ⓘ Cichons diagram. In set theory, Cichons diagram or Cichons diagram is a table of 10 infinite cardinal numbers related to the set theory of the reals displaying ..

                                     

ⓘ Cichons diagram

In set theory, Cichons diagram or Cichons diagram is a table of 10 infinite cardinal numbers related to the set theory of the reals displaying the provable relations between these cardinal characteristics of the continuum. All these cardinals are greater than or equal to ℵ 1 {\displaystyle \aleph _{1}}, the smallest uncountable cardinal, and they are bounded above by 2 ℵ 0 {\displaystyle 2^{\aleph _{0}}}, the cardinality of the continuum. Four cardinals describe properties of the ideal of sets of measure zero; four more describe the corresponding properties of the ideal of meager sets.

                                     

1. Definitions

Let I be an ideal of a fixed infinite set X, containing all finite subsets of X. We define the following "cardinal coefficients" of I:

  • add ⁡ I = min { | A |: A ⊆ I ∧ ⋃ A ∉ I }. {\displaystyle \operatorname {add} I=\min\{|{\mathcal {A}}|:{\mathcal {A}}\subseteq I\wedge \bigcup {\mathcal {A}}\notin I{\big \}}.}
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 ℵ 0 {\displaystyle \aleph _{0}} ; if I is a σ-ideal, then addI ≥ ℵ 1 {\displaystyle \aleph _{1}}.
  • cov ⁡ I = min { | A |: A ⊆ I ∧ ⋃ A = X }. {\displaystyle \operatorname {cov} I=\min\{|{\mathcal {A}}|:{\mathcal {A}}\subseteq I\wedge \bigcup {\mathcal {A}}=X{\big \}}.}
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 addI ≤ covI.
  • non ⁡ I = min { | A |: A ⊆ X ∧ A ∉ I }, {\displaystyle \operatorname {non} I=\min\{|A|:A\subseteq X\ \wedge \ A\notin I{\big \}},}
The "uniformity number" of I sometimes also written unif ⁡ I {\displaystyle \operatorname {unif} I}) is the size of the smallest set not in I. By our assumption on I, addI ≤ nonI.
  • cof ⁡ I = min { | B |: B ⊆ I ∧ ∀ A ∈ I ∃ B ∈ B A ⊆ B }. {\displaystyle \operatorname {cof} I=\min\{|{\mathcal {B}}|:{\mathcal {B}}\subseteq I\wedge \forall A\in I\exists B\in {\mathcal {B}}A\subseteq B{\big \}}.}
The "cofinality" of I is the cofinality of the partial order I, ⊆. It is easy to see that we must have nonI ≤ cofI and covI ≤ cofI.

Furthermore, the "bounding number" or "unboundedness number" b {\displaystyle {\mathfrak {b}}} and the "dominating number" d {\displaystyle {\mathfrak {d}}} are defined as follows:

  • b = min { | F |: F ⊆ N ∧ ∀ g ∈ N ∃ f ∈ F ∃ ∞ n ∈ N g n < f n) }, {\displaystyle {\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} }gn
                                     
  • Cichon may refer to: Aleksander Cichon Przemyslaw Cichon Thomas Cichon Cichon s diagram named after Jacek Cichon
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  • Peters, Ltd., Wellesley, MA, 1995. xii 546 pp. ISBN 1 - 56881 - 044 - X Cichon s diagram Baire property Tomek Bartoszynski at the Mathematics Genealogy Project
  • This is a major area of study in the set theory of the real line see Cichon diagram MA has a tendency to set most interesting cardinal invariants equal
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