We have other options that amuse me more: CARDINAL NUMBER (14), BACON NUMBER (11) and CALL NUMBER (10).
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In Rp one pitch class is different, in R0 all are different, and in R1 and R2 four interval classes are the same. Rp is defined for sets S1 and S2 of cardinal number n and S3 of cardinal number n-1 as:Forte, Allen (1977). The Structure of Atonal Music, p.47. .
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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.
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The cardinal number of is the Hilbert dimension of . Thus every Hilbert space is isometrically isomorphic to a sequence space for some set .
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In mathematics, a shrewd cardinal is a certain kind of large cardinal number introduced by ., extending the definition of indescribable cardinals. A cardinal number κ is called λ-shrewd if for every proposition φ, and set A ⊆ Vκ with (Vκ+λ, ∈, A) ⊧ φ there exists an α, λ' < κ with (Vα+λ', ∈, A ∩ Vα) ⊧ φ. It is called shrewd if it is λ-shrewd for every λ (including λ > κ).
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' This still applies in cases where a relatively indeterminate subject is genderized, such as the Spanish todos a una [voz] ('all at once', literally 'all at one [voice]'). It should be rewritten in Portuguese without any cardinal number. For example, todos juntos 'all together'. On the other hand, in Portuguese, cardinal number 'two' inflects with gender (dois if masculine, duas if feminine), while in Spanish dos is used for both.
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Beyond the basic cardinals and ordinals, Japanese has other types of numerals. Distributive numbers are formed regularly from a cardinal number, a counter word, and the suffix , as in .
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In order to state the paradox it is necessary to understand that the cardinal numbers admit an ordering, so that one can speak about one being greater or less than another. Then Cantor's paradox is: :Theorem: There is no greatest cardinal number. This fact is a direct consequence of Cantor's theorem on the cardinality of the power set of a set. :Proof: Assume the contrary, and let C be the largest cardinal number.
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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: [κ]<ω → κ there is a set H of order type κ such that for each n, f restricted to n-element subsets of H omits at least one value in κ. 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.
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In this section it is assumed that every set can be endowed with a group structure . Let be a set. Let be the Hartogs number of . This is the least cardinal number such that there is no injection from into .
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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.
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In mathematics, a Q-indescribable cardinal is a certain kind of large cardinal number that is hard to describe in some language Q. There are many different types of indescribable cardinals corresponding to different choices of languages Q. They were introduced by . A cardinal number κ is called Π-indescribable if for every Πm proposition φ, and set A ⊆ Vκ with (Vκ+n, ∈, A) ⊧ φ there exists an α < κ with (Vα+n, ∈, A ∩ Vα) ⊧ φ. Here one looks at formulas with m-1 alternations of quantifiers with the outermost quantifier being universal. Σ-indescribable cardinals are defined in a similar way.
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For expressing fractions, one should use the sentence pattern like "cardinal number + hun-chi + cardinal number"; for example, gō͘ hun-chi it (五分之一) for "one fifth" (1/5). Note that the colloquial set of numerals is used in fractional numerals with still the exception of numerals 1 and 2, which should use the literary set as it and jī. For expressing decimals, one should only use the literary numeral set with tiám (點) for the decimal mark. For example, one may say π equals sam tiám it- sù-it-ngó͘-kiú-jī-lio̍k-ngó͘-sam (3.141592653).
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The cardinal number precedes the noun (e.g., shlosha yeladim), except for the number one which succeeds it (e.g., yeled echad). The number two is special: shnayim (m.) and shtayim (f.) become shney (m.) and shtey (f.) when followed by the noun they count.
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In mathematical logic the Löwenheim number of an abstract logic is the smallest cardinal number for which a weak downward Löwenheim–Skolem theorem holds.Zhang 2002 page 77 They are named after Leopold Löwenheim, who proved that these exist for a very broad class of logics.
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In set theory, a Rowbottom cardinal, introduced by , is a certain kind of large cardinal number. An uncountable cardinal number κ is said to be Rowbottom if for every function f: [κ]<ω -> λ (where λ < κ) there is a set H of order type κ that is quasi-homogeneous for f, i.e., for every n, the f-image of the set of n-element subsets of H has countably many elements. Every Ramsey cardinal is Rowbottom, and every Rowbottom cardinal is Jónsson. By a theorem of Kleinberg, the theories ZFC + “there is a Rowbottom cardinal” and ZFC + “there is a Jónsson cardinal” are equiconsistent.
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In mathematical set theory, an Ulam matrix is an array of subsets of a cardinal number with certain properties. Ulam matrices were introduced by in his work on measurable cardinals: they may be used, for example, to show that a real-valued measurable cardinal is weakly inaccessible.
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Obviously 1 is additively indecomposable, since 0+0<1. No finite ordinal other than 1 is additively indecomposable. Also, \omega is additively indecomposable, since the sum of two finite ordinals is still finite. More generally, every infinite initial ordinal (an ordinal corresponding to a cardinal number) is additively indecomposable.
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Assuming the axiom of choice and, given an infinite cardinal κ and a finite cardinal μ greater than 1, there may or may not be a cardinal λ satisfying \mu^\lambda = \kappa. However, if such a cardinal exists, it is infinite and less than κ, and any finite cardinality ν greater than 1 will also satisfy u^\lambda = \kappa. The logarithm of an infinite cardinal number κ is defined as the least cardinal number μ such that κ ≤ 2μ. Logarithms of infinite cardinals are useful in some fields of mathematics, for example in the study of cardinal invariants of topological spaces, though they lack some of the properties that logarithms of positive real numbers possess.
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Scott's trick assigns representatives differently, using the fact that for every set A there is a least rank γA in the cumulative hierarchy when some set of the same cardinality as A appears. Thus one may define the representative of the cardinal number of A to be the set of all sets of rank γA that have the same cardinality as A. This definition assigns a representative to every cardinal number even when not every set can be well-ordered (an assumption equivalent to the axiom of choice). It can be carried out in Zermelo-Fraenkel set theory, without using the axiom of choice, but making essential use of the axiom of regularity.
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In mathematics, a subcompact cardinal is a certain kind of large cardinal number. A cardinal number κ is subcompact if and only if for every A ⊂ H(κ+) there is a non-trivial elementary embedding j:(H(μ+), B) → (H(κ+), A) (where H(κ+) is the set of all sets of cardinality hereditarily less than κ+) with critical point μ and j(μ) = κ. Analogously, κ is a quasicompact cardinal if and only if for every A ⊂ H(κ+) there is a non-trivial elementary embedding j:(H(κ+), A) → (H(μ+), B) with critical point κ and j(κ) = μ. H(λ) consists of all sets whose transitive closure has cardinality less than λ.
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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 which Ramsey cardinals generalize to the uncountable case. Let [κ]<ω denote the set of all finite subsets of κ. An uncountable cardinal number κ is called Ramsey if, for every function :f: [κ]<ω → {0, 1} there is a set A of cardinality κ that is homogeneous for f. That is, for every n, 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 stationary subset of κ.
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Using the Von Neumann definition of ordinals, every ordinal is the well- ordered set of all smaller ordinals. The union of a nonempty set of ordinals that has no greatest element is then always a limit ordinal. Using Von Neumann cardinal assignment, every infinite cardinal number is also a limit ordinal.
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The aleph numbers are indexed by ordinal numbers. Under the assumption of the axiom of choice, this transfinite sequence includes every cardinal number. If one rejects that axiom, the situation is more complicated, with additional infinite cardinals that are not alephs. Cardinality is studied for its own sake as part of set theory.
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Before writing Principles, Russell became aware of Cantor's proof that there was no greatest cardinal number, which Russell believed was mistaken.The Autobiography of Bertrand Russell, George Allen and Unwin Ltd., 1971, p.147. The Cantor's paradox in turn was shown (for example by Crossley) to be a special case of the Russell Paradox.
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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.
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In mathematics, an unfoldable cardinal is a certain kind of large cardinal number. Formally, a cardinal number κ is λ-unfoldable if and only if for every transitive model M of cardinality κ of ZFC-minus-power set such that κ is in M and M contains all its sequences of length less than κ, there is a non-trivial elementary embedding j of M into a transitive model with the critical point of j being κ and j(κ) ≥ λ. A cardinal is unfoldable if and only if it is an λ-unfoldable for all ordinals λ. A cardinal number κ is strongly λ-unfoldable if and only if for every transitive model M of cardinality κ of ZFC-minus- power set such that κ is in M and M contains all its sequences of length less than κ, there is a non-trivial elementary embedding j of M into a transitive model "N" with the critical point of j being κ, j(κ) ≥ λ, and V(λ) is a subset of N. Without loss of generality, we can demand also that N contains all its sequences of length λ.
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The axiom of choice is equivalent to the statement that every set can be well-ordered, i.e. that every cardinal has an initial ordinal. In this case, it is traditional to identify the cardinal number with its initial ordinal, and we say that the initial ordinal is a cardinal. The \alpha-th infinite initial ordinal is written \omega_\alpha.
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The following tables list the cardinal number names and symbols for the numbers 0 through 10 in various languages and scripts of the world. Where possible, each language's native writing system is used, along with transliterations in Latin script and other important writing systems where applicable. In some languages, numbers will be illustrated through to 20.
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The set of natural numbers is an infinite set. By definition, this kind of infinity is called countable infinity. All sets that can be put into a bijective relation to the natural numbers are said to have this kind of infinity. This is also expressed by saying that the cardinal number of the set is aleph-naught ().
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From 5th to 99th, ordinals are formed simply by declining the corresponding cardinal number as a regular adjective. If the last syllable is stressed, a closed long e or o becomes open. Thus: pêti/pêta/pêto (5th), šêsti/šêsta/šêsto (6th), sêdmi/sêdma/sêdmo (7th), ... devétindevétdeseti/a/o (99th). 100th and 1000th are formed the same way: stôti/a/o, tísoči/a/o.
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Let X be a discrete topological space, and let \omega X be an Alexandroff one-point compactification of X. A Hausdorff space P is polyadic if for some cardinal number \lambda, there exists a continuous surjective function f : \omega X^\lambda \rightarrow P, where \omega X^\lambda is the product space obtained by multiplying \omega X with itself \lambda times.
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For dealing with infinite sets, the natural numbers have been generalized to the ordinal numbers and to the cardinal numbers. The former gives the ordering of the set, while the latter gives its size. For finite sets, both ordinal and cardinal numbers are identified with the natural numbers. In the infinite case, many ordinal numbers correspond to the same cardinal number.
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Every sentence must contain formal words to designate what Husserl calls "formal categories". There are two kinds of categories: meaning categories and formal-ontological categories. Meaning categories relate judgments; they include forms of conjunction, disjunction, forms of plural, among others. Formal-ontological categories relate objects and include notions such as set, cardinal number, ordinal number, part and whole, relation, and so on.
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The cardinal numbers in Frater: 1 - uni 2 - bi 3 - tri 4 - kuadri 5 - kuinti 6 - ses 7 - sep 8 - okta 9 - nona 10 - deka 11 - dekauni 12 - dekabi 13 - dekatri 20 - bideka 24 - bidekakuadri 30 - trideka 40 - kuadrideka 85 - oktadekakuinti 100 - senti 367 - trisenti-sesdeka-sep 600 - sessenti 1000 - mil 1000000 - milion Ordinal numbers are formed by placing the cardinal number after the noun.
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An important special case is where the topology on X is the discrete topology; in this case X can be identified with a cardinal number κ and C(X) with the real algebra Rκ of functions from κ to R. The hyperreal fields we obtain in this case are called ultrapowers of R and are identical to the ultrapowers constructed via free ultrafilters in model theory.
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Mirsky's theorem extends immediately to infinite partially ordered sets with finite height. However, the relation between the length of a chain and the number of antichains in a partition into antichains does not extend to infinite cardinalities: for every infinite cardinal number κ, there exist partially ordered sets that have no infinite chain and that do not have an antichain partition with κ or fewer antichains .
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For larger multiples a cardinal number and a counter are used instead, such as "five portions" or "a portion five times the normal size" instead of "a quintuple portion". In espresso servings, the Italian solo, doppio, and triplo are sometimes used, with doppio being most common. The Latin multipliers simplex, duplex, triplex etc. have occasional use in English, primarily in technical use, though duplex is more common.
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Every strongly inaccessible cardinal is also weakly inaccessible, as every strong limit cardinal is also a weak limit cardinal. If the generalized continuum hypothesis holds, then a cardinal is strongly inaccessible if and only if it is weakly inaccessible. \aleph_0 (aleph-null) is a regular strong limit cardinal. Assuming the axiom of choice, every other infinite cardinal number is regular or a (weak) limit.
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If λ is any ordinal, κ is λ-strong means that κ is a cardinal number and there exists an elementary embedding j from the universe V into a transitive inner model M with critical point κ and :V_\lambda\subseteq M That is, M agrees with V on an initial segment. Then κ is strong means that it is λ-strong for all ordinals λ.
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In the United States, dates are traditionally written in the "month-day-year" order, with neither increasing nor decreasing order of significance. This order is used in both the traditional all-numeric date (e.g., "1/21/16" or "01/21/2016") and the expanded form (e.g., "January 21, 2016"—usually spoken with the year as a cardinal number and the day as an ordinal number, e.g.
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The idea is that κ cannot be distinguished (looking from below) from smaller cardinals by any formula of n+1-th order logic with m-1 alternations of quantifiers even with the advantage of an extra unary predicate symbol (for A). This implies that it is large because it means that there must be many smaller cardinals with similar properties. The cardinal number κ is called totally indescribable if it is Π-indescribable for all positive integers m and n. If α is an ordinal, the cardinal number κ is called α-indescribable if for every formula φ and every subset U of Vκ such that φ(U) holds in Vκ+α there is a some λ<κ such that φ(U ∩ Vλ) holds in Vλ+α. If α is infinite then α-indescribable ordinals are totally indescribable, and if α is finite they are the same as Π-indescribable ordinals.
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Any nonempty finite set has a group structure as a cyclic group generated by any element. Under the assumption of the axiom of choice, every infinite set is equipotent with a unique cardinal number which equals an aleph. Using the axiom of choice, one can show that for any family of sets (A). Moreover, by Tarski's theorem on choice, another equivalent of the axiom of choice, for all finite (B).
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In mathematics, specifically in axiomatic set theory, a Hartogs number is a particular kind of ordinal number. In particular, if X is any set, then the Hartogs number of X is the least ordinal α such that there is no injection from α into X. If X can be well-ordered then the cardinal number of α is a minimal cardinal greater than that of X. If X cannot be well-ordered then there cannot be an injection from X to α. However, the cardinal number of α is still a minimal cardinal not less than or equal to the cardinality of X. (If we restrict to cardinal numbers of well-orderable sets then that of α is the smallest that is not not less than or equal to that of X.) The map taking X to α is sometimes called Hartogs's function. This mapping is used to construct the aleph numbers, which are all the cardinal numbers of infinite well- orderable sets.
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The theory of accessible categories is a part of mathematics, specifically of category theory. It attempts to describe categories in terms of the "size" (a cardinal number) of the operations needed to generate their objects. The theory originates in the work of Grothendieck completed by 1969, and Gabriel and Ulmer (1971). It has been further developed in 1989 by Michael Makkai and Robert Paré, with motivation coming from model theory, a branch of mathematical logic.
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Edward Nelson's internal set theory enriches the Zermelo–Fraenkel set theory syntactically by introducing a unary predicate "standard". In this approach, infinitesimals are (non-"standard") elements of the set of the real numbers (rather than being elements of an extension thereof, as in Robinson's theory). The continuum hypothesis posits that the cardinality of the set of the real numbers is \aleph_1; i.e. the smallest infinite cardinal number after \aleph_0, the cardinality of the integers.
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In model theory, a graph is just a structure. But in that case, there is no limitation on the number of edges: it can be any cardinal number, see continuous graph. In computational biology, power graph analysis introduces power graphs as an alternative representation of undirected graphs. In geographic information systems, geometric networks are closely modeled after graphs, and borrow many concepts from graph theory to perform spatial analysis on road networks or utility grids.
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For infinite sets, the picture is more complicated, leading to the concept of cardinal number—a way to distinguish the various sizes of infinite sets. A bijective function from a set to itself is also called a permutation, and the set of all permutations of a set forms a symmetry group. Bijective functions are essential to many areas of mathematics including the definitions of isomorphism, homeomorphism, diffeomorphism, permutation group, and projective map.
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If X and Y are finite sets, then there exists a bijection between the two sets X and Y if and only if X and Y have the same number of elements. Indeed, in axiomatic set theory, this is taken as the definition of "same number of elements" (equinumerosity), and generalising this definition to infinite sets leads to the concept of cardinal number, a way to distinguish the various sizes of infinite sets.
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One krona is subdivided into 100 öre (singular; plural öre or ören, where the former is always used after a cardinal number, hence "50 öre", but otherwise the latter is often preferred in contemporary speech). However, all öre coins have been discontinued as of 30 September 2010. Goods can still be priced in öre, but all sums are rounded to the nearest krona when paying with cash. The word öre is ultimately derived from the Latin word for gold.
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If the axiom of choice holds, then every cardinal κ has a successor, denoted κ+, where κ+ > κ and there are no cardinals between κ and its successor. (Without the axiom of choice, using Hartogs' theorem, it can be shown that for any cardinal number κ, there is a minimal cardinal κ+ such that \kappa^+ leq\kappa. ) For finite cardinals, the successor is simply κ + 1. For infinite cardinals, the successor cardinal differs from the successor ordinal.
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225) Stephen Kleene notes that any of the six inequality restrictions on the range of the variable y is permitted, i.e. y < z, y ≤ z, w < y < z, w < y ≤ z, w ≤ y < z and w ≤ y ≤ z. "When the indicated range contains no y such that R(y) [is "true"], the value of the "μy" expression is the cardinal number of the range" (p. 226); this is why the default "z" appears in the definition above.
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However, in digital communication using computers and writing emails and SMS, the Latin script has been proposed to replace the Cyrillic. A Bulgarian Latin alphabet, the so-called shlyokavitsa, is already often employed for convenience for emails and SMS messages. Ciphers are used to denote Bulgarian sounds that cannot be represented with a single Latin character (for example, a "4" represents a "ч" because they look alike and the Bulgarian word for the cardinal number four, чѐтири čѐtiri, starts with a "ч").
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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 on a cardinal , or more generally on any set. For a cardinal , it can be described as a subdivision of all of its subsets into large and small sets such that itself is large, and all singletons are small, complements of small sets are large and vice versa. The intersection of fewer than large sets is again large.
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A regnal year is a year of the reign of a sovereign, from the Latin regnum meaning kingdom, rule. Regnal years considered the date as an ordinal, not a cardinal number. For example, a monarch could have a first year of rule, a second year of rule, a third year of rule, and so on, but not a zeroth year of rule. Applying this ancient epoch system to modern calculations of time, which include zero, is what led to the debate over when the third millennium began.
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Write κ, λ for ordinals, m for a cardinal number and n for a natural number. introduced the notation :\kappa\rightarrow(\lambda)^n_m as a shorthand way of saying that every partition of the set [κ]n of n-element subsets of \kappa into m pieces has a homogeneous set of order type λ. A homogeneous set is in this case a subset of κ such that every n-element subset is in the same element of the partition. When m is 2 it is often omitted.
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In journalism and blogging, a listicle is a short-form of writing that uses a list as its thematic structure, but is fleshed out with sufficient copy to be published as an article. A typical listicle will prominently feature a cardinal number in its title, with subsequent subheadings within the text itself reflecting this schema. The word is a portmanteau derived from list and article. It has also been suggested that the word evokes "popsicle", emphasising the fun but "not too nutritious" nature of the listicle.
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It is a consequence of the axiom of choice that the chromatic number is well-defined for infinite graphs, but for these graphs the chromatic number might itself be an infinite cardinal number. A subgraph of a graph is another graph obtained from a subset of its vertices and a subset of its edges. If the larger graph is colored, the same coloring can be used for the subgraph. Therefore, the chromatic number of a subgraph cannot be larger than the chromatic number of the whole graph.
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More generally, a cardinal number κ is called λ-Πm-shrewd if for every Πm proposition φ, and set A ⊆ Vκ with (Vκ+λ, ∈, A) ⊧ φ there exists an α, λ' < κ with (Vα+λ', ∈, A ∩ Vα) ⊧ φ. Here one looks at formulas with m-1 alternations of quantifiers with the outermost quantifier being universal. For finite n, an n-Πm-shrewd cardinals is the same thing as a Πmn-indescribable cardinal. If κ is a subtle cardinal, then the set of κ-shrewd cardinals is stationary in κ.
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These sets are then taken to "be" cardinal numbers, by definition. In Zermelo-Fraenkel set theory with the axiom of choice, one way of assigning representatives to cardinal numbers is to associate each cardinal number with the least ordinal number of the same cardinality. These special ordinals are the ℵ numbers. But if the axiom of choice is not assumed, for some cardinal numbers it may not be possible to find such an ordinal number, and thus the cardinal numbers of those sets have no ordinal number as representatives.
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In mathematical logic, the Löwenheim–Skolem theorem is a theorem on the existence and cardinality of models, named after Leopold Löwenheim and Thoralf Skolem. The precise formulation is given below. It implies that if a countable first-order theory has an infinite model, then for every infinite cardinal number κ it has a model of size κ, and that no first-order theory with an infinite model can have a unique model up to isomorphism. As a consequence, first-order theories are unable to control the cardinality of their infinite models.
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Given a concrete category (C, U) and a cardinal number N, let UN be the functor C → Set determined by UN(c) = (U(c))N. Then a subfunctor of UN is called an N-ary predicate and a natural transformation UN → U an N-ary operation. The class of all N-ary predicates and N-ary operations of a concrete category (C,U), with N ranging over the class of all cardinal numbers, forms a large signature. The category of models for this signature then contains a full subcategory which is equivalent to C.
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In model theory, a branch of mathematical logic, a complete first-order theory T is called stable in λ (an infinite cardinal number), if the Stone space of every model of T of size ≤ λ has itself size ≤ λ. T is called a stable theory if there is no upper bound for the cardinals κ such that T is stable in κ. The stability spectrum of T is the class of all cardinals κ such that T is stable in κ. For countable theories there are only four possible stability spectra.
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This sequence has order type \omega, so \omega+\omega is the limit of a sequence of type less than \omega+\omega whose elements are ordinals less than \omega+\omega; therefore it is singular. \aleph_1 is the next cardinal number greater than \aleph_0, so the cardinals less than \aleph_1 are countable (finite or denumerable). Assuming the axiom of choice, the union of a countable set of countable sets is itself countable. So \aleph_1 cannot be written as the sum of a countable set of countable cardinal numbers, and is regular.
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A calendar era assigns a cardinal number to each sequential year, using a reference point in the past as the beginning of the era. The worldwide standard is the Anno Domini, although some prefer the term Common Era because it has no explicit reference to Christianity. It was introduced in the 6th century and was intended to count years from the nativity of Jesus. Richards does not explicitly say that Anno Domini is the worldwide standard, but does say on page 585 that the Gregorian calendar is used throughout the world for secular purposes.
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Georgian, Latin, and Romanian are notable languages with distributive numerals; see Romanian distributive numbers. In Japanese numerals, distributive forms are formed regularly from a cardinal number, a counter word, and the suffix , as in . In Turkish, one of the -ar/-er suffixes (chosen according to vowel harmony) are added to the end of a cardinal numeral, as in "birer" (one of each) and "dokuzar" (nine of each). If the numeral ends with a vowel, a letter ş comes to the middle; as in "ikişer" (two of each) and "altışar" (six of each).
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Then (in the von Neumann formulation of cardinality) C is a set and therefore has a power set 2C which, by Cantor's theorem, has cardinality strictly larger than C. Demonstrating a cardinality (namely that of 2C) larger than C, which was assumed to be the greatest cardinal number, falsifies the definition of C. This contradiction establishes that such a cardinal cannot exist. Another consequence of Cantor's theorem is that the cardinal numbers constitute a proper class. That is, they cannot all be collected together as elements of a single set. Here is a somewhat more general result.
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The multiplicative case (abbreviated ' or ') is a grammatical case used for marking a number of something ("three times"). The case is found in the Hungarian language,Mentioned in: István Kenesei, Anna Fenyvesi, Robert Michael Vago, Hungarian, page xxviii, 1998 - 472 pages [ Google book search] for example nyolc (eight), nyolcszor (eight times), however it is not considered a real case in modern Hungarian linguistics because of its adverb-forming nature. The case appears also in Finnish as an adverbial (adverb-forming) case. Used with a cardinal number it denotes the number of actions; for example, viisi (five) -> viidesti (five times).
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A bijective function, f: X → Y, from set X to set Y demonstrates that the sets have the same cardinality, in this case equal to the cardinal number 4. Aleph null, the smallest infinite cardinal 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.
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The use of Scott's trick for cardinal numbers shows how the method is typically employed. The initial definition of a cardinal number is an equivalence class of sets, where two sets are equivalent if there is a bijection between them. The difficulty is that almost every equivalence class of this relation is a proper class, and so the equivalence classes themselves cannot be directly manipulated in set theories, such as Zermelo-Fraenkel set theory, that only deal with sets. It is often desirable in the context of set theory to have sets that are representatives for the equivalence classes.
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When the numerator is one, it may be omitted (as in "a tenth" or "each quarter"). The entire fraction may be expressed as a single composition, in which case it is hyphenated, or as a number of fractions with a numerator of one, in which case they are not. (For example, "two-fifths" is the fraction and "two fifths" is the same fraction understood as 2 instances of .) Fractions should always be hyphenated when used as adjectives. Alternatively, a fraction may be described by reading it out as the numerator "over" the denominator, with the denominator expressed as a cardinal number.
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Generalizing finite and (ordinary) infinite sequences which are maps from the positive integers leads to mappings from ordinal numbers to transfinite sequences. Cardinal numbers define the size of sets, meaning how many members they contain, and can be standardized by choosing the first ordinal number of a certain size to represent the cardinal number of that size. The smallest ordinal infinity is that of the positive integers, and any set which has the cardinality of the integers is countably infinite. If a set is too large to be put in one-to-one correspondence with the positive integers, it is called uncountable.
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When the Peano axioms were first proposed, Bertrand Russell and others agreed that these axioms implicitly defined what we mean by a "natural number". An illustration of 'interpretation' is Russell's own definition of 'cardinal number'. The uninterpreted system in this case is Peano's axioms for the number system, whose three primitive ideas and five axioms, Peano believed, were sufficient to enable one to derive all the properties of the system of natural numbers. Actually, Russell maintains, Peano's axioms define any progression of the form x_0,x_1,x_2,\ldots,x_n,\ldots of which the series of the natural numbers is one instance.
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Every well-ordered set is uniquely order isomorphic to a unique ordinal number, called the order type of the well-ordered set. The position of each element within the ordered set is also given by an ordinal number. In the case of a finite set, the basic operation of counting, to find the ordinal number of a particular object, or to find the object with a particular ordinal number, corresponds to assigning ordinal numbers one by one to the objects. The size (number of elements, cardinal number) of a finite set is equal to the order type.
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However, only a rather large cardinal number can be both and thus weakly inaccessible. An ordinal is a weakly inaccessible cardinal if and only if it is a regular ordinal and it is a limit of regular ordinals. (Zero, one, and \aleph_0 are regular ordinals, but not limits of regular ordinals.) A cardinal which is weakly inaccessible and also a strong limit cardinal is strongly inaccessible. The assumption of the existence of a strongly inaccessible cardinal is sometimes applied in the form of the assumption that one can work inside a Grothendieck universe, the two ideas being intimately connected.
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Therefore, by the De Bruijn–Erdős theorem, P itself also has a w-colorable incomparability graph, and thus has the desired partition into chains . However, the theorem does not extend so simply to partially ordered sets in which the width, and not just the cardinality of the set, is infinite. In this case the size of the largest antichain and the minimum number of chains needed to cover the partial order may be very different from each other. In particular, for every infinite cardinal number κ there is an infinite partially ordered set of width ℵ0 whose partition into the fewest chains has κ chains .
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Chilean Telethon's logo The 1991 Chilean telethon was the tenth version of the solidarity campaign conducted in Chile, which took place on 29 and 30 November 1991. The theme of this version was "Thank You" as a way of thanks to the Chilean public who had participated in the campaign for 13 years. It was the last edition of the Chilean telethon that was identified with a cardinal number from the next edition it began to be identified as Teletón and the year it was made. The final total on the night, released by the Banco de Chile and read by Javier Miranda (TV host) was CL$ 1,584,289,345.
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While stress is phonemic, minimal pairs are rare, and marking the stress in written Dutch is always optional, but it is sometimes recommended to distinguish homographs that differ only in stress. While it is common practice to distinguish een (indefinite article) from één (the cardinal number one),The current collection at nl.wiktionary this distinction is not so much about stress as it is about the pronunciation of the vowel ( versus ), and while the former is always unstressed, the latter may or may not be stressed. Stress also distinguishes some verbs, as stress placement on prefixes also carries a grammatical distinction, such as in vóórkomen ('to occur') and voorkómen ('to prevent').
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In mathematics, an Erdős cardinal, also called a partition cardinal is a certain kind of large cardinal number introduced by . The Erdős cardinal is defined to be the least cardinal such that for every function there is a set of order type that is homogeneous for (if such a cardinal exists). In the notation of the partition calculus, the Erdős cardinal is the smallest cardinal such that : Existence of zero sharp implies that the constructible universe satisfies "for every countable ordinal , there is an -Erdős cardinal". In fact, for every indiscernible satisfies "for every ordinal , there is an -Erdős cardinal in (the Levy collapse to make countable)".
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In set theory, Cantor's paradox states that there is no set of all cardinalities. This is derived from the theorem that there is no greatest cardinal number. In informal terms, the paradox is that the collection of all possible "infinite sizes" is not only infinite, but so infinitely large that its own infinite size cannot be any of the infinite sizes in the collection. The difficulty is handled in axiomatic set theory by declaring that this collection is not a set but a proper class; in von Neumann–Bernays–Gödel set theory it follows from this and the axiom of limitation of size that this proper class must be in bijection with the class of all sets.
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Hanson called this the isomeric relationship, and defined two such sets as isomeric.Hanson, Howard (1960). Harmonic Materials of Modern Music (New York: Appleton-Century-Crofts), p. 22\. . See: isomer. According to Michiel Schuijer (2008), the hexachord theorem, that any two pitch-class complementary hexachords have the same interval vector, even if they are not equivalent under transposition and inversion, was first proposed by Milton Babbitt, and, "the discovery of the relation," was, "reported," by David Lewin in 1960 as an example of the complement theorem: that the difference between pitch-class intervals in two complementary pitch-class sets is equal to the difference between the cardinal number of the sets (given two hexachords, this difference is 0).
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In mathematics, the dimension theorem for vector spaces states that all bases of a vector space have equally many elements. This number of elements may be finite or infinite (in the latter case, it is a cardinal number), and defines the dimension of the vector space. Formally, the dimension theorem for vector spaces states that :Given a vector space , any two bases have the same cardinality. As a basis is a generating set that is linearly independent, the theorem is a consequence of the following theorem, which is also useful: :In a vector space , if is a generating set, and is a linearly independent set, then the cardinality of is not larger than the cardinality of .
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In addition, some special fraction can be expressed in other simpler forms. For percentage, one can still use the sentence pattern of hun-chi as pah hun-chi cha̍p (百分之十) for "ten percent" in most situations; however, for native speakers, the suffix -siâⁿ (成) for "n×10 percents" is used more commonly, so the "twenty percents" should be nn̄g-siâⁿ (兩成). Note that the numeral set used with the suffix -siâⁿ is totally the colloquial one with no exception. In Taiwan, the term pha-sian-to͘ is also used for fractional numerals, but one should use the sentence term as "cardinal number + ê pha-sian-to͘"; for example, chhit-cha̍p ê pha-sian-to͘ (70%).
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Although concepts of degeneracy and coloring number are frequently considered in the context of finite graphs, the original motivation for was the theory of infinite graphs. For an infinite graph G, one may define the coloring number analogously to the definition for finite graphs, as the smallest cardinal number α such that there exists a well-ordering of the vertices of G in which each vertex has fewer than α neighbors that are earlier in the ordering. The inequality between coloring and chromatic numbers holds also in this infinite setting; state that, at the time of publication of their paper, it was already well known. The degeneracy of random subsets of infinite lattices has been studied under the name of bootstrap percolation.
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In mathematics, subtle cardinals and ethereal cardinals are closely related kinds of large cardinal number. A cardinal κ is called subtle if for every closed and unbounded C ⊂ κ and for every sequence A of length κ for which element number δ (for an arbitrary δ), Aδ ⊂ δ there are α, β, belonging to C, with α < β, such that Aα = Aβ ∩ α. A cardinal κ is called ethereal if for every closed and unbounded C ⊂ κ and for every sequence A of length κ for which element number δ (for an arbitrary δ), Aδ ⊂ δ and Aδ has the same cardinal as δ, there are α, β, belonging to C, with α < β, such that card(α) = card(Aβ ∩ Aα). Subtle cardinals were introduced by .
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If V is a standard model of ZFC and κ is an inaccessible in V, then: Vκ is one of the intended models of Zermelo–Fraenkel set theory; and Def(Vκ) is one of the intended models of Mendelson's version of Von Neumann–Bernays–Gödel set theory which excludes global choice, replacing limitation of size by replacement and ordinary choice; and Vκ+1 is one of the intended models of Morse–Kelley set theory. Here Def (X) is the Δ0 definable subsets of X (see constructible universe). However, κ does not need to be inaccessible, or even a cardinal number, in order for Vκ to be a standard model of ZF (see below). Suppose V is a model of ZFC.
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The coalitions belonging to W are winning; the others are losing. A simple game W is monotonic if S \in W and S\subseteq T imply T \in W. It is proper if S \in W implies N\setminus S otin W. It is strong if S otin W implies N\setminus S \in W. A veto player (vetoer) is an individual that belongs to all winning coalitions. A simple game is nonweak if it has no veto player. It is finite if there is a finite set (called a carrier) T \subseteq N such that for all coalitions S, we have S \in W iff S\cap T \in W. Let X be a (finite or infinite) set of alternatives, whose cardinal number (the number of elements) \\# X is at least two.
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This means that, for every set S of cardinality \kappa, and every partition of the ordered pairs of elements of S into two subsets P_1 and P_1, there exists either a subset S_1\subset S of cardinality \kappa or a subset S_2\subset S of cardinality \alef_0, such that all pairs of elements of S_i belong to P_i. Here, P_1 can be interpreted as the edges of a graph having S as its vertex set, in which S_1 (if it exists) is a clique of cardinality \kappa, and S_2 (if it exists) is a countably infinite independent set. If S is taken to be the cardinal number \kappa itself, the theorem can be formulated in terms of ordinal numbers with the notation \kappa\rightarrow(\kappa,\omega)^2, meaning that S_2 (when it exists) has order type \omega.
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Since the cardinal numbers are well-ordered by indexing with the ordinal numbers (see Cardinal number, formal definition), this also establishes that there is no greatest ordinal number; conversely, the latter statement implies Cantor's paradox. By applying this indexing to the Burali-Forti paradox we obtain another proof that the cardinal numbers are a proper class rather than a set, and (at least in ZFC or in von Neumann–Bernays–Gödel set theory) it follows from this that there is a bijection between the class of cardinals and the class of all sets. Since every set is a subset of this latter class, and every cardinality is the cardinality of a set (by definition!) this intuitively means that the "cardinality" of the collection of cardinals is greater than the cardinality of any set: it is more infinite than any true infinity. This is the paradoxical nature of Cantor's "paradox".
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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: [κ] 2 → {0, 1} there is a set of cardinality κ that is homogeneous for f. In this context, [κ] 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]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.
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The classes of successor ordinals and limit ordinals (of various cofinalities) as well as zero exhaust the entire class of ordinals, so these cases are often used in proofs by transfinite induction or definitions by transfinite recursion. Limit ordinals represent a sort of "turning point" in such procedures, in which one must use limiting operations such as taking the union over all preceding ordinals. In principle, one could do anything at limit ordinals, but taking the union is continuous in the order topology and this is usually desirable. If we use the Von Neumann cardinal assignment, every infinite cardinal number is also a limit ordinal (and this is a fitting observation, as cardinal derives from the Latin cardo meaning hinge or turning point): the proof of this fact is done by simply showing that every infinite successor ordinal is equinumerous to a limit ordinal via the Hotel Infinity argument.
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The English word dozen comes from the old form douzaine, a French word meaning "a group of twelve" ("Assemblage de choses de même nature au nombre de douze" (translation: A group of twelve things of the same nature), as defined in the eighth edition of the Dictionnaire de l'Académie française). This French word is a derivation from the cardinal number douze ("twelve", from Latin duodĕcim) and the collective suffix -aine (from Latin -ēna), a suffix also used to form other words with similar meanings such as quinzaine (a group of fifteen), vingtaine (a group of twenty), centaine (a group of one hundred), etc. These French words have synonymous cognates in Spanish: docena, quincena, veintena, centena, etc. English dozen, French douzaine, Catalan dotzena, Persian dowjin "دوجین", Arabic durzen "درزن", Turkish "düzine", German Dutzend, Dutch dozijn, Italian dozzina and Polish tuzin, are also used as indefinite quantifiers to mean "about twelve" or "many" (as in "a dozen times", "dozens of people").
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Whereas the notion of cardinal number is associated with a set with no particular structure on it, the ordinals are intimately linked with the special kind of sets that are called well-ordered (so intimately linked, in fact, that some mathematicians make no distinction between the two concepts). A well-ordered set is a totally ordered set (given any two elements one defines a smaller and a larger one in a coherent way), in which every non- empty subset of the set has a least element. In particular, there is no infinite decreasing sequence. (However, there may be infinite increasing sequences.) Ordinals may be used to label the elements of any given well- ordered set (the smallest element being labelled 0, the one after that 1, the next one 2, "and so on"), and to measure the "length" of the whole set by the least ordinal that is not a label for an element of the set.
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In written languages, an ordinal indicator is a character, or group of characters, following a numeral denoting that it is an ordinal number, rather than a cardinal number. In English orthography, this corresponds to the suffixes -st, -nd, -rd, -th in written ordinals (represented either on the line 1st, 2nd, 3rd, 4th or as superscript, 1st, 2nd, 3rd, 4th). Also commonly encountered are the superscript or superior (and often underlined) masculine ordinal indicator, , and feminine ordinal indicator, , originally from Romance, but via the cultural influence of Italian by the 18th century, widely used in the wider cultural sphere of Western Europe, as in 1º primo and 1ª prima "first, chief; prime quality". The practice of underlined (or doubly underlined) superscripted abbreviations was common in 19th-century writing (not limited to ordinal indicators in particular, and also extant in the numero sign ), and was also found in handwritten English until at least the late 19th century (e.g.
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Formally, a measurable cardinal is an uncountable cardinal number κ such that there exists a κ-additive, non-trivial, 0-1-valued measure on the power set of κ. (Here the term κ-additive means that, for any sequence Aα, α<λ of cardinality λ < κ, Aα being pairwise disjoint sets of ordinals less than κ, the measure of the union of the Aα equals the sum of the measures of the individual Aα.) Equivalently, κ is measurable means that it is the critical point of a non-trivial elementary embedding of the universe V into a transitive class M. This equivalence is due to Jerome Keisler and Dana Scott, and uses the ultrapower construction from model theory. Since V is a proper class, a technical problem that is not usually present when considering ultrapowers needs to be addressed, by what is now called Scott's trick. Equivalently, κ is a measurable cardinal if and only if it is an uncountable cardinal with a κ-complete, non-principal ultrafilter.
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The Czech Republic hosts two different standardized forms of date and time writing. The Rules of Czech Orthography are mandatory for the educational system. These rules are based on tradition and are widely used by common people. The date is written in "day month year" order, each part separated by a space. Day and month are written as ordinal numbers and year as a cardinal number (1. 12. 2009). The month can be replaced by its full name in genitive case (1. prosince 2009). Writing the month in Roman digits (1. XII. 2009) is considered archaic. The time of day format is dot separated hours and minutes without a space (3.15). However to express time period the colon must be used (3:15). The second format is defined by the Czech State Norm (ČSN 01 6910) based on ISO standards. It accepts the ISO format (2009-12-01 and 03:15) and allows simplified traditional formatting and/or globalised formatting such as leading zeroes or omitted spaces (01.12.2009).
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In musical composition, the opus number is the "work number" that is assigned to a musical composition, or to a set of compositions, to indicate the chronological order of the composer's production. Opus numbers are used to distinguish among compositions with similar titles; the word is abbreviated as "Op." for a single work, or "Opp." when referring to more than one work. To indicate the specific place of a given work within a music catalogue, the opus number is paired with a cardinal number; for example, Beethoven's Piano Sonata No. 14 in C-sharp minor (1801) (nicknamed Moonlight Sonata) is "Opus 27, No. 2", whose work-number identifies it as a companion piece to "Opus 27, No. 1" (Piano Sonata No. 13 in E-flat major, 1800–01), paired in same opus number, with both being subtitled Sonata quasi una Fantasia, the only two of the kind in all of Beethoven's 32 piano sonatas. Furthermore, the Piano Sonata, Op. 27 No. 2, in C-sharp minor is also catalogued as "Sonata No. 14", because it is the fourteenth sonata composed by Ludwig van Beethoven.
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