Definitions

Cichon's diagram

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 invariants. All these cardinals are greater or equal than aleph_1, the smallest uncountable cardinal, and they are bounded above by 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).

Definitions

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

  • {rm add}(I)=min.

    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 aleph_0; if I is a σ-ideal, then add(I)≥aleph_1.

    • {rm cov}(I)=min
    .

    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).

    • {rm non}(I)=min
    ,

    The "uniformity number" of I (sometimes also written {rm unif}(I)) is the size of the smallest set not in I. By our assumption on I, add(I) ≤ non(I).

    • {rm cof}(I)=min

    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).

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

    • {mathfrak b}=minbig
    ,
  • {mathfrak d}=minbig
  • ,

    where "exists^infty nin{mathbb N}" means: "there are infinitely many natural numbers n such that...", and "forall^infty nin{mathbb N}" means "for all except finitely many natural numbers n we have...".

    Diagram

    Let {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 {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 "ab".

    > >
    {rm cov}({mathcal L}) longrightarrow {rm non}({mathcal K}) longrightarrow {rm cof}({mathcal K}) longrightarrow {rm cof}({mathcal L}) longrightarrow 2^{aleph_0}
    Bigguparrow uparrow uparrow Bigguparrow
    {mathfrak b} longrightarrow {mathfrak d}
    uparrow uparrow
    aleph_1 longrightarrow {rm add}({mathcal L}) longrightarrow {rm add}({mathcal K}) longrightarrow {rm cov}({mathcal K}) longrightarrow {rm non}({mathcal L})

    In addition, the following relations hold:

    {rm add}({mathcal K})=min{{rm cov}({mathcal K}),{mathfrak b}} and {rm cof}({mathcal K})=max{{rm non}({mathcal K}),{mathfrak d}}.

    It turns out that the relations described by the diagram, together with the inequalities mentioned above, are "all" the relations between these cardinals that are provable in ZFC, in the following sense: whenever we assign the cardinals aleph_1 and aleph_2 to the 10 cardinals in Cichoń's diagram in a way that is consistent with the diagram (that is, there is never an arrow from aleph_2 to aleph_1) and with the two additional relations, then this assignment is realized in some model of ZFC.

    Some inequalities in the diagram (such as "add ≤ cov") follow immediately from the definitions. The inequalities {rm cov}({mathcal K}) le {rm non}({mathcal L}) and {rm cov}({mathcal L}) le {rm non}({mathcal K}) 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 Wrocław mathematician Jacek Cichoń.

    The continuum hypothesis, of 2^{aleph_0} being equal to aleph_1, would make all of these arrows equalities.

    Slightly weaker than the CH, Martin's axiom would imply that all cardinals in the diagram, except perhaps aleph_1, are equal to 2^{aleph_0}.

    References

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