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Scale (descriptive Set Theory)
In the mathematical discipline of descriptive set theory, a scale is a certain kind of object defined on a set of points in some Polish space (for example, a scale might be defined on a set of real numbers). Scales were originally isolated as a concept in the theory of uniformization, but have found wide applicability in descriptive set theory, with applications such as establishing bounds on the possible lengths of wellorderings of a given complexity, and showing (under certain assumptions) that there are largest countable sets of certain complexities. Formal definition Given a pointset ''A'' contained in some product space :A\subseteq X=X_0\times X_1\times\ldots X_ where each ''Xk'' is either the Baire space or a countably infinite discrete set, we say that a ''norm'' on ''A'' is a map from ''A'' into the ordinal numbers. Each norm has an associated prewellordering, where one element of ''A'' precedes another element if the norm of the first is less than the norm of the second. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Descriptive Set Theory
In mathematical logic, descriptive set theory (DST) is the study of certain classes of "well-behaved" 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 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 \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 of generality in considering Polish spaces of certain restricted for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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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 isomo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Uniformization (set Theory)
In set theory, a branch of mathematics, the axiom of uniformization is a weak form of the axiom of choice. It states that if R is a subset of X\times Y, where X and Y are Polish spaces, then there is a subset f of R that is a partial function from X to Y, and whose domain (the set of all x such that f(x) exists) equals : \\, Such a function is called a uniformizing function for R, or a uniformization of R. To see the relationship with the axiom of choice, observe that R can be thought of as associating, to each element of X, a subset of Y. A uniformization of R then picks exactly one element from each such subset, whenever the subset is non-empty. Thus, allowing arbitrary sets ''X'' and ''Y'' (rather than just Polish spaces) would make the axiom of uniformization equivalent to the axiom of choice. A pointclass \boldsymbol is said to have the uniformization property if every relation R in \boldsymbol can be uniformized by a partial function in \boldsymbol. The uniformizati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Relation (mathematics)
In mathematics, a relation on a Set (mathematics), set may, or may not, hold between two given set members. For example, ''"is less than"'' is a relation on the set of natural numbers; it holds e.g. between 1 and 3 (denoted as 1 is an asymmetric relation, but ≥ is not. Again, the previous 3 alternatives are far from being exhaustive; as an example over the natural numbers, the relation defined by is neither symmetric nor antisymmetric, let alone asymmetric. ; : for all , if and then . A transitive relation is irreflexive if and only if it is asymmetric. For example, "is ancestor of" is a transitive relation, while "is parent of" is not. ; : for all , if then or . This property is sometimes called "total", which is distinct from the definitions of "total" given in the section . ; : for all , or . This property is sometimes called "total", which is distinct from the definitions of "total" given in the section . ; : every nonempty subset of contains a Maximal and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Prewellordering Property
In set theory, a prewellordering on a set X is a preorder \leq on X (a transitive and strongly connected relation on X) that is wellfounded in the sense that the relation x \leq y \land y \nleq x is wellfounded. If \leq is a prewellordering on X, then the relation \sim defined by x \sim y \text x \leq y \land y \leq x is an equivalence relation on X, and \leq induces a wellordering on the quotient X / \sim. The order-type of this induced wellordering is an ordinal, referred to as the length of the prewellordering. A norm on a set X is a map from X into the ordinals. Every norm induces a prewellordering; if \phi : X \to Ord is a norm, the associated prewellordering is given by x \leq y \text \phi(x) \leq \phi(y) Conversely, every prewellordering is induced by a unique regular norm (a norm \phi : X \to Ord is regular if, for any x \in X and any \alpha < \phi(x), there is such that ). Prewellordering property ...
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Pointclass
In the mathematical field of descriptive set theory, a pointclass is a collection of sets of points, where a ''point'' is ordinarily understood to be an element of some 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 measurability (and indeed universal measurability), ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Axiom Of Choice
In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collection of sets, each containing at least one element, it is possible to construct a new set by arbitrarily choosing one element from each set, even if the collection is infinite. Formally, it states that for every indexed family (S_i)_ of nonempty sets, there exists an indexed set (x_i)_ such that x_i \in S_i for every i \in I. The axiom of choice was formulated in 1904 by Ernst Zermelo in order to formalize his proof of the well-ordering theorem. In many cases, a set arising from choosing elements arbitrarily can be made without invoking the axiom of choice; this is, in particular, the case if the number of sets from which to choose the elements is finite, or if a canonical rule on how to choose the elements is available – some distinguis ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Prewellordering
In set theory, a prewellordering on a set X is a preorder \leq on X (a transitive and strongly connected relation on X) that is wellfounded in the sense that the relation x \leq y \land y \nleq x is wellfounded. If \leq is a prewellordering on X, then the relation \sim defined by x \sim y \text x \leq y \land y \leq x is an equivalence relation on X, and \leq induces a wellordering on the quotient X / \sim. The order-type of this induced wellordering is an ordinal, referred to as the length of the prewellordering. A norm on a set X is a map from X into the ordinals. Every norm induces a prewellordering; if \phi : X \to Ord is a norm, the associated prewellordering is given by x \leq y \text \phi(x) \leq \phi(y) Conversely, every prewellordering is induced by a unique regular norm (a norm \phi : X \to Ord is regular if, for any x \in X and any \alpha < \phi(x), there is such that ). Prewellordering prope ...
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Baire Space (set Theory)
In set theory, the Baire space is the set of all infinite sequences of natural numbers with a certain topology. This space is commonly used in descriptive set theory, to the extent that its elements are often called "reals". It is denoted NN, ωω, by the symbol \mathcal or also ωω, not to be confused with the countable ordinal obtained by ordinal exponentiation. The Baire space is defined to be the Cartesian product of countably infinitely many copies of the set of natural numbers, and is given the product topology (where each copy of the set of natural numbers is given the discrete topology). The Baire space is often represented using the tree of finite sequences of natural numbers. The Baire space can be contrasted with Cantor space, the set of infinite sequences of binary digits. Topology and trees The product topology used to define the Baire space can be described more concretely in terms of trees. The basic open sets of the product topology are cylinder ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Set (mathematics)
A set is the mathematical model for a collection of different things; a set contains ''elements'' or ''members'', which can be mathematical objects of any kind: numbers, symbols, points in space, lines, other geometrical shapes, variables, or even other sets. The set with no element is the empty set; a set with a single element is a singleton. A set may have a finite number of elements or be an infinite set. Two sets are equal if they have precisely the same elements. Sets are ubiquitous in modern mathematics. Indeed, set theory, more specifically Zermelo–Fraenkel set theory, has been the standard way to provide rigorous foundations for all branches of mathematics since the first half of the 20th century. History The concept of a set emerged in mathematics at the end of the 19th century. The German word for set, ''Menge'', was coined by Bernard Bolzano in his work '' Paradoxes of the Infinite''. Georg Cantor, one of the founders of set theory, gave the following ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Countable Set
In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbers; this means that each element in the set may be associated to a unique natural number, or that the elements of the set can be counted one at a time, although the counting may never finish due to an infinite number of elements. In more technical terms, assuming the axiom of countable choice, a set is ''countable'' if its cardinality (its number of elements) is not greater than that of the natural numbers. A countable set that is not finite is said countably infinite. The concept is attributed to Georg Cantor, who proved the existence of uncountable sets, that is, sets that are not countable; for example the set of the real numbers. A note on terminology Although the terms "countable" and "countably infinite" as defined here are quit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Wellordering
In mathematics, a well-order (or well-ordering or well-order relation) on a set ''S'' is a total order on ''S'' with the property that every non-empty subset of ''S'' has a least element in this ordering. The set ''S'' together with the well-order relation is then called a well-ordered set. In some academic articles and textbooks these terms are instead written as wellorder, wellordered, and wellordering or well order, well ordered, and well ordering. Every non-empty well-ordered set has a least element. Every element ''s'' of a well-ordered set, except a possible greatest element, has a unique successor (next element), namely the least element of the subset of all elements greater than ''s''. There may be elements besides the least element which have no predecessor (see below for an example). A well-ordered set ''S'' contains for every subset ''T'' with an upper bound a least upper bound, namely the least element of the subset of all upper bounds of ''T'' in ''S''. If ≤ i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |