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 than or equal to $\aleph_1$, 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 a fixed infinite set X, containing all finite subsets of X. We define the following "cardinal coefficients" of I:

• ${\rm add}(I)=\min\{|{\mathcal A}|: {\mathcal A}\subseteq I \wedge \bigcup{\mathcal A}\notin I\big\}$.

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

• ${\rm cov}(I)=\min\{|{\mathcal A}|:{\mathcal A}\subseteq I \wedge\bigcup{\mathcal A}=X\big\}$.

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 add(I) ≤ cov(I).

• ${\rm non}(I)=\min\{|A|:A\subseteq X\ \wedge\ A\notin I\big\}$,

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

• ${\rm cof}(I)=\min\{|{\mathcal B}|:{\mathcal B}\subseteq I \wedge (\forall A\in I)(\exists B\in {\mathcal B})(A\subseteq B)\big\}.$

The "cofinality" of I is the cofinality of the partial order (I, ⊆). It is easy to see that we must have 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:

• ${\mathfrak b}=\min\big\{|F|:F\subseteq{\mathbb N}^{\mathbb N}\ \wedge\ (\forall g\in {\mathbb N}^{\mathbb N})(\exists f\in F)(\exists^\infty n\in{\mathbb N})(g(n)<f(n))\big\}$,
• ${\mathfrak d}=\min\big\{|F|:F\subseteq{\mathbb N}^{\mathbb N}\ \wedge\ (\forall g\in{\mathbb N}^{\mathbb N})(\exists f\in F)(\forall^\infty n\in{\mathbb N})(g(n)<f(n))\big\}$,

where "$\exists^\infty n\in{\mathbb N}$" means: "there are infinitely many natural numbers n such that...", and "$\forall^\infty n\in{\mathbb N}$" 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 meaning that a ≤ b):

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

In addition, the following relations hold:

${\rm add}({\mathcal K})=\min\{{\rm cov}({\mathcal K}),{\mathfrak b}\}$ and ${\rm cof}({\mathcal K})=\max\{{\rm non}({\mathcal K}),{\mathfrak d}\}$.^[1]

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 sense. Let A be any assignment of the cardinals $\aleph_1$ and $\aleph_2$ to the 10 cardinals in Cichoń's diagram. Then, if A is consistent with the diagram in that there is no arrow from $\aleph_2$ to $\aleph_1$, and if A also satisfies the two additional relations, then A can be 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.^[2]

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

Martin's axiom, a weakening of CH, implies that all cardinals in the diagram (except perhaps $\aleph_1$) are equal to $2^{\aleph_0}$.

References

1. ^ Bartoszyński, Tomek (2009), "Invariants of Measure and Category", in Foreman, Matthew, Handbook of Set Theory, Springer-Verlag, pp. 491–555, arXiv:math/9910015, doi:10.1007/978-1-4020-5764-9_8, ISBN 9781402048432 
2. ^ Fremlin, David H. (1984), "Cichon's diagram", Sémin. Initiation Anal. 23ème Année-1983/84, Publ. Math. Univ. Pierre Marie Curie, 66, Zbl 0559.03029, Exp. No.5, 13 p., .