, we will never be able to produce a choice function for all of X. Next we might try specifying the least element from each set.
In the product topology, the closure of a product of subsets is equal to the product of the closures.
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Anonymous2013-09-01 0:55
The observation here is that one can define a function to select from an infinite number of pairs of shoes by stating for example, to choose the left shoe. Without the axiom of choice, one cannot assert that such a function exists for pairs of socks, because left and right socks are (presumably) indistinguishable from each other.
A (covariant) functor F from a category C to a category D, written F : C → D, consists of:
for each object x in C, an object F(x) in D; and
for each morphism f : x → y in C, a morphism F(f) : F(x) → F(y),
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Anonymous2013-09-01 2:26
Isomorphism: f : X → Y is called an isomorphism if there exists a morphism g : Y → X such that f ∘ g = idY and g ∘ f = idX. If a morphism has both left-inverse and right-inverse, then the two inverses are equal, so f is an isomorphism, and g is called simply the inverse of f. Inverse morphisms, if they exist, are unique. The inverse g is also an isomorphism with inverse f. Two objects with an isomorphism between them are said to be isomorphic or equivalent.
Representable functors: We can generalize the previous example to any category C. To every pair X, Y of objects in C one can assign the set Hom(X,Y) of morphisms from X to Y. This defines a functor to Set which is contravariant in the first argument and covariant in the second, i.e. it is a functor Cop × C → Set. If f : X1 → X2 and g : Y1 → Y2 are morphisms in C, then the group homomorphism Hom(f,g) : Hom(X2,Y1) → Hom(X1,Y2) is given by φ ↦ g o φ o f.