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| In the mathematical discipline of [[set theory]], a '''cardinal characteristic of the continuum''' is an infinite [[cardinal number]] that may consistently lie strictly between [[Aleph null|<math>\aleph_0</math>]] (the [[cardinality]] of the set of [[natural number]]s), and the [[cardinality of the continuum]], that is, the cardinality of the set <math>\mathbb R</math> of all [[real number]]s. The latter cardinal is denoted <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 what relations between them are provable, and constructing models of set theory for various [[consistency|consistent]] configurations of them.
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| == Background ==
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| [[Cantor's diagonal argument]] shows that <math>\mathfrak c</math> is strictly greater than <math>\aleph_0</math>, but it does not specify whether it is the ''least'' cardinal greater than <math>\aleph_0</math> (that is, <math>\aleph_1</math>). Indeed the assumption that <math>\mathfrak c=\aleph_1</math> is the well-known [[Continuum Hypothesis]], which was shown to be independent of the standard [[Zermelo–Fraenkel set theory|ZFC]] axioms for set theory by [[Paul Cohen (mathematician)|Paul Cohen]]. If the Continuum Hypothesis fails and so <math>\mathfrak c</math> is at least <math>\aleph_2</math>, natural questions arise about the cardinals strictly between <math>\aleph_0</math> and <math>\mathfrak c</math>, for example regarding Lebesgue measurability. By considering the least cardinal with some property, one may get a definition for an uncountable cardinal that is consistently less than <math>\mathfrak c</math>. Generally one only considers definitions for cardinals that are provably 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 are all equal to <math>\aleph_1</math>.
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| == Examples == | |
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| As is standard, we denote by <math>\omega</math> the least infinite [[Ordinal number|ordinal]], which has cardinality <math>\aleph_0</math> and may be identified with the set of all natural numbers.
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| A number of cardinal characteristics naturally arise as [[Cardinal function|cardinal invariants]] for [[Ideal (set theory)|ideals]] which are closely connected with the structure of the reals, such as the ideal of [[Null set#Lebesgue measure|Lebesgue null sets]] and the ideal of [[meagre set]]s.
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| === non(N) ===
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| The cardinal characteristic non(<math>\mathcal N</math>) is the least cardinality of a [[non-measurable set]]; equivalently, it is the least cardinality of a set that is not a [[Null set#Lebesgue measure|Lebesgue null set]]. | |
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| ===Bounding number <math>\mathfrak b</math> and dominating number <math>\mathfrak d</math>===
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| We denote by <math>\omega^\omega</math> the set of functions from <math>\omega</math> to <math>\omega</math>. For any two functions <math>f:\omega\to\omega</math> and <math>g:\omega\to\omega</math> we denote by <math>f\leq^*g</math> the statement that for all but finitely many <math>n\in\omega, f(n)\leq g(n)</math>. The bounding number <math>\mathfrak b</math> is the least cardinality of an unbounded set in this relation, that is, <math>\mathfrak b = \min(\{|F| : F\subseteq\omega^\omega\land\forall f:\omega\to\omega\exists g\in F(g\nleq^*f)\}).</math>
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| The dominating number <math>\mathfrak d</math> is the least cardinality of a set of functions from <math>\omega</math> to <math>\omega</math> such that every such function is dominated by (that is, <math>\leq^*</math>) a member of that set, that is,
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| <math>\mathfrak d = \min(\{|F| : F\subseteq\omega^\omega\land\forall f:\omega\to\omega\exists g\in F(f\leq^*g)\}).</math>
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| Clearly any such dominating set <math>F</math> is unbounded, so <math>\mathfrak b</math> is at most <math>\mathfrak d</math>, and a diagonalisation argument shows that <math>\mathfrak b>\aleph_0</math>. Of course if <math>\mathfrak c=\aleph_1</math> this implies that <math>\mathfrak b=\mathfrak d=\aleph_1</math>, but Hechler<ref>Stephen Hechler. On the existence of certain cofinal subsets of <math>{}^\omega\omega</math>. In T. Jech (ed), ''Axiomatic Set Theory, Part II.'' Volume 13(2) of ''Proc. Symp. Pure Math.'', pp 155–173. American Mathematical Society, 1974</ref> has shown that it is also consistent to have <math>\mathfrak b</math> strictly less than <math>\mathfrak d</math>.
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| === Splitting number <math>\mathfrak s</math> and reaping number <math>\mathfrak r</math> ===
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| We denote by <math>[\omega]^\omega</math> the set of all infinite subsets of <math>\omega</math>. For any <math>a,b\in[\omega]^\omega</math>, we say that <math>a</math> ''splits'' <math>b</math> if both <math>b\cap a</math> and <math>b\setminus a</math> are infinite. The ''splitting number'' <math>\mathfrak s</math> is the least cardinality of a subset <math>S</math> of <math>[\omega]^\omega</math> such that for all <math>b\in[\omega]^\omega</math>, there is some <math>a\in S</math> such that <math>a</math> splits <math>b</math>. That is, <math>\mathfrak s = \min(\{|S| : S\subseteq[\omega]^\omega\land\forall b\in[\omega]^\omega\exists a\in S(|b\cap a|=\aleph_0\land |b\setminus a|=\aleph_0)\}).</math>
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| The ''reaping number'' <math>\mathfrak r</math> is the least cardinality of a subset <math>R</math> of <math>[\omega]^\omega</math> such that no element <math>a</math> of <math>[\omega]^\omega</math> splits every element of <math>R</math>. That is, <math>\mathfrak r = \min(\{|R| : R\subseteq[\omega]^\omega\land\forall a\in[\omega]^\omega\exists b\in R(|b\cap a|<\aleph_0\lor|b\setminus a|<\aleph_0)\}).</math>
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| ===Ultrafilter number <math>\mathfrak u</math>===
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| The ultrafilter number <math>\mathfrak u</math> is defined to be the least cardinality of a [[Filter base#Filter on a set|filter base]] of an [[ultrafilter]] on <math>\omega</math>. Kunen<ref>[[Kenneth Kunen]]. ''Set Theory An Introduction to Independence Proofs''. Studies in Logic and the Foundations of Mathematics vol. 102, Elsevier, 1980</ref> gave a model of set theory
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| in which <math>\mathfrak u=\aleph_1</math> but <math>\mathfrak c=\aleph_{\aleph_1}</math>, and using a [[countable support iteration]] of [[List of forcing notions#Sacks forcing|Sacks forcings]], Baumgartner and Laver<ref>[[James Earl Baumgartner]] and [[Richard Laver]]. Iterated perfect-set forcing. ''Annals of Mathematical Logic'' '''17''' (1979) pp 271–288.</ref>
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| demonstrated a model in which <math>\mathfrak u=\aleph_1</math> and <math>\mathfrak c=\aleph_2</math>.
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| ===Almost disjointness number <math>\mathfrak{a}</math>===
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| Two subsets <math>A</math> and <math>B</math> of <math>\omega</math> are said to be ''almost disjoint'' if <math>|A\cap B|</math> is finite, and a family of subsets of <math>\omega</math> is said to be almost disjoint if its members are pairwise almost disjoint. A ''maximal almost disjoint'' (mad) family of subsets of <math>\omega</math> is thus an almost disjoint family <math>\mathcal{A}</math>
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| such that for every subset <math>X</math> of <math>\omega</math>, there is a set <math>A\in\mathcal{A}</math> such that <math>A</math> and <math>X</math> are not almost disjoint
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| (that is, their intersection is infinite). The almost disjointness number <math>\mathfrak{a}</math> is the least cardinality of an infinite maximal almost disjoint family.
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| A basic result<ref>[[Eric van Douwen]]. The Integers and Topology. In K. Kunen and J.E. Vaughan (eds) ''Handbook of Set-Theoretic Topology. North-Holland, Amsterdam, 1984.</ref> is that | |
| <math>\mathfrak{b}\leq\mathfrak{a}</math>; Shelah<ref>[[Saharon Shelah]]. On cardinal invariants of the continuum. In J. Baumgartner, D. Martin and S. Shelah (eds) ''Axiomatic Set Theory'', Contemporary Mathematics 31, American Mathematical Society, 1984, pp 183-207.</ref> showed that it is consistent to have the strict inequality <math>\mathfrak{b}<\mathfrak{a}</math>.
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| == Cichoń's diagram ==
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| A well known diagram of cardinal characteristics is [[Cichoń's diagram]], showing all relations provable in [[Zermelo–Fraenkel set theory|ZFC]] between 10 cardinal characteristics.
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| == References ==
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| {{Reflist}}
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| == Further reading ==
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| * [[Tomek Bartoszyński]] and Haim Judah. ''Set Theory On the Structure of the Real Line''. A K Peters, 1995.
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| *{{cite book | last = Vaughan | first = Jerry E. | chapter = Chapter 11: Small uncountable cardinals and topology | editor1-last = van Mill | editor1-first = Jan | editor2-last = Reed | editor2-first = George M. | title = Open Problems in Topology | pages = 196–218 | year = 1990 | publisher = [[North-Holland Publishing Company]] | location = Amsterdam | isbn = 0-444-88768-7 | url = http://www.uncg.edu/~vaughanj/Publications/smallcd.pdf | format = PDF | accessdate = December 5, 2011}}
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| *{{cite book | last = Blass | first = Andreas | authorlink = Andreas Blass | chapter = Chapter 6 : Combinatorial Cardinal Characteristics of the Continuum | editor1-last = Foreman | editor1-first = Matthew | editor1-link = Matthew Foreman | editor2-last = Kanamori | editor2-first = Akihiro | editor2-link = Akihiro Kanamori | title = Handbook of Set Theory | volume = 1 | pages = 395–490 | date = January 12, 2010 | publisher = [[Springer Science+Business Media|Springer]] | isbn = 1-4020-4843-2 | url = http://www.math.lsa.umich.edu/~ablass/hbk.pdf | format = PDF | accessdate = December 5, 2011}}
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| *{{cite book | last = Bartoszyński | first = Tomek | chapter = Chapter 7 : Invariants of Measure and Category | editor1-last = Foreman | editor1-first = Matthew | editor2-last = Kanamori | editor2-first = Akihiro | title = Handbook of Set Theory | volume = 1 | pages = 491–556 | date = January 12, 2010 | publisher = Springer | isbn = 1-4020-4843-2 | arxiv = math.LO/9910015}}
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| [[Category:Cardinal numbers]]
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