The Cichon’s Diagram is a diagram that shows the relationships among ten small cardinals – four cardinals associated with the -ideal of sets of Lebesgue measure zero, four cardinals associated with the -ideal of sets of meager sets, the bounding number , and the dominating number . What makes this interesting is that elements of analysis, topology and set theory flow into the same spot. Here’s the diagram.
In this diagram, means . The preceding three posts (the first post, the second post and the third post) give the necessary definitions and background to understand the diagram. In addition to the above diagram, the following relationships also hold.
The Cardinal Characteristics of a -Ideal
For any -ideal on a set , there are four associated cardinals – , , and . The first one is the additivity number, which is the least number of elements of whose union is not an element of . The second cardinal is called the uniformity number, which is the least cardinality of a subset of that is not an element of . The third cardinal is called the covering number, which is the least cardinality of a subfamily of that is also a covering of . The fourth cardinal is called the cofinality number, which is the least cardinality of a subfamily of that is cofinal in . For more information, see the first post. The four cardinals are related in a way that is depicted in the following diagram. Again, means .
Figure 3 – Cardinal Characteristics of a -Ideal
Figure 3 explains the basic orientation of the Cichon’s Diagram. Filling it with three -ideals produces the Cichon’s Diagram.
The Three -Ideals in the Cichon’s Diagram
Let be the -ideal of bounded subsets of . It is known that (this is called the bounding number) and (this is called the dominating number). The ideal is discussed in this previous post. Let be the -ideal of meager subsets of the real line (this is discussed in this previous post). Let be the -ideal of Lebesgue measure zero subsets of the real line.
Thus the Cichon’s Diagram (Figure 1 above) houses information about three -ideals. The two numbers for the -ideal are situated in the middle of the diagram ( and ). The four numbers for the -ideal are situated in the center portion of the diagram. The four numbers for the -ideal are located on the left side and the right side. The Cichon’s Diagram (Figure 1) is flanked by on the lower left and by continnum on the upper right.
More on the Cichon’s Diagram
One interesting aspect of the Cichon’s Diagram: it is a small diagram with small cardinals where elements of analysis (measure) and topology (category) come together. The following diagram shows the path that includes both the bounding number and the dominating number.
The path circled in the above diagram involves all three -ideals. It is also one of the longest increasing paths in the diagram.
There are fifteen arrows in Figure 1. The proofs of these arrows (or inequalities) require varying degrees of effort. Three are basic information – , and . Because is a -ideal, its additivity number must be uncountable. By definition, . The -ideal has a cofinal subfamily consisting of Borel sets. Thus .
Four of the arrows follow from the relative magnitude of the four cardinals of a -ideal as shown in Figure 3 – , , and .
Three of the arrows are proved in this previous post – , and . The last inequality follows from this fact: if is an unbounded set, then there exist many meager subsets of the real line whose union is a non-meager set, essentially a result in Miller .
The proofs of the remaining five arrows can be found in  – , , , and . The proofs of two additional relationships displayed in Figure 2 can also be found in .
The fifteen arrows in the Cichon’s Diagram represent the only inequalities among the ten cardinals (not counting and ) that are provable in ZFC  and . As illustration, we give an example of non-ZFC provable relation in the next section.
An Example of an Inequality Not Provable in ZFC
In the following diagram, the cardinals and are encircled. These two numbers are not connected by arrows.
We sketch out a proof that no inequalities can be established between and . First Martin’s Axiom (MA) implies that . Topologically, the statement MA () means that any compact Hausdorff space that satisfies the countable chain condition cannot be the union of or fewer many nowhere dense sets. The Martin’s Axiom (MA) is the statement that MA () holds for all less than . It follows that MA implies that cannot be less than and thus . It is always the case that the bounding number is .
On the other hand, in Laver’s model  for the Borel conjecture, . In Laver’s model, every subset of the real line that is of strong measure zero is countable. Since any set with the Rothberger property is of strong measure zero, every subset of the real line that has the Rothberger property is countable in Laver’s model. Let be the least cardinality of a subset of the real line that does not have the Rothberger property. Thus in Laver’s model, . It is well known that ; see Theorem 5 in . Thus in Laver’s model, .
In Laver’s model, . Note that implies that there is an uncountable subset of the real line that is concentrated about , the set of all rational numbers; see Theorem 10.2 in . Any concentrated set is of strong measure zero; see Theorem 3.1 in . Thus it must be the case that in Laver’s model.
The Cichon’s Diagram is a remarkable diagram. It blends elements of analysis and topology into a small diagram. The fifteen arrows shown in the diagram are obviously far from the end of the story. The Cichon’s Diagram had been around for a long time. Much had been written about it. The article  posted some questions about the diagram. See , ,  and  for further information on the cardinals in the diagram.
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