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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 of having cardinality and for , in the sense that for any in , . A |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Transfinite Number
In mathematics, transfinite numbers are numbers that are "infinite" in the sense that they are larger than all finite numbers, yet not necessarily absolutely infinite. These include the transfinite cardinals, which are cardinal numbers used to quantify the size of infinite sets, and the transfinite ordinals, which are ordinal numbers used to provide an ordering of infinite sets. The term ''transfinite'' was coined by Georg Cantor in 1895, who wished to avoid some of the implications of the word ''infinite'' in connection with these objects, which were, nevertheless, not ''finite''. Few contemporary writers share these qualms; it is now accepted usage to refer to transfinite cardinals and ordinals as infinite numbers. Nevertheless, the term "transfinite" also remains in use. Definition Any finite natural number can be used in at least two ways: as an ordinal and as a cardinal. Cardinal numbers specify the size of sets (e.g., a bag of five marbles), whereas ordinal numbers specify t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Large Cardinal
In the mathematical field of set theory, a large cardinal property is a certain kind of property of transfinite cardinal numbers. Cardinals with such properties are, as the name suggests, generally very "large" (for example, bigger than the least α such that α=ωα). The proposition that such cardinals exist cannot be proved in the most common axiomatization of set theory, namely ZFC, and such propositions can be viewed as ways of measuring how "much", beyond ZFC, one needs to assume to be able to prove certain desired results. In other words, they can be seen, in Dana Scott's phrase, as quantifying the fact "that if you want more you have to assume more". There is a rough convention that results provable from ZFC alone may be stated without hypotheses, but that if the proof requires other assumptions (such as the existence of large cardinals), these should be stated. Whether this is simply a linguistic convention, or something more, is a controversial point among distinct philo ... [...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 then the following are equivalent for a cardinal \kappa: # \kappa is a regular cardinal. # If \kappa = \sum_ \lambda_i and \lambda_i < \kappa for all , then . # If , and if and for all , then . # The |
Uncountable Set
In mathematics, an uncountable set (or uncountably infinite set) is an infinite set that contains too many elements to be countable. The uncountability of a set is closely related to its cardinal number: a set is uncountable if its cardinal number is larger than that of the set of all natural numbers. Characterizations There are many equivalent characterizations of uncountability. A set ''X'' is uncountable if and only if any of the following conditions hold: * There is no injective function (hence no bijection) from ''X'' to the set of natural numbers. * ''X'' is nonempty and for every ω-sequence of elements of ''X'', there exists at least one element of X not included in it. That is, ''X'' is nonempty and there is no surjective function from the natural numbers to ''X''. * The cardinality of ''X'' is neither finite nor equal to \aleph_0 (aleph-null, the cardinality of the natural numbers). * The set ''X'' has cardinality strictly greater than \aleph_0. The first three ... [...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 for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Powerset
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 . The notation , meaning the set of all functions from S to a given set of two elements (e.g., ), is used because the powerset of can be identified with, equivalent to, or bijective to the set of all the functions from to the given two elements set. 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homogeneous (large Cardinal Property)
In set theory and in the context of a large cardinal property, a subset, ''S'', of ''D'' is homogeneous for a function ''f'' if ''f'' is constant in finite subsets of ''S''. More precisely, given a set ''D'', let \mathcal_(D) be the set of all finite subsets of ''D'' (see Powerset#Subsets of limited cardinality) and let f: \mathcal_(D) \to B be a function defined in this set. On these conditions, ''S'' is homogeneous for ''f'' if, for every natural number ''n'', ''f'' is constant in the set \mathcal_(S). That is, ''f'' is constant on the unordered ''n''-tuples of elements of ''S''. See also *Ramsey's theorem *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 establishes that ω enjoys a certain property that Ramsey cardinals generalize to the uncountable case. Let ... {{settheory-stub Large cardinals ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stationary Subset
In mathematics, specifically set theory and model theory, a stationary set is a set that is not too small in the sense that it intersects all club sets, and is analogous to a set of non-zero measure in measure theory. There are at least three closely related notions of stationary set, depending on whether one is looking at subsets of an ordinal, or subsets of something of given cardinality, or a powerset. Classical notion If \kappa is a cardinal of uncountable cofinality, S \subseteq \kappa, and S intersects every club set in \kappa, then S is called a stationary set.Jech (2003) p.91 If a set is not stationary, then it is called a thin set. This notion should not be confused with the notion of a thin set in number theory. If S is a stationary set and C is a club set, then their intersection S \cap C is also stationary. This is because if D is any club set, then C \cap D is a club set, thus (S \cap C) \cap D = S \cap (C \cap D) is non empty. Therefore, (S \cap C) must be station ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is called the ''length'' of the sequence. Unlike a set, the same elements can appear multiple times at different positions in a sequence, and unlike a set, the order does matter. Formally, a sequence can be defined as a function from natural numbers (the positions of elements in the sequence) to the elements at each position. The notion of a sequence can be generalized to an indexed family, defined as a function from an ''arbitrary'' index set. For example, (M, A, R, Y) is a sequence of letters with the letter 'M' first and 'Y' last. This sequence differs from (A, R, M, Y). Also, the sequence (1, 1, 2, 3, 5, 8), which contains the number 1 at two different positions, is a valid sequence. Sequences can be ''finite'', as in these examples, or ''infi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Weakly Compact Cardinal
In mathematics, a weakly compact cardinal is a certain kind of cardinal number introduced by ; weakly compact cardinals are large cardinals, meaning that their existence cannot be proven from the standard axioms of set theory. (Tarski originally called them "not strongly incompact" cardinals.) Formally, a cardinal κ is defined to be weakly compact if it is uncountable and for every function ''f'': º 2 → there is a set of cardinality κ that is homogeneous for ''f''. In this context, º 2 means the set of 2-element subsets of κ, and a subset ''S'' of κ is homogeneous for ''f'' if and only if either all of 'S''sup>2 maps to 0 or all of it maps to 1. The name "weakly compact" refers to the fact that if a cardinal is weakly compact then a certain related infinitary language satisfies a version of the compactness theorem; see below. Every weakly compact cardinal is a reflecting cardinal, and is also a limit of reflecting cardinals. This means also that weakly compact ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Consistency Strength
In mathematical logic, two theories are equiconsistent if the consistency of one theory implies the consistency of the other theory, and vice versa. In this case, they are, roughly speaking, "as consistent as each other". In general, it is not possible to prove the absolute consistency of a theory ''T''. Instead we usually take a theory ''S'', believed to be consistent, and try to prove the weaker statement that if ''S'' is consistent then ''T'' must also be consistent—if we can do this we say that ''T'' is ''consistent relative to S''. If ''S'' is also consistent relative to ''T'' then we say that ''S'' and ''T'' are equiconsistent. Consistency In mathematical logic, formal theories are studied as mathematical objects. Since some theories are powerful enough to model different mathematical objects, it is natural to wonder about their own consistency. Hilbert proposed a program at the beginning of the 20th century whose ultimate goal was to show, using mathematical methods ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
