It can be proved that the cardinality of the real numbers is greater than that of the natural numbers just described. This can be visualized using Cantor's diagonal argument; classic questions of cardinality (for instance the continuum hypothesis) are concerned with discovering whether there is some cardinal between some pair of other infinite cardinals. In more recent times mathematicians have been describing the properties of larger and larger cardinals.
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Anonymous2013-08-31 8:07
The Indian mathematical text Surya Prajnapti (c. 3rd–4th century BCE) classifies all numbers into three sets: enumerable, innumerable, and infinite. Each of these was further subdivided into three orders:
The IEEE floating-point standard (IEEE 754) specifies the positive and negative infinity values. These are defined as the result of arithmetic overflow, division by zero, and other exceptional operations.
For example, properties of the natural and real numbers can be derived within set theory, as each number system can be identified with a set of equivalence classes under a suitable equivalence relation whose field is some infinite set.
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Anonymous2013-08-31 10:23
Although originally controversial, the axiom of choice is now used without reservation by most mathematicians, and it is included in Zermelo–Fraenkel set theory with the axiom of choice (ZFC), the standard form of axiomatic set theory.
The status of the axiom of choice varies between different varieties of constructive mathematics.
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Anonymous2013-08-31 12:18
Cantor's work initially polarized the mathematicians of his day. While Karl Weierstrass and Dedekind supported Cantor, Leopold Kronecker, now seen as a founder of mathematical constructivism, did not. Cantorian set theory eventually became widespread, due to the utility of Cantorian concepts, such as one-to-one correspondence among sets, his proof that there are more real numbers than integers, and the "infinity of infinities" ("Cantor's paradise") resulting from the power set operation. This utility of set theory led to the article "Mengenlehre" contributed in 1898 by Arthur Schoenflies to Klein's encyclopedia.
After having finished ML series right after reading through Muramasa, I decided to change the pace and started reading LOVELY x C^TION. The "datesim" part's implementation is underwhelming. Can't really say much about the story since I just started reading it a few hours ago, but I'm already in love with Amagase. Moe overload
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Anonymous2013-08-31 13:03
Multiplication is non-decreasing in both arguments: κ ≤ μ → (κ·ν ≤ μ·ν and ν·κ ≤ ν·μ).
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Anonymous2013-08-31 13:40
You know who wasn't stupid? The guy who invented the pet rock.
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Anonymous2013-08-31 13:48
Dedekind's approach was essentially to adopt the idea of one-to-one correspondence as a standard for comparing the size of sets, and to reject the view of Galileo (which derived from Euclid) that the whole cannot be the same size as the part. An infinite set can simply be defined as one having the same size as at least one of its proper parts; this notion of infinity is called Dedekind infinite. The diagram gives an example: viewing lines as infinite sets of points, the left half of the lower blue line can be mapped in a one-to-one manner (green correspondences) to the higher blue line, and, in turn, to the whole lower blue line (red correspondences); therefore the whole lower blue line and its left half have the same cardinality, i.e. "size".
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Anonymous2013-08-31 14:25
Is anyone willing to explain the story of Umineko as simply put as possible?
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Anonymous2013-08-31 14:34
Set theory begins with a fundamental binary relation between an object o and a set A. If o is a member (or element) of A, write o ∈ A. Since sets are objects, the membership relation can relate sets as well.
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Anonymous2013-08-31 15:10
I made a big mistake bringing Marisa with me. There aren't any partner spellcards on the floor (only in secret rooms, but I have found only one). I prepared her to do massive damage with spellcards, so she sucks at one-on-one battle.
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Anonymous2013-08-31 15:19
A cardinal invariant is a property of the real line measured by a cardinal number. For example, a well-studied invariant is the smallest cardinality of a collection of meagre sets of reals whose union is the entire real line.
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Anonymous2013-08-31 16:04
For any set A there is a function f such that for any non-empty subset B of A, f(B) lies in B.
Trichotomy: If two sets are given, then either they have the same cardinality, or one has a smaller cardinality than the other.
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Anonymous2013-08-31 17:34
There are models of Zermelo-Fraenkel set theory in which the axiom of choice is false. We will abbreviate "Zermelo-Fraenkel set theory plus the negation of the axiom of choice" by ZF¬C. For certain models of ZF¬C, it is possible to prove the negation of some standard facts. Note that any model of ZF¬C is also a model of ZF, so for each of the following statements, there exists a model of ZF in which that statement is true.
Assuming the axiom of choice and, given an infinite cardinal σ and a cardinal μ, there exists a cardinal κ such that μ + κ = σ if and only if μ ≤ σ. It will be unique (and equal to σ) if and only if μ < σ.
\sum_{i=0}^{\infty} \, f(i) = \infty means that the sum of the infinite series diverges in the specific sense that the partial sums grow without bound.
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Anonymous2013-08-31 21:36
Modern understanding of infinity began in 1867–71, with Cantor's work on number theory. An 1872 meeting between Cantor and Richard Dedekind influenced Cantor's thinking and culminated in Cantor's 1874 paper.
For example, the existence of sufficiently large cardinals implies that there is an inner model satisfying the axiom of determinacy (and thus not satisfying the axiom of choice).
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Anonymous2013-08-31 23:07
Given any set X of pairwise disjoint non-empty sets, there exists at least one set C that contains exactly one element in common with each of the sets in X.
Statements in this class include the statement that P = NP, the Riemann hypothesis, and many other unsolved mathematical problems. When one attempts to solve problems in this class, it makes no difference whether ZF or ZFC is employed if the only question is the existence of a proof. It is possible, however, that there is a shorter proof of a theorem from ZFC than from ZF.
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Anonymous2013-09-01 0:38
A uniform space is compact if and only if it is complete and totally bounded.
The expression hom(a, b) — alternatively expressed as homC(a, b), mor(a, b), or C(a, b) — denotes the hom-class of all morphisms from a to b.
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Anonymous2013-09-01 2:07
The study of morphisms and of the structures (called objects) over which they are defined, is central to category theory. Much of the terminology of morphisms, as well as the intuition underlying them, comes from concrete categories, where the objects are simply sets with some additional structure, and morphisms are structure-preserving functions. In category theory, morphisms are sometimes also called arrows.
Presheaves: If X is a topological space, then the open sets in X form a partially ordered set Open(X) under inclusion. Like every partially ordered set, Open(X) forms a small category by adding a single arrow U → V if and only if U \subseteq V. Contravariant functors on Open(X) are called presheaves on X. For instance, by assigning to every open set U the associative algebra of real-valued continuous functions on U, one obtains a presheaf of algebras on X.