Measurable Cardinal

	Measurable Cardinal In mathematics, a measurable cardinal is a certain kind of large cardinal number. In order to define the concept, one introduces a two-valued measure (mathematics), measure on a cardinal ''κ'', or more generally on any set. For a cardinal ''κ'', it can be described as a subdivision of all of its subsets into large and small sets such that ''κ'' itself is large, ∅ and all singleton (mathematics), singletons (with ''α'' ∈ ''κ'') are small, set complement, complements of small sets are large and vice versa. The intersection of fewer than ''κ'' large sets is again large. It turns out that uncountable cardinals endowed with a two-valued measure are large cardinals whose existence cannot be proved from ZFC. The concept of a measurable cardinal was introduced by Stanisław Ulam in 1930. Definition Formally, a measurable cardinal is an uncountable cardinal number ''κ'' such that there exists a ''κ''-additive, non-trivial, 0-1-valued measure (mathematics), measure ... [...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]
	Dana Scott Dana Stewart Scott (born October 11, 1932) is an American logician who is the emeritus Hillman University Professor of Computer Science, Philosophy, and Mathematical Logic at Carnegie Mellon University; he is now retired and lives in Berkeley, California. His work on automata theory earned him the Turing Award in 1976, while his collaborative work with Christopher Strachey in the 1970s laid the foundations of modern approaches to the semantics of programming languages. He has also worked on modal logic, topology, and category theory. Early career He received his B.A. in Mathematics from the University of California, Berkeley, in 1954. He wrote his Ph.D. thesis on ''Convergent Sequences of Complete Theories'' under the supervision of Alonzo Church while at Princeton, and defended his thesis in 1958. Solomon Feferman (2005) writes of this period: After completing his Ph.D. studies, he moved to the University of Chicago, working as an instructor there until 1960. In 1959, h ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Club Set In mathematics, particularly in mathematical logic and set theory, a club set is a subset of a limit ordinal that is closed under the order topology, and is unbounded (see below) relative to the limit ordinal. The name ''club'' is a contraction of "closed and unbounded". Formal definition Formally, if \kappa is a limit ordinal, then a set C\subseteq\kappa is ''closed'' in \kappa if and only if for every \alpha < \kappa, if $\sup(C \cap \alpha) = \alpha \neq 0,$ then $\alpha \in C.$ Thus, if the limit of some sequence from $C$ is less than $\kappa,$ then the limit is also in $C.$ If $\kappa$ is a limit ordinal and $C \subseteq \kappa$ then $C$ is unbounded in $\kappa$ if for any $\alpha < \kappa,$ there is some $\beta \in C$ such that $\alpha < \ ... [...More Info...] [...Related Items...]$ OR: [Wikipedia] [Google] [Baidu]
	Axiom Of Determinacy In mathematics, the axiom of determinacy (abbreviated as AD) is a possible axiom for set theory introduced by Jan Mycielski and Hugo Steinhaus in 1962. It refers to certain two-person topological games of length ω. AD states that every game of a certain type is determined; that is, one of the two players has a winning strategy. Steinhaus and Mycielski's motivation for AD was its interesting consequences, and suggested that AD could be true in the smallest natural model L(R) of a set theory, which accepts only a weak form of the axiom of choice (AC) but contains all real and all ordinal numbers. Some consequences of AD followed from theorems proved earlier by Stefan Banach and Stanisław Mazur, and Morton Davis. Mycielski and Stanisław Świerczkowski contributed another one: AD implies that all sets of real numbers are Lebesgue measurable. Later Donald A. Martin and others proved more important consequences, especially in descriptive set theory. In 1988, John R. S ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Successor Cardinal In set theory, one can define a successor operation on cardinal numbers in a similar way to the successor operation on the ordinal numbers. The cardinal successor coincides with the ordinal successor for finite cardinals, but in the infinite case they diverge because every infinite ordinal and its successor have the same cardinality (a bijection can be set up between the two by simply sending the last element of the successor to 0, 0 to 1, etc., and fixing ω and all the elements above; in the style of Hilbert's Hotel Infinity). Using the von Neumann cardinal assignment and the axiom of choice (AC), this successor operation is easy to define: for a cardinal number ''κ'' we have :\kappa^+ = \left, \inf \\ , where ON is the class of ordinals. That is, the successor cardinal is the cardinality of the least ordinal into which a set of the given cardinality can be mapped one-to-one, but which cannot be mapped one-to-one back into that set. That the set above is nonempty follows from ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Zermelo–Fraenkel Set Theory In set theory, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in the early twentieth century in order to formulate a theory of sets free of paradoxes such as Russell's paradox. Today, Zermelo–Fraenkel set theory, with the historically controversial axiom of choice (AC) included, is the standard form of axiomatic set theory and as such is the most common foundation of mathematics. Zermelo–Fraenkel set theory with the axiom of choice included is abbreviated ZFC, where C stands for "choice", and ZF refers to the axioms of Zermelo–Fraenkel set theory with the axiom of choice excluded. Informally, Zermelo–Fraenkel set theory is intended to formalize a single primitive notion, that of a hereditary well-founded set, so that all entities in the universe of discourse are such sets. Thus the axioms of Zermelo–Fraenkel set theory refer only to pure sets and prevent its models fr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Ramsey Cardinal In mathematics, a Ramsey cardinal is a certain kind of large cardinal number introduced by and named after Frank P. Ramsey, whose theorem, called Ramsey's theorem establishes that ω enjoys a certain property that Ramsey cardinals generalize to the uncountable case. Let 'κ''sup><ω denote the set of all finite subsets of ''κ''. A cardinal number ''κ'' is called Ramsey if, for every function :''f'': 'κ''sup><ω → there is a set ''A'' of cardinality ''κ'' that is homogeneous for ''f''. That is, for every ''n'', the function ''f'' is constant on the subsets of cardinality ''n'' from ''A''. A cardinal ''κ'' is called ineffably Ramsey if ''A'' can be chosen to be a stationary subset of ''κ''. A cardinal ''κ'' is called virtually Ramsey if for every function :''f'': 'κ''sup><ω → there is ''C'', a closed and unbounded subset of ''κ'', so that for every ''λ'' in ''C'' of uncountable [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Ineffable Cardinal In the mathematics of transfinite numbers, an ineffable cardinal is a certain kind of large cardinal number, introduced by . In the following definitions, \kappa will always be a regular uncountable cardinal number. A cardinal number \kappa is called almost ineffable if for every f: \kappa \to \mathcal(\kappa) (where \mathcal(\kappa) is the powerset of \kappa) with the property that f(\delta) is a subset of \delta for all ordinals \delta < \kappa, there is a subset $S$ of $\kappa$ having cardinality $\kappa$ and homogeneous for $f$ , in the sense that for any $\delta_1 < \delta_2$ in $S$ , $f(\delta_1) = f(\delta_2) \cap \delta_1$ . A [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Inaccessible Cardinal In set theory, a cardinal number is a strongly inaccessible cardinal if it is uncountable, regular, and a strong limit cardinal. A cardinal is a weakly inaccessible cardinal if it is uncountable, regular, and a weak limit cardinal. Since about 1950, "inaccessible cardinal" has typically meant "strongly inaccessible cardinal" whereas before it has meant "weakly inaccessible cardinal". Weakly inaccessible cardinals were introduced by . Strongly inaccessible cardinals were introduced by and ; in the latter they were referred to along with \aleph_0 as ''Grenzzahlen'' ( English "limit numbers"). Every strongly inaccessible cardinal is a weakly inaccessible cardinal. The generalized continuum hypothesis implies that all weakly inaccessible cardinals are strongly inaccessible as well. The two notions of an inaccessible cardinal \kappa describe a cardinality \kappa which can not be obtained as the cardinality of a result of typical set-theoretic operations involving only sets of c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Strong 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]
	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]
	Scott's Trick In set theory, Scott's trick is a method for giving a definition of equivalence classes for equivalence relations on a proper class (Jech 2003:65) by referring to levels of the cumulative hierarchy. The method relies on the axiom of regularity but not on the axiom of choice. It can be used to define representatives for ordinal numbers in ZF, Zermelo–Fraenkel set theory without the axiom of choice (Forster 2003:182). The method was introduced by . Beyond the problem of defining set representatives for ordinal numbers, Scott's trick can be used to obtain representatives for cardinal numbers and more generally for isomorphism types, for example, order types of linearly ordered sets (Jech 2003:65). It is credited to be indispensable (even in the presence of the axiom of choice) when taking ultrapowers of proper classes in model theory. (Kanamori 1994:47) Application to cardinalities The use of Scott's trick for cardinal numbers shows how the method is typically employed. Th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]