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
, the smallest uncountable cardinal, and they are bounded above by
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"
" means: "there are infinitely many natural numbers n such that …", and "
" means "for all except finitely many natural numbers n we have …".
be the σ-ideal of those subsets of the real line that are meager (or "of the first category") in the euclidean topology, and let
be the σ-ideal of those subsets of the real line that are of Lebesgue measure zero.
Then the following inequalities hold: Where an arrow from
In addition, the following relations hold:
and 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 limited sense.
to the 10 cardinals in Cichoń's diagram.
Then if A is consistent with the diagram's relations, and if A also satisfies the two additional relations, then A can be realized in some model of ZFC.
For larger continuum sizes, the situation is less clear.
It is consistent with ZFC that all of the Cichoń's diagram cardinals are simultaneously different apart from
[2][3][4] Some inequalities in the diagram (such as "add ≤ cov") follow immediately from the definitions.
are classical theorems and follow from the fact that the real line can be partitioned into a meager set and a set of measure zero.
The British mathematician David Fremlin named the diagram after the Polish mathematician from Wrocław, Jacek Cichoń [pl].
, would make all of these relations equalities.
Martin's axiom, a weakening of the continuum hypothesis, implies that all cardinals in the diagram (except perhaps
Similar diagrams can be drawn for cardinal characteristics of higher cardinals
strongly inaccessible, which assort various cardinals between