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, the smallest uncountable cardinal, and they are bounded above by, 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.

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:
Furthermore, the "bounding number" or "unboundedness number" and the "dominating number" are defined as follows:
where "" means: "there are infinitely many natural numbers n such that...", and "" means "for all except finitely many natural numbers n we have...".

Diagram

Let be the σ-ideal of those subsets of the real line which are meager in the euclidean topology, and let
be the σ-ideal of those subsets of the real line which are of Lebesgue measure zero. Then the following inequalities hold :


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. Let A be any assignment of the cardinals and to the 10 cardinals in Cichoń's diagram. Then, if A is consistent with the diagram in that there is no arrow from to, 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 and , but it remains open whether all combinations of the cardinal orderings consistent with the diagram are consistent.
Some inequalities in the diagram follow immediately from the definitions. The inequalities and
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.

Remarks

The British mathematician David Fremlin named the diagram after the Polish mathematician from Wrocław,.
The continuum hypothesis, of being equal to, would make all of these arrows equalities.
Martin's axiom, a weakening of CH, implies that all cardinals in the diagram are equal to.