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...o show that a [[real-valued measurable]] cardinal is [[Weakly inaccessible cardinal|weakly inaccessible]].<ref>{{citation | last1=Jech | first1=Thomas | author Suppose that κ and λ are cardinal numbers, and let <math>\mathcal F</math> be a <math>\lambda</math>-complete filter ...

{{Short description|Smallest cardinal number for which a weak downward Löwenheim–Skolem theorem holds}} ...ic]] the '''Löwenheim number''' of an [[abstract logic]] is the smallest [[cardinal number]] for which a weak downward [[Löwenheim–Skolem theorem]] holds.<ref> ...

...umbers, Scott's trick can be used to obtain representatives for [[cardinal numbers]] and more generally for [[Isomorphism class|isomorphism types]], for examp ...es for the equivalence classes. These sets are then taken to "be" cardinal numbers, by definition. ...

...), preprint.</ref>^p.20 and are also called ''additive principal numbers''. The [[class (set theory)|class]] of additively indecomposable ordinals ...finite set|infinite]] [[initial ordinal]] (an ordinal corresponding to a [[cardinal number]]) is additively indecomposable. ...

| caption = spiral figure representing both finite and transfinite ordinal numbers ...ity with the [[cardinality of the continuum|cardinality of the set of real numbers]] can neither be proved nor disproved within the standard version of [[axio ...

{{Short description|The real numbers or their cardinality}} ...{\aleph_0}\!</math>, the cardinality of the [[power set]] of the [[natural numbers]]. ...

...the same as that of <math>\mathbb{R}</math>—the set of [[Real number|real numbers]].<ref>{{Cite web|title=first uncountable ordinal in nLab|url=https://ncatl ...rdinal, we define <math>\omega_\alpha</math> as the initial ordinal of the cardinal <math>\aleph_\alpha</math>. ...

...d <math>2^{\aleph_0}</math> or <math>\mathfrak c</math>. A variety of such cardinal characteristics arise naturally, and much work has been done in determining ...greater than <math>\aleph_0</math> and at most <math>\mathfrak c</math> as cardinal characteristics of the continuum, so if the Continuum Hypothesis holds they ...

...math>< M</math>). Rathjen uses this to diagonalise over the [[Inaccessible cardinal|weakly inaccessible]] hierarchy. [[Category:Ordinal numbers]] ...

...han the [[Successor cardinal|successor]] <math>\kappa^{+}</math> of some [[cardinal number]] <math>\kappa</math> can be written as the union of sets <math>X_1, ...C. | last2=Rado | first2=R. | title=The pigeon-hole principle for ordinal numbers | doi=10.1112/plms/s3-15.1.750 | mr=0190003 | year=1965 | journal= Proceedi ...

...is the set of possible outcomes. E.g., it may be the set of positive real numbers, representing the possible annual [[gross domestic product]]. It is normali == Cardinal definition == ...

If ''r'' ≥ 0 is finite and ''κ'' is an [[infinite cardinal]], then | title = Partition relations for cardinal numbers ...

...tional equation that holds for the real numbers also holds for the complex numbers.<ref>{{Citation|surname=Dauben|given=Joseph W.|year=1979|title=Georg Cantor ...or functions of two variables, this suggests that it holds for all complex numbers as well.<ref name = "Gamelin" >Gamelin, T. ''Complex Analysis'', UTM Series ...

...exists) has [[order type]] <math>\omega</math>. For uncountable [[regular cardinal]]s <math>\kappa</math> (and some other cardinals) this can be strengthened ...

...spline wavelets.gif|thumb|right|Animation showing the compactly supported cardinal B-spline wavelets of orders 1, 2, 3, 4 and 5.]] ...ts. These special wavelets are also called '''B-spline wavelets''' and '''cardinal B-spline wavelets'''.<ref>{{cite book|last1=Charles K Chui|title=An Introdu ...

...>V_\kappa</math> in the [[von Neumann hierarchy]], for some [[inaccessible cardinal]] <math>\kappa</math>.<ref name="HamkinsSolberg22">{{cite arXiv|eprint=2009 ...hold in a model if and only if the model is isomorphic to the true natural numbers.<ref name="MaddyVaananen23" />^{pp.&nbsp;5–6}<ref name="HamkinsSolb ...

...er 6 section 8}} that generalizes the [[order type]] ''η'' of the rational numbers. ...e ''η''₀ set (up to isomorphism) is the ordered set of rational numbers. ...

...ral numbers. Natural numbers including 0 are also sometimes called ''whole numbers''.<ref> ...ath>\mathbb{Z}</math>): Positive and [[negative number|negative]] counting numbers, as well as zero: {..., −3, −2, −1, 0, 1, 2, 3, ...}. ...

...alled [[Meagre set|meagre]]. The rational numbers, while dense in the real numbers, are meagre as a subset of the reals. ...erally, a topological space is called κ-resolvable for a [[Cardinal number|cardinal]] κ if it contains κ pairwise disjoint dense sets. ...

...hesis'' may also refer to the assertion that every uncountable set of real numbers can be placed in bijective correspondence with the set of all reals. This [[Category:Cardinal numbers]] ...