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Easton's Theorem
In set theory, Easton's theorem is a result on the possible cardinal numbers of powersets. (extending a result of Robert M. Solovay) showed via forcing that the only constraints on permissible values for 2''κ'' when ''κ'' is a regular cardinal are : \kappa < \operatorname(2^\kappa) (where cf(''α'') is the of ''α'') and : Statement If ''G'' is a class function whose domain consists of ordinals and whose range consists of ordinals such that # ''G'' is non-decreasing, # the |
Set Theory
Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly concerned with those that are relevant to mathematics as a whole. The modern study of set theory was initiated by the German mathematicians Richard Dedekind and Georg Cantor in the 1870s. In particular, Georg Cantor is commonly considered the founder of set theory. The non-formalized systems investigated during this early stage go under the name of '' naive set theory''. After the discovery of paradoxes within naive set theory (such as Russell's paradox, Cantor's paradox and the Burali-Forti paradox) various axiomatic systems were proposed in the early twentieth century, of which Zermelo–Fraenkel set theory (with or without the axiom of choice) is still the best-known and most studied. Set theory is commonly employed as a foundational ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Generalized Continuum Hypothesis
In mathematics, the continuum hypothesis (abbreviated CH) is a hypothesis about the possible sizes of infinite sets. It states that or equivalently, that In Zermelo–Fraenkel set theory with the axiom of choice (ZFC), this is equivalent to the following equation in aleph numbers: 2^=\aleph_1, or even shorter with beth numbers: \beth_1 = \aleph_1. The continuum hypothesis was advanced by Georg Cantor in 1878, and establishing its truth or falsehood is the first of Hilbert's 23 problems presented in 1900. The answer to this problem is independent of ZFC, so that either the continuum hypothesis or its negation can be added as an axiom to ZFC set theory, with the resulting theory being consistent if and only if ZFC is consistent. This independence was proved in 1963 by Paul Cohen, complementing earlier work by Kurt Gödel in 1940. The name of the hypothesis comes from the term '' the continuum'' for the real numbers. History Cantor believed the continuum hypothesis to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cardinal Numbers
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] |
Theorems In The Foundations Of Mathematics
In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of the axioms and previously proved theorems. In the mainstream of mathematics, the axioms and the inference rules are commonly left implicit, and, in this case, they are almost always those of Zermelo–Fraenkel set theory with the axiom of choice, or of a less powerful theory, such as Peano arithmetic. A notable exception is Wiles's proof of Fermat's Last Theorem, which involves the Grothendieck universes whose existence requires the addition of a new axiom to the set theory. Generally, an assertion that is explicitly called a theorem is a proved result that is not an immediate consequence of other known theorems. Moreover, many authors qualify as ''theorems'' only the most important results, and use the terms ''lemma'', ''proposition'' and ' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Set Theory
Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly concerned with those that are relevant to mathematics as a whole. The modern study of set theory was initiated by the German mathematicians Richard Dedekind and Georg Cantor in the 1870s. In particular, Georg Cantor is commonly considered the founder of set theory. The non-formalized systems investigated during this early stage go under the name of '' naive set theory''. After the discovery of paradoxes within naive set theory (such as Russell's paradox, Cantor's paradox and the Burali-Forti paradox) various axiomatic systems were proposed in the early twentieth century, of which Zermelo–Fraenkel set theory (with or without the axiom of choice) is still the best-known and most studied. Set theory is commonly employed as a foundational ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Beth Number
In mathematics, particularly in set theory, the beth numbers are a certain sequence of infinite cardinal numbers (also known as transfinite numbers), conventionally written \beth_0,\ \beth_1,\ \beth_2,\ \beth_3,\ \dots, where \beth is the second Hebrew letter ( beth). The beth numbers are related to the aleph numbers (\aleph_0,\ \aleph_1,\ \dots), but unless the generalized continuum hypothesis is true, there are numbers indexed by \aleph that are not indexed by \beth. Definition Beth numbers are defined by transfinite recursion: * \beth_0=\aleph_0, * \beth_=2^, * \beth_=\sup\, where \alpha is an ordinal and \lambda is a limit ordinal. The cardinal \beth_0=\aleph_0 is the cardinality of any countably infinite set such as the set \mathbb of natural numbers, so that \beth_0=, \mathbb, . Let \alpha be an ordinal, and A_\alpha be a set with cardinality \beth_\alpha=, A_\alpha, . Then, *\mathcal(A_\alpha) denotes the power set of A_\alpha (i.e., the set of all subsets of A_\alp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Aleph Number
In mathematics, particularly in set theory, the aleph numbers are a sequence of numbers used to represent the cardinality (or size) of infinite sets that can be well-ordered. They were introduced by the mathematician Georg Cantor and are named after the symbol he used to denote them, the Hebrew letter aleph (\,\aleph\,). The cardinality of the natural numbers is \,\aleph_0\, (read ''aleph-nought'' or ''aleph-zero''; the term ''aleph-null'' is also sometimes used), the next larger cardinality of a well-orderable set is aleph-one \,\aleph_1\;, then \,\aleph_2\, and so on. Continuing in this manner, it is possible to define a cardinal number \,\aleph_\alpha\, for every ordinal number \,\alpha\;, as described below. The concept and notation are due to Georg Cantor, who defined the notion of cardinality and realized that infinite sets can have different cardinalities. The aleph numbers differ from the infinity (\,\infty\,) commonly found in algebra and calculus, in that the alephs ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Singular Cardinal Hypothesis
In set theory, the singular cardinals hypothesis (SCH) arose from the question of whether the least cardinal number for which the generalized continuum hypothesis (GCH) might fail could be a singular cardinal. According to Mitchell (1992), the singular cardinals hypothesis is: :If ''κ'' is any singular strong limit cardinal, then 2''κ'' = ''κ''+. Here, ''κ''+ denotes the successor cardinal of ''κ''. Since SCH is a consequence of GCH, which is known to be consistent with ZFC, SCH is consistent with ZFC. The negation of SCH has also been shown to be consistent with ZFC, if one assumes the existence of a sufficiently large cardinal number. In fact, by results of Moti Gitik, ZFC + the negation of SCH is equiconsistent with ZFC + the existence of a measurable cardinal ''κ'' of Mitchell order ''κ''++. Another form of the SCH is the following statement: :2cf(''κ'') \kappa^+ —a violation of the SCH. Gitik, building on work of Woo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Continuum Function
In mathematics, the continuum function is \kappa\mapsto 2^\kappa, i.e. raising 2 to the power of κ using cardinal exponentiation. Given a cardinal number, it is the cardinality of the power set of a set of the given cardinality. See also * Continuum hypothesis *Cardinality of the continuum * Beth number * Easton's theorem *Gimel function In axiomatic set theory, the gimel function is the following function mapping cardinal numbers to cardinal numbers: :\gimel\colon\kappa\mapsto\kappa^ where cf denotes the cofinality function; the gimel function is used for studying the continuum ... Cardinal numbers {{settheory-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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(\prod A/D) denote the cofinality of the ordered set of functions \prod A where the ordering is defined as follows: f Main results Obviously, pcf(''A'') consists of regular cardinals. Considering ultrafilters concentrated on elements of ''A'', we get that . Shelah pr ...[...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Singular Cardinal
Singular may refer to: * Singular, the grammatical number that denotes a unit quantity, as opposed to the plural and other forms * Singular homology * SINGULAR, an open source Computer Algebra System (CAS) * Singular or sounder, a group of boar, see List of animal names * Singular matrix, a matrix that is not invertible * Singular measure, a measure or probability distribution whose support has zero Lebesgue (or other) measure * Singular cardinal, an infinite cardinal number that is not a regular cardinal * The property of a ''singularity'' or ''singular point'' in various meanings; see Singularity (other) * Singular (band), a Thai jazz pop duo *'' Singular: Act I'', a 2018 studio album by Sabrina Carpenter *'' Singular: Act II'', a 2019 studio album by Sabrina Carpenter See also * Singulair, Merck trademark for the drug Montelukast * Cingular Wireless AT&T Mobility LLC, also known as AT&T Wireless and marketed as simply AT&T, is an American telecommunications company ... [...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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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