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Wadge Hierarchy
In descriptive set theory, within mathematics, Wadge degrees are levels of complexity for sets of reals. Sets are compared by continuous reductions. The Wadge hierarchy is the structure of Wadge degrees. These concepts are named after William W. Wadge. Wadge degrees Suppose A and B are subsets of Baire space ωω. Then A is Wadge reducible to B or A ≤W B if there is a continuous function f on ωω with A = f^ /math>. The Wadge order is the preorder or quasiorder on the subsets of Baire space. Equivalence classes of sets under this preorder are called Wadge degrees, the degree of a set A is denoted by A">math>Asub>W. The set of Wadge degrees ordered by the Wadge order is called the Wadge hierarchy. Properties of Wadge degrees include their consistency with measures of complexity stated in terms of definability. For example, if A ≤W B and B is a countable intersection of open sets, then so is A. The same works for all levels of the Borel hierarchy and the difference hier ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Descriptive Set Theory
In mathematical logic, descriptive set theory (DST) is the study of certain classes of "well-behaved" set (mathematics), subsets of the real line and other Polish spaces. As well as being one of the primary areas of research in set theory, it has applications to other areas of mathematics such as functional analysis, ergodic theory, the study of operator algebras and Group action (mathematics), group actions, and mathematical logic. Polish spaces Descriptive set theory begins with the study of Polish spaces and their Borel sets. A Polish space is a second-countable topological space that is metrizable with a complete metric. Heuristically, it is a complete separable metric space whose metric has been "forgotten". Examples include the real line \mathbb, the Baire space (set theory), Baire space \mathcal, the Cantor space \mathcal, and the Hilbert cube I^. Universality properties The class of Polish spaces has several universality properties, which show that there is no loss ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pointclass
In the mathematical field of descriptive set theory, a pointclass is a collection of Set (mathematics), sets of point (mathematics), points, where a ''point'' is ordinarily understood to be an element of some perfect set, perfect Polish space. In practice, a pointclass is usually characterized by some sort of ''definability property''; for example, the collection of all open sets in some fixed collection of Polish spaces is a pointclass. (An open set may be seen as in some sense definable because it cannot be a purely arbitrary collection of points; for any point in the set, all points sufficiently close to that point must also be in the set.) Pointclasses find application in formulating many important principles and theorems from set theory and real analysis. Strong set-theoretic principles may be stated in terms of the determinacy of various pointclasses, which in turn implies that sets in those pointclasses (or sometimes larger ones) have regularity properties such as Lebesgue m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cofinality
In mathematics, especially in order theory, the cofinality cf(''A'') of a partially ordered set ''A'' is the least of the cardinalities of the cofinal subsets of ''A''. Formally, :\operatorname(A) = \inf \ This definition of cofinality relies on the axiom of choice, as it uses the fact that every non-empty set of cardinal numbers has a least member. The cofinality of a partially ordered set ''A'' can alternatively be defined as the least ordinal ''x'' such that there is a function from ''x'' to ''A'' with cofinal image. This second definition makes sense without the axiom of choice. If the axiom of choice is assumed, as will be the case in the rest of this article, then the two definitions are equivalent. Cofinality can be similarly defined for a directed set and is used to generalize the notion of a subsequence in a net. Examples * The cofinality of a partially ordered set with greatest element is 1 as the set consisting only of the greatest element is cofinal (and must be ... [...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 In mathematics, a cardinal number, or ... [...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. The ordinals 1, 2, and 3 are the first three successor ordinals and the ordinals ω+1, ω+2 and ω+3 are the first three infinite successor ordinals. 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 defin ...[...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Even Ordinal
Even may refer to: General * Even (given name), a Norwegian male personal name * Even (surname), a Breton surname * Even (people), an ethnic group from Siberia and Russian Far East **Even language, a language spoken by the Evens * Odd and Even, a solitaire game which is played with two decks of playing cards *Evening, the period of a day that begins at the end of daylight and overlaps with the beginning of night Science and technology *In mathematics, the term ''even'' is used in several senses related to ''odd'': ** even and odd numbers, an integer is even if dividing by two yields an integer ** even and odd functions, a function is even if ''f''(−''x'') = ''f''(''x'') for all ''x'' ** even and odd permutations, a permutation of a finite set is even if it is composed of an even number of transpositions **Singly even number, an integer divisible by 2 but not divisible by 4 * Even code, if the Hamming weight of all of a binary code's codewords is even Entertainment *E ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Closure (mathematics)
In mathematics, a subset of a given set (mathematics), set is closed under an Operation (mathematics), operation on the larger set if performing that operation on members of the subset always produces a member of that subset. For example, the natural numbers are closed under addition, but not under subtraction: is not a natural number, although both 1 and 2 are. Similarly, a subset is said to be closed under a ''collection'' of operations if it is closed under each of the operations individually. The closure of a subset is the result of a closure operator applied to the subset. The ''closure'' of a subset under some operations is the smallest superset that is closed under these operations. It is often called the ''span'' (for example linear span) or the ''generated set''. Definitions Let be a set (mathematics), set equipped with one or several methods for producing elements of from other elements of .Operation (mathematics), Operations and (partial function, partial) multivar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Veblen Function
In mathematics, the Veblen functions are a hierarchy of normal functions ( continuous strictly increasing functions from ordinals to ordinals), introduced by Oswald Veblen in . If ''φ''0 is any normal function, then for any non-zero ordinal ''α'', ''φ''''α'' is the function enumerating the common fixed points of ''φ''''β'' for ''β''<''α''. These functions are all normal. Veblen hierarchy In the special case when ''φ''0(''α'')=ω''α'' this family of functions is known as the Veblen hierarchy. The function ''φ''1 is the same as the ε function: ''φ''1(''α'')= ε''α''. If then .M. Rathjen[...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Θ (set Theory)
In set theory, \varTheta (pronounced like the letter theta) is the least nonzero ordinal \alpha such that there is no surjection from the reals onto \alpha. \varTheta has been studied in connection with strong partition cardinals and the axiom of determinacy. The axiom of determinacy is equivalent to the existence of unboundedly many strong partition cardinals below \varTheta, in the sense that every cardinal below \varTheta has a strong partition cardinal above it. This does not preclude the possibility that a single strong partition cardinal, above \varTheta, suffices for all cardinals below \varTheta, but the existence of such a cardinal would have additional consequences. If the reals can be well-ordered, then \varTheta is simply (2^)^+, the cardinal successor of the cardinality of the continuum. Any set may be well-ordered assuming the axiom of choice (AC). However, Θ is often studied in contexts where the axiom of choice fails, such as models of the axiom of determin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Well Founded
In mathematics, a binary relation is called well-founded (or wellfounded or foundational) on a set or, more generally, a class if every non-empty subset has a minimal element with respect to ; that is, there exists an such that, for every , one does not have . In other words, a relation is well-founded if: (\forall S \subseteq X)\; \neq \varnothing \implies (\exists m \in S) (\forall s \in S) \lnot(s \mathrel m) Some authors include an extra condition that is set-like, i.e., that the elements less than any given element form a set. Equivalently, assuming the axiom of dependent choice, a relation is well-founded when it contains no infinite descending chains, meaning there is no infinite sequence of elements of such that for every natural number . In order theory, a partial order is called well-founded if the corresponding strict order is a well-founded relation. If the order is a total order then it is called a well-order. In set theory, a set is called a well-fou ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Donald A
Donald is a Scottish masculine given name. It is derived from the Goidelic languages, Gaelic name ''Dòmhnall''.. This comes from the Proto-Celtic language, Proto-Celtic *''Dumno-ualos'' ("world-ruler" or "world-wielder"). The final -''d'' in ''Donald'' is partly derived from a misinterpretation of the Gaelic pronunciation by English speakers. A short form of Donald is Don (given name), Don, and pet forms of Donald include Donnie and Donny. The feminine given name Donella (other) , Donella is derived from Donald. ''Donald'' has cognates in other Celtic languages: Irish language, Modern Irish ''Dónal'' (anglicised as ''Donal'' and ''Donall'');. Scottish Gaelic ''Dòmhnall'', ''Domhnull'' and ''Dòmhnull''; Welsh language, Welsh ''Dyfnwal (other), Dyfnwal'' and Cumbric ''Dumnagual''. Although the feminine given name ''Donna (given name), Donna'' is sometimes used as a feminine form of ''Donald'', the names are not etymologically related. Variations King ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lipschitz Continuous
In mathematical analysis, Lipschitz continuity, named after Germany, German mathematician Rudolf Lipschitz, is a strong form of uniform continuity for function (mathematics), functions. Intuitively, a Lipschitz continuous function is limited in how fast it can change: there exists a real number such that, for every pair of points on the graph of this function, the absolute value of the slope of the line connecting them is not greater than this real number; the smallest such bound is called the ''Lipschitz constant'' of the function (and is related to the ''modulus of continuity, modulus of uniform continuity''). For instance, every function that is defined on an interval and has a bounded first derivative is Lipschitz continuous. In the theory of differential equations, Lipschitz continuity is the central condition of the Picard–Lindelöf theorem which guarantees the existence and uniqueness of the solution to an initial value problem. A special type of Lipschitz continuity, cal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
