|
Huge Cardinal
In mathematics, a cardinal number \kappa is called huge if there exists an elementary embedding j : V \to M from V into a transitive inner model M with critical point \kappa and :^M \subset M. Here, ^\alpha M is the class of all sequences of length \alpha whose elements are in M. Huge cardinals were introduced by . Variants In what follows, j^n refers to the n-th iterate of the elementary embedding j, that is, j composed with itself n times, for a finite ordinal n. Also, ^M is the class of all sequences of length less than \alpha whose elements are in M. Notice that for the "super" versions, \gamma should be less than j(\kappa), not . κ is almost n-huge if and only if there is j : V \to M with critical point \kappa and :^M \subset M. κ is super almost n-huge if and only if for every ordinal γ there is j : V \to M with critical point \kappa, \gamma< j(\kappa), and : κ is n-huge if and only if there is with cri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Rank Into Rank
A rank is a position in a hierarchy. It can be formally recognized—for example, cardinal, chief executive officer, general, professor—or unofficial. People Formal ranks * Academic rank * Corporate title * Diplomatic rank * Hierarchy of the Catholic Church * Imperial, royal and noble ranks * Military rank * Police rank Unofficial ranks * Social class * Social position * Social status Either * Seniority Mathematics * Rank (differential topology) * Rank (graph theory) * Rank (linear algebra), the dimension of the vector space generated (or spanned) by a matrix's columns * Rank (set theory) * Rank (type theory) * Rank of an abelian group, the cardinality of a maximal linearly independent subset * Rank of a free module * Rank of a greedoid, the maximal size of a feasible set * Rank of a group, the smallest cardinality of a generating set for the group * Rank of a Lie group – see Cartan subgroup * Rank of a matroid, the maximal size of an independent set * Rank of a pa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
List Of Large Cardinal Properties
This page includes a list of large cardinal properties in the mathematical field of set theory. It is arranged roughly in order of the consistency strength of the axiom asserting the existence of cardinals with the given property. Existence of a cardinal number κ of a given type implies the existence of cardinals of most of the types listed above that type, and for most listed cardinal descriptions φ of lesser consistency strength, ''V''κ satisfies "there is an unbounded class of cardinals satisfying φ". The following table usually arranges cardinals in order of consistency strength, with size of the cardinal used as a tiebreaker. In a few cases (such as strongly compact cardinals) the exact consistency strength is not known and the table uses the current best guess. * "Small" cardinals: 0, 1, 2, ..., \aleph_0, \aleph_1,..., \kappa = \aleph_, ... (see Aleph number) * worldly cardinals * weakly and strongly inaccessible, α-inaccessible, and hyper inaccessible cardinals ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rank-into-rank
In set theory, a branch of mathematics, a rank-into-rank embedding is a large cardinal property defined by one of the following four axioms given in order of increasing consistency strength. (A set of rank < \lambda is one of the elements of the set of the von Neumann hierarchy.) *Axiom I3: There is a nontrivial elementary embedding of into itself. *Axiom I2: There is a nontrivial elementary embedding of into a transitive class that includes where is the first fixed point above the critical point (set theory), critical point. *Axiom I1: There is a nontrivial elementary embedding of into itself. *Axiom I0: There is a nontrivial elementary embedding of into itself with critical point below . These are essentially the strongest known large cardinal axioms not known to be inconsistent in ZFC; the axiom ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kunen's Inconsistency Theorem
In set theory, a branch of mathematics, Kunen's inconsistency theorem, proved by , shows that several plausible large cardinal axioms are inconsistent with the axiom of choice. Some consequences of Kunen's theorem (or its proof) are: *There is no non-trivial elementary embedding of the universe ''V'' into itself. In other words, there is no Reinhardt cardinal. *If ''j'' is an elementary embedding of the universe ''V'' into an inner model ''M'', and λ is the smallest fixed point of ''j'' above the critical point κ of ''j'', then ''M'' does not contain the set ''j'' "λ (the image of ''j'' restricted to λ). *There is no ω-huge cardinal. *There is no non-trivial elementary embedding of ''V''λ+2 into itself. It is not known if Kunen's theorem still holds in ZF (ZFC without the axiom of choice), though showed that there is no definable elementary embedding from ''V'' into ''V''. That is there is no formula ''J'' in the language of set theory s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Supercompact Cardinal
In set theory, a supercompact cardinal is a type of large cardinal independently introduced by Solovay and Reinhardt. They display a variety of reflection properties. Formal definition If \lambda is any ordinal, \kappa is \lambda-supercompact means that there exists an elementary embedding j from the universe V into a transitive inner model M with critical point \kappa, j(\kappa)>\lambda and :^\lambda M\subseteq M \,. That is, M contains all of its \lambda-sequences. Then \kappa is supercompact means that it is \lambda-supercompact for all ordinals \lambda. Alternatively, an uncountable cardinal \kappa is supercompact if for every A such that \vert A\vert\geq\kappa there exists a normal measure over , in the following sense. is defined as follows: : := \. An ultrafilter U over is ''fine'' if it is \kappa-complete and \ \in U, for every a \in A. A normal measure over is a fine ultrafilter U over with the additional property that every function f: \to A such th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vopěnka's Principle
In mathematics, Vopěnka's principle is a large cardinal axiom. The intuition behind the axiom is that the set-theoretical universe is so large that in every proper class, some members are similar to others, with this similarity formalized through elementary embeddings. Vopěnka's principle was first introduced by Petr Vopěnka and independently considered by H. Jerome Keisler, and was written up by . According to , Vopěnka's principle was originally intended as a joke: Vopěnka was apparently unenthusiastic about large cardinals and introduced his principle as a bogus large cardinal property, planning to show later that it was not consistent. However, before publishing his inconsistency proof he found a flaw in it. Definition Vopěnka's principle asserts that for every proper class of binary relations (each with set-sized domain), there is one elementarily embeddable into another. This cannot be stated as a single sentence of ZFC as it involves a quantification over classes. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
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] |
Function Composition
In mathematics, the composition operator \circ takes two function (mathematics), functions, f and g, and returns a new function h(x) := (g \circ f) (x) = g(f(x)). Thus, the function is function application, applied after applying to . (g \circ f) is pronounced "the composition of and ". Reverse composition, sometimes denoted f \mapsto g , applies the operation in the opposite order, applying f first and g second. Intuitively, reverse composition is a chaining process in which the output of function feeds the input of function . The composition of functions is a special case of the composition of relations, sometimes also denoted by \circ. As a result, all properties of composition of relations are true of composition of functions, such as #Properties, associativity. Examples * Composition of functions on a finite set (mathematics), set: If , and , then , as shown in the figure. * Composition of functions on an infinite set: If (where is the set of all real numbers) is ... [...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 '' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
