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Derivative (set Theory)
In axiomatic set theory, a function ''f'' : Ord → Ord is called normal (or a normal function) if and only if it is continuous (with respect to the order topology) and strictly monotonically increasing. This is equivalent to the following two conditions: # For every limit ordinal ''γ'' (i.e. ''γ'' is neither zero nor a successor), it is the case that ''f''(''γ'') = sup . # For all ordinals ''α'' < ''β'', it is the case that ''f''(''α'') < ''f''(''β''). Examples A simple normal function is given by (see ). But is ''not'' normal because it is not continuous at any limit ordinal; that is, the inverse image of the one-point open set is the set , which is not open when ''λ'' is a limit ordinal. If ''β'' is a fixed ordinal, then the functions , (for ), and (for ) are all normal. More important exa ...[...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Axiomatic Set Theory
Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly concerned with those that are relevant to mathematics as a whole. The modern study of set theory was initiated by the German mathematicians Richard Dedekind and Georg Cantor in the 1870s. In particular, Georg Cantor is commonly considered the founder of set theory. The non-formalized systems investigated during this early stage go under the name of '' naive set theory''. After the discovery of paradoxes within naive set theory (such as Russell's paradox, Cantor's paradox and the Burali-Forti paradox) various axiomatic systems were proposed in the early twentieth century, of which Zermelo–Fraenkel set theory (with or without the axiom of choice) is still the best-known and most studied. Set theory is commonly employed as a foundational s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ordinal Number
In set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, th, etc.) aimed to extend enumeration to infinite sets. A finite set can be enumerated by successively labeling each element with the least natural number that has not been previously used. To extend this process to various infinite sets, ordinal numbers are defined more generally as linearly ordered labels that include the natural numbers and have the property that every set of ordinals has a least element (this is needed for giving a meaning to "the least unused element"). This more general definition allows us to define an ordinal number \omega that is greater than every natural number, along with ordinal numbers \omega + 1, \omega + 2, etc., which are even greater than \omega. A linear order such that every subset has a least element is called a well-order. The axiom of choice implies that every set can be well-ordered, and given two well-ordered sets, one is isomorph ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Continuous Function
In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in value, known as '' discontinuities''. More precisely, a function is continuous if arbitrarily small changes in its value can be assured by restricting to sufficiently small changes of its argument. A discontinuous function is a function that is . Up until the 19th century, mathematicians largely relied on intuitive notions of continuity, and considered only continuous functions. The epsilon–delta definition of a limit was introduced to formalize the definition of continuity. Continuity is one of the core concepts of calculus and mathematical analysis, where arguments and values of functions are real and complex numbers. The concept has been generalized to functions between metric spaces and between topological spaces. The latter are t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Order Topology
In mathematics, an order topology is a certain topology that can be defined on any totally ordered set. It is a natural generalization of the topology of the real numbers to arbitrary totally ordered sets. If ''X'' is a totally ordered set, the order topology on ''X'' is generated by the subbase of "open rays" :\ :\ for all ''a, b'' in ''X''. Provided ''X'' has at least two elements, this is equivalent to saying that the open intervals :(a,b) = \ together with the above rays form a base for the order topology. The open sets in ''X'' are the sets that are a union of (possibly infinitely many) such open intervals and rays. A topological space ''X'' is called orderable or linearly orderable if there exists a total order on its elements such that the order topology induced by that order and the given topology on ''X'' coincide. The order topology makes ''X'' into a completely normal Hausdorff space. The standard topologies on R, Q, Z, and N are the order topologies. In ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Monotonic Function
In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of order theory. In calculus and analysis In calculus, a function f defined on a subset of the real numbers with real values is called ''monotonic'' if and only if it is either entirely non-increasing, or entirely non-decreasing. That is, as per Fig. 1, a function that increases monotonically does not exclusively have to increase, it simply must not decrease. A function is called ''monotonically increasing'' (also ''increasing'' or ''non-decreasing'') if for all x and y such that x \leq y one has f\!\left(x\right) \leq f\!\left(y\right), so f preserves the order (see Figure 1). Likewise, a function is called ''monotonically decreasing'' (also ''decreasing'' or ''non-increasing'') if, whenever x \leq y, then f\!\left(x\right) \geq f\!\left(y\r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Limit Ordinal
In set theory, a limit ordinal is an ordinal number that is neither zero nor a successor ordinal. Alternatively, an ordinal λ is a limit ordinal if there is an ordinal less than λ, and whenever β is an ordinal less than λ, then there exists an ordinal γ such that β 0, are limits of limits, etc. Properties The classes of successor ordinals and limit ordinals (of various cofinalities) as well as zero exhaust the entire class of ordinals, so these cases are often used in proofs by transfinite induction or definitions by transfinite recursion. Limit ordinals represent a sort of "turning point" in such procedures, in which one must use limiting operations such as taking the union over all preceding ordinals. In principle, one could do anything at limit ordinals, but taking the union is continuous in the order topology and this is usually desirable. If we use the von Neumann cardinal assignment, every infinite cardinal number is also a limit ordinal (and this is a fitting o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Supremum
In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set P is a greatest element in P that is less than or equal to each element of S, if such an element exists. Consequently, the term ''greatest lower bound'' (abbreviated as ) is also commonly used. The supremum (abbreviated sup; plural suprema) of a subset S of a partially ordered set P is the least element in P that is greater than or equal to each element of S, if such an element exists. Consequently, the supremum is also referred to as the ''least upper bound'' (or ). The infimum is in a precise sense dual to the concept of a supremum. Infima and suprema of real numbers are common special cases that are important in analysis, and especially in Lebesgue integration. However, the general definitions remain valid in the more abstract setting of order theory where arbitrary partially ordered sets are considered. The concepts of infimum and supremum are close to minimum ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ordinal Arithmetic
In the mathematical field of set theory, ordinal arithmetic describes the three usual operations on ordinal numbers: addition, multiplication, and exponentiation. Each can be defined in essentially two different ways: either by constructing an explicit well-ordered set that represents the result of the operation or by using transfinite recursion. Cantor normal form provides a standardized way of writing ordinals. In addition to these usual ordinal operations, there are also the "natural" arithmetic of ordinals and the nimber operations. Addition The union of two disjoint well-ordered sets ''S'' and ''T'' can be well-ordered. The order-type of that union is the ordinal that results from adding the order-types of ''S'' and ''T''. If two well-ordered sets are not already disjoint, then they can be replaced by order-isomorphic disjoint sets, e.g. replace ''S'' by × ''S'' and ''T'' by × ''T''. This way, the well-ordered set ''S'' is written "to the left" of the well-ordere ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Aleph Number
In mathematics, particularly in set theory, the aleph numbers are a sequence of numbers used to represent the cardinality (or size) of infinite sets that can be well-ordered. They were introduced by the mathematician Georg Cantor and are named after the symbol he used to denote them, the Hebrew letter aleph (\,\aleph\,). The cardinality of the natural numbers is \,\aleph_0\, (read ''aleph-nought'' or ''aleph-zero''; the term ''aleph-null'' is also sometimes used), the next larger cardinality of a well-orderable set is aleph-one \,\aleph_1\;, then \,\aleph_2\, and so on. Continuing in this manner, it is possible to define a cardinal number \,\aleph_\alpha\, for every ordinal number \,\alpha\;, as described below. The concept and notation are due to Georg Cantor, who defined the notion of cardinality and realized that infinite sets can have different cardinalities. The aleph numbers differ from the infinity (\,\infty\,) commonly found in algebra and calculus, in that the alephs ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cardinal Number
In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality (size) of sets. The cardinality of a finite set is a natural number: the number of elements in the set. The '' transfinite'' cardinal numbers, often denoted using the Hebrew symbol \aleph ( aleph) followed by a subscript, describe the sizes of infinite sets. Cardinality is defined in terms of bijective functions. Two sets have the same cardinality if, and only if, there is a one-to-one correspondence (bijection) between the elements of the two sets. 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Beth Number
In mathematics, particularly in set theory, the beth numbers are a certain sequence of infinite cardinal numbers (also known as transfinite numbers), conventionally written \beth_0,\ \beth_1,\ \beth_2,\ \beth_3,\ \dots, where \beth is the second Hebrew letter ( beth). The beth numbers are related to the aleph numbers (\aleph_0,\ \aleph_1,\ \dots), but unless the generalized continuum hypothesis is true, there are numbers indexed by \aleph that are not indexed by \beth. Definition Beth numbers are defined by transfinite recursion: * \beth_0=\aleph_0, * \beth_=2^, * \beth_=\sup\, where \alpha is an ordinal and \lambda is a limit ordinal. The cardinal \beth_0=\aleph_0 is the cardinality of any countably infinite set such as the set \mathbb of natural numbers, so that \beth_0=, \mathbb, . Let \alpha be an ordinal, and A_\alpha be a set with cardinality \beth_\alpha=, A_\alpha, . Then, *\mathcal(A_\alpha) denotes the power set of A_\alpha (i.e., the set of all subsets of A_\alp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Successor Ordinal
In set theory, the successor of an ordinal number ''α'' is the smallest ordinal number greater than ''α''. An ordinal number that is a successor is called a successor ordinal. Properties Every ordinal other than 0 is either a successor ordinal or a limit ordinal.. In Von Neumann's model Using von Neumann's ordinal numbers (the standard model of the ordinals used in set theory), the successor ''S''(''α'') of an ordinal number ''α'' is given by the formula :S(\alpha) = \alpha \cup \. Since the ordering on the ordinal numbers is given by ''α'' < ''β'' if and only if ''α'' ∈ ''β'', it is immediate that there is no ordinal number between α and ''S''(''α''), and it is also clear that ''α'' < ''S''(''α''). Ordinal addition The successor operation can be used to define ordinal addition rigorously via |
