PCF Theory

	PCF Theory PCF theory is the name of a mathematical theory, introduced by Saharon , that deals with the cofinality of the ultraproducts of ordered sets. It gives strong upper bounds on the cardinalities of power sets of singular cardinals, and has many more applications as well. The abbreviation "PCF" stands for "possible cofinalities". Main definitions If ''A'' is an infinite set of regular cardinals, ''D'' is an ultrafilter on ''A'', then we let \operatorname\left(\prod A/D\right) denote the cofinality of the ordered set of functions \prod A where the ordering is defined as follows: f if $\\in D$ . pcf(''A'') is the set of cofinalities that occur if we consider all ultrafilters on ''A'', that is, $\operatorname(A)=\left\.$ Main results Obviously, pcf(''A'') consists of regular cardinals. Considering ultrafilters concentrated on elements of ''A'', we get that $A\subseteq \operatorname(A)$ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...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]
	Ultraproduct The ultraproduct is a mathematical construction that appears mainly in abstract algebra and mathematical logic, in particular in model theory and set theory. An ultraproduct is a quotient of the direct product of a family of structures. All factors need to have the same signature. The ultrapower is the special case of this construction in which all factors are equal. For example, ultrapowers can be used to construct new fields from given ones. The hyperreal numbers, an ultrapower of the real numbers, are a special case of this. Some striking applications of ultraproducts include very elegant proofs of the compactness theorem and the completeness theorem, Keisler's ultrapower theorem, which gives an algebraic characterization of the semantic notion of elementary equivalence, and the Robinson–Zakon presentation of the use of superstructures and their monomorphisms to construct nonstandard models of analysis, leading to the growth of the area of nonstandard analysis, which was ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Ordered Set In mathematics, especially order theory, a partial order on a set is an arrangement such that, for certain pairs of elements, one precedes the other. The word ''partial'' is used to indicate that not every pair of elements needs to be comparable; that is, there may be pairs for which neither element precedes the other. Partial orders thus generalize total orders, in which every pair is comparable. Formally, a partial order is a homogeneous binary relation that is reflexive, antisymmetric, and transitive. A partially ordered set (poset for short) is an ordered pair P=(X,\leq) consisting of a set X (called the ''ground set'' of P) and a partial order \leq on X. When the meaning is clear from context and there is no ambiguity about the partial order, the set X itself is sometimes called a poset. Partial order relations The term ''partial order'' usually refers to the reflexive partial order relations, referred to in this article as ''non-strict'' partial orders. However some a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Power Set In mathematics, the power set (or powerset) of a set is the set of all subsets of , including the empty set and itself. In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is postulated by the axiom of power set. The powerset of is variously denoted as , , , \mathbb(S), or . Any subset of is called a ''family of sets'' over . Example If is the set , then all the subsets of are * (also denoted \varnothing or \empty, the empty set or the null set) * * * * * * * and hence the power set of is . Properties If is a finite set with the cardinality (i.e., the number of all elements in the set is ), then the number of all the subsets of is . This fact as well as the reason of the notation denoting the power set are demonstrated in the below. : An indicator function or a characteristic function of a subset of a set with the cardinality is a function from to the two-element set , denoted as , ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Regular Cardinal In set theory, a regular cardinal is a cardinal number that is equal to its own cofinality. More explicitly, this means that \kappa is a regular cardinal if and only if every unbounded subset C \subseteq \kappa has cardinality \kappa. Infinite well-ordered cardinals that are not regular are called singular cardinals. Finite cardinal numbers are typically not called regular or singular. In the presence of the axiom of choice, any cardinal number can be well-ordered, and so the following are equivalent: # \kappa is a regular cardinal. # If \kappa = \textstyle\sum_ \lambda_i and \lambda_i < \kappa for all $i$ , then $, I, \ge \kappa$ . # If $S = \textstyle\bigcup_ S_i$ , and if $, I, < \kappa$ and $, S_i, < \kappa$ for all $i$ , then $, S, < \kappa$ . That is, every union of fewer than $\kappa$ sets smaller than $\kappa$ is smaller than $\kappa$ . # The [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Cardinal Number In mathematics, a cardinal number, or cardinal for short, is what is commonly called the number of elements of a set. In the case of a finite set, its cardinal number, or cardinality is therefore a natural number. For dealing with the case of infinite sets, the infinite cardinal numbers have been introduced, which are often denoted with the Hebrew letter \aleph (aleph) marked with subscript indicating their rank among the infinite cardinals. 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 number of elements. In the case of infinite sets, the behavior is more complex. A fundamental theorem due to Georg Cantor shows that it is possible for two infinite sets to have different cardinalities, and in particular the cardinality of the set of real numbers is gre ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Ultrafilter In the Mathematics, mathematical field of order theory, an ultrafilter on a given partially ordered set (or "poset") P is a certain subset of P, namely a Maximal element, maximal Filter (mathematics), filter on P; that is, a proper filter on P that cannot be enlarged to a bigger proper filter on P. If X is an arbitrary set, its power set (X), ordered by set inclusion, is always a Boolean algebra (structure), Boolean algebra and hence a poset, and ultrafilters on (X) are usually called X.If X happens to be partially ordered, too, particular care is needed to understand from the context whether an (ultra)filter on (X) or an (ultra)filter just on X is meant; both kinds of (ultra)filters are quite different. Some authors use "(ultra)filter ''of'' a partial ordered set" vs. "''on'' an arbitrary set"; i.e. they write "(ultra)filter on X" to abbreviate "(ultra)filter of (X)". An ultrafilter on a set X may be considered as a finitely additive 0-1-valued measure (mathematics), measure on ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Limit Cardinal In mathematics, limit cardinals are certain cardinal numbers. A cardinal number ''λ'' is a weak limit cardinal if ''λ'' is neither a successor cardinal nor zero. This means that one cannot "reach" ''λ'' from another cardinal by repeated successor operations. These cardinals are sometimes called simply "limit cardinals" when the context is clear. A cardinal ''λ'' is a strong limit cardinal if ''λ'' cannot be reached by repeated powerset operations. This means that ''λ'' is nonzero and, for all ''κ'' < ''λ'', 2^''κ'' < ''λ''. Every strong limit cardinal is also a weak limit cardinal, because ''κ''⁺ ≤ 2^''κ'' for every cardinal ''κ'', where ''κ''⁺ denotes the successor cardinal of ''κ''. The first infinite cardinal, $\aleph_0$ ( aleph-naught ), is a strong limit cardinal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Jónsson Cardinal In set theory, a Jónsson cardinal (named after Bjarni Jónsson) is a certain kind of large cardinal number. An uncountable cardinal number κ is said to be ''Jónsson'' if for every function f : kappa \to \kappa there is a set H of order type \kappa such that for each n, f restricted to n-element subsets of H omits at least one value in \kappa. Every Rowbottom cardinal is Jónsson. By a theorem of Eugene M. Kleinberg, the theories ZFC + “there is a Rowbottom cardinal” and ZFC + “there is a Jónsson cardinal” are equiconsistent. William Mitchell proved, with the help of the Dodd-Jensen core model that the consistency of the existence of a Jónsson cardinal implies the consistency of the existence of a Ramsey cardinal, so that the existence of Jónsson cardinals and the existence of Ramsey cardinals are equiconsistent.Mitchell, William J.: "Jonsson Cardinals, Erdos Cardinals and the Core Model", Journal of Symbolic Logic 64(3):1065-1086, 1999. In general, Jónsson cardin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Jónsson Algebra Jónsson is a surname of Icelandic origin, meaning ''son of Jón''. In Icelandic names, the name is not strictly a surname, but a patronymic. The name refers to: * Arnar Jónsson (actor) (born 1943), Icelandic actor * Arnar Jónsson (basketball) (born 1983), Icelandic basketball player * Arngrímur Jónsson “The Learned” (1568–1648), Icelandic scholar Bjarni Jónsson (1920–2016), Icelandic mathematician and logician Bjarni Jónsson (artist) Bjarni Jónsson (September 15, 1934 – January 8, 2008) was an Icelandic painter. He was notable for his paintings of old Icelandic fishing boats, many of which have been added to the National Museum of Iceland, ''Þjóðminjasafn Íslands''. ... (1934–2008), Icelandic painter * Bjarni Jónsson (footballer) (born 1965), Icelandic international footballer * Björn Jónsson (1846–1912), Icelandic prime minister * Eggert Jónsson (born 1988), Icelandic footballer * Einar Jónsson (1874–1954), Icelandic sculptor * Emil Jónsson ( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Chang's Conjecture In model theory, a branch of mathematical logic, Chang's conjecture, attributed to Chen Chung Chang by , states that every model of type (ω2,ω1) for a countable language has an elementary submodel of type (ω1, ω). A model is of type (α,β) if it is of cardinality α and a unary relation is represented by a subset of cardinality β. The usual notation is (\omega_2,\omega_1)\twoheadrightarrow(\omega_1,\omega). The axiom of constructibility implies that Chang's conjecture fails. Silver proved the consistency of Chang's conjecture from the consistency of an ω1- Erdős cardinal. Hans-Dieter Donder showed a weak version of the reverse implication: if CC is not only consistent but actually holds, then ω2 is ω1-Erdős in K. More generally, Chang's conjecture for two pairs (α,β), (γ,δ) of cardinals is the claim that every model of type (α,β) for a countable language has an elementary submodel of type (γ,δ). The consistency of (\omega_3,\omega_2)\twoheadrightarrow(\omega_2 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]

Main results