Cichoń's diagram

In set theory, Cichoń's diagram or Cichon's 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 (first category sets). add ⁡ ( K ) = min { cov ⁡ ( K ) , b } {displaystyle operatorname {add} ({mathcal {K}})=min{operatorname {cov} ({mathcal {K}}),{mathfrak {b}}}} and cof ⁡ ( K ) = max { non ⁡ ( K ) , d } . {displaystyle operatorname {cof} ({mathcal {K}})=max{operatorname {non} ({mathcal {K}}),{mathfrak {d}}}.} In set theory, Cichoń's diagram or Cichon's 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 (first category sets). 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: Furthermore, the 'bounding number' or 'unboundedness number' b {displaystyle {mathfrak {b}}} and the 'dominating number' d {displaystyle {mathfrak {d}}} are defined as follows: where ' ∃ ∞ n ∈ N {displaystyle exists ^{infty }nin {mathbb {N} }} ' means: 'there are infinitely many natural numbers n such that...', and ' ∀ ∞ n ∈ N {displaystyle forall ^{infty }nin {mathbb {N} }} ' means 'for all except finitely many natural numbers n we have...'. Let K {displaystyle {mathcal {K}}} be the σ-ideal of those subsets of the real line which are meager (or 'of the first category') in the euclidean topology, and let L {displaystyle {mathcal {L}}} 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 a to b is to be read as meaning that a ≤ b):