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| In set theory,
| | Royal Votaw is my name but I by no means really favored that name. Interviewing is what I do in my working day occupation. Some time in the past I chose to live in Arizona but I need to transfer for my family. One of his favorite hobbies is playing crochet but he hasn't produced a dime with it.<br><br>My site; [http://Www.mantuademocrats.com/ActivityFeed/MyProfile/tabid/60/userId/76489/Default.aspx http://Www.mantuademocrats.com] |
| '''Cichoń's diagram''' or '''Cichon's diagram''' is a table of 10 infinite [[cardinal number]]s related to the [[set theory of the reals]] displaying the provable relations between these
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| [[Cardinal characteristic of the continuum|cardinal characteristics of the continuum]]. All these cardinals are greater than or equal to <math>\aleph_1</math>, the smallest uncountable cardinal, and they are bounded above by <math>2^{\aleph_0}</math>, the [[cardinality of the continuum]]. Four cardinals describe properties of the [[ideal (order theory)|ideal]] of sets of [[Lebesgue measure|measure zero]]; four more describe the corresponding properties of the ideal of [[meagre set|meager sets (first category sets)]].
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| ==Definitions==
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| Let ''I'' be an [[ideal (set theory)|ideal]] of a fixed infinite set ''X'', containing all finite subsets of ''X''. We define the following "cardinal coefficients" of ''I'':
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| *<math>{\rm add}(I)=\min\{|{\mathcal A}|: {\mathcal A}\subseteq I \wedge \bigcup{\mathcal A}\notin I\big\}</math>.
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| ::The "additivity" of ''I'' is the smallest number of sets from ''I'' whose union is not in ''I'' any more. As any ideal is closed under finite unions, this number is always at least <math>\aleph_0</math>; if ''I'' is a σ-ideal, then add(''I'')≥<math>\aleph_1</math>.
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| *<math>{\rm cov}(I)=\min\{|{\mathcal A}|:{\mathcal A}\subseteq I \wedge\bigcup{\mathcal A}=X\big\}</math>.
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| :: The "covering number" of ''I'' is the smallest number of sets from ''I'' whose union is all of ''X''. As ''X'' itself is not in ''I'', we must have add(''I'') ≤ cov(''I'').
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| *<math>{\rm non}(I)=\min\{|A|:A\subseteq X\ \wedge\ A\notin I\big\}</math>,
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| :: The "uniformity number" of ''I'' (sometimes also written <math>{\rm unif}(I)</math>) is the size of the smallest set not in ''I''. By our assumption on ''I'', add(''I'') ≤ non(''I'').
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| *<math>{\rm cof}(I)=\min\{|{\mathcal B}|:{\mathcal B}\subseteq I \wedge (\forall A\in I)(\exists B\in {\mathcal B})(A\subseteq B)\big\}.</math>
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| :: The "cofinality" of ''I'' is the [[cofinality]] of the [[partial order]] (''I'', ⊆). It is easy to see that we must have non(''I'') ≤ cof(''I'') and cov(''I'') ≤ cof(''I'').
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| Furthermore, the "bounding number" or "unboundedness number" <math>{\mathfrak b}</math> and the "dominating number" <math>{\mathfrak d}</math> are defined as follows:
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| *<math>{\mathfrak b}=\min\big\{|F|:F\subseteq{\mathbb N}^{\mathbb N}\ \wedge\ (\forall g\in {\mathbb N}^{\mathbb N})(\exists f\in F)(\exists^\infty n\in{\mathbb N})(g(n)<f(n))\big\}</math>,
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| *<math>{\mathfrak d}=\min\big\{|F|:F\subseteq{\mathbb N}^{\mathbb N}\ \wedge\ (\forall g\in{\mathbb N}^{\mathbb N})(\exists f\in F)(\forall^\infty n\in{\mathbb N})(g(n)<f(n))\big\}</math>,
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| where "<math>\exists^\infty n\in{\mathbb N}</math>" means: "there are infinitely many natural numbers ''n'' such that...", and "<math>\forall^\infty n\in{\mathbb N}</math>" means "for all except finitely many natural numbers ''n'' we have...".
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| ==Diagram==
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| Let <math>{\mathcal K}</math> be the σ-ideal of those subsets of the real line which are [[meager set|meager]] (or "of the first category") in the [[euclidean topology]], and let
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| <math>{\mathcal L}</math> be the σ-ideal of those subsets of the real line which are of [[Lebesgue measure]] zero. Then the following inequalities hold (where an arrow from {{math|''a''}} to {{math|''b''}} is to be read as meaning that {{math|''a'' ≤ ''b''}}):
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| <center> | |
| {| align="center" |
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| |-
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| | || || <math>{\rm cov}({\mathcal L})</math> || <math>\longrightarrow</math>
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| | <math>{\rm non}({\mathcal K})</math> || <math>\longrightarrow</math>
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| | <math>{\rm cof}({\mathcal K})</math> ||<math>\longrightarrow</math>
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| | <math>{\rm cof}({\mathcal L})</math> ||<math>\longrightarrow</math> ||<math>2^{\aleph_0}</math>
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| |- align=center |
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| | rowspan=3 | <math> \Bigg\uparrow </math> ||
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| | <math>\uparrow</math> ||
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| | <math>\uparrow</math> ||
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| | rowspan=3 | <math> \Bigg\uparrow </math>
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| |- align=center |
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| | || ||
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| |<math>{\mathfrak b}</math> || <math>\longrightarrow</math>
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| |<math>{\mathfrak d} </math>
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| |- align=center |
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| | <math>\uparrow</math> ||
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| | <math>\uparrow</math>
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| |-
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| |<math>\aleph_1</math> ||<math>\longrightarrow</math>
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| |<math>{\rm add}({\mathcal L})</math> ||<math>\longrightarrow</math>
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| |<math>{\rm add}({\mathcal K})</math> ||<math>\longrightarrow</math>
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| |<math>{\rm cov}({\mathcal K})</math> ||<math>\longrightarrow</math>
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| |<math>{\rm non}({\mathcal L})</math>
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| |}
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|
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| </center>
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| In addition, the following relations hold:
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| <center>
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| <math>{\rm add}({\mathcal K})=\min\{{\rm cov}({\mathcal K}),{\mathfrak b}\}</math> and
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| <math>{\rm cof}({\mathcal K})=\max\{{\rm non}({\mathcal K}),{\mathfrak d}\}</math>.<ref name="survey">
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| {{citation
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| | last = Bartoszyński | first = Tomek | authorlink = Tomek Bartoszyński
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| | year = 2009
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| | contribution = Invariants of Measure and Category
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| | editor-last = Foreman |editor-first=Matthew
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| | title = Handbook of Set Theory
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| | pages = 491–555
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| | publisher = Springer-Verlag
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| | arxiv = math/9910015
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| | doi = 10.1007/978-1-4020-5764-9_8
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| | isbn = 978-1-4020-4843-2
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| }}</ref>
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| </center>
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| It turns out that the inequalities described by the diagram, together with the relations mentioned above, are all the relations between these cardinals that are provable in ZFC, in the following sense. Let ''A'' be any assignment of the cardinals <math>\aleph_1</math> and <math>\aleph_2</math> to the 10 cardinals in Cichoń's diagram. Then, if ''A'' is consistent with the diagram in that there is no arrow from <math>\aleph_2</math> to <math>\aleph_1</math>, and if ''A'' also satisfies the two additional relations, then ''A'' can be realized in some model of [[ZFC]].
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| Some inequalities in the diagram (such as "add ≤ cov") follow immediately from the definitions. The inequalities <math>{\rm cov}({\mathcal K}) \le {\rm non}({\mathcal L})</math> and
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| <math>{\rm cov}({\mathcal L}) \le {\rm non}({\mathcal K})</math> are classical theorems
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| <!-- Sierpinski?? -->and follow from the fact that the real line can be partitioned into a meager set and a set of measure zero.
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| ==Remarks==
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| The British mathematician [[David Fremlin]] named the diagram after the Wrocław mathematician [[Jacek Cichoń]].<ref>{{citation|last=Fremlin|first=David H.|contribution=Cichon's diagram|series=Publ. Math. [[Pierre and Marie Curie University]]|volume=66|title=Sémin. Initiation Anal. 23ème Année-1983/84|id=Exp. No.5, 13 p.,|year=1984|zbl=0559.03029}}.</ref>
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| The [[continuum hypothesis]], of <math>2^{\aleph_0}</math> being equal to <math>\aleph_1</math>, would make all of these arrows equalities.
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| [[Martin's axiom]], a weakening of CH, implies that all cardinals in the diagram (except perhaps <math>\aleph_1</math>) are equal to <math>2^{\aleph_0}</math>.
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| <!-- add some stuff: Bartoszynski. Miller. Bartoszynski-Judah. Pawlikowski. Pawlikowski-Reclaw -->
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| == References ==
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| <references/>
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| {{DEFAULTSORT:Cichon's diagram}}
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| [[Category:Cardinal numbers]]
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| [[Category:Descriptive set theory]]
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Royal Votaw is my name but I by no means really favored that name. Interviewing is what I do in my working day occupation. Some time in the past I chose to live in Arizona but I need to transfer for my family. One of his favorite hobbies is playing crochet but he hasn't produced a dime with it.
My site; http://Www.mantuademocrats.com