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Cichoń’s diagram. (English) Zbl 0559.03029
Publ. Math. Univ. Pierre Marie Curie 66, Sémin. Initiation Anal. 23ème Année-1983/84, Exp. No.5, 13 p. (1984).
In this note I collect results of F. Rothberger, A. W. Miller, T. Bartoszyński, J. Raisonnier and J. Stern on the relationships between ten cardinal numbers lying between $\omega\sb 1$ and the continuum. If ${\cal I}$ is an ideal of sets such that $\cup {\cal I}=X\not\in {\cal I}$, write add(${\cal I})=\min \{\#({\cal E}):$ ${\cal E}\subseteq {\cal I}$, $\cup {\cal E}\not\in {\cal I}\}$; non(${\cal I})=\min \{\#(A):$ $A\subseteq X$, $A\not\in {\cal I}\}$; cov(${\cal I})=\min \{\#({\cal E}):$ ${\cal E}\subseteq {\cal I}$, $\cup {\cal E}=X\}$; cf(${\cal I})=\min \{\#({\cal E}):$ ${\cal E}\subseteq {\cal I}$, ${\cal E}$ is cofinal with ${\cal I}\}$. Write ${\cal N}$ for the ideal of Lebesgue negligible subsets of ${\bbfR}$, ${\cal M}$ for the ideal of meagre subsets of ${\bbfR}$, ${\cal K}$ for the $\sigma$-ideal of subsets of ${\bbfN}\sp{{\bbfN}}$ generated by the compact sets. We find that non(${\cal K})=add({\cal K})$ and that cov(${\cal K})=cf({\cal K})$; call these cardinals ${\frak b}$ and ${\frak d}$ respectively. All what is known about the cardinals associated with ${\cal N}$, ${\cal M}$ and ${\cal K}$ is captured by a diagram, developed by {\it J. Cichoń}, {\it A. Kamburelis} and {\it J. Pawlikowski} [Proc. Am. Math. Soc. 94, 142-146 (1985)]. In detail, the following is known: $\omega\sb 1\le add({\cal N})\le add({\cal M})\le {\frak b}\le {\frak d}\le cf({\cal M})\le cf({\cal N})\le {\frak c}$, but cov(${\cal N})>{\frak b}$ and cov(${\cal N})<{\frak b}$ are both relatively consistent with ZFC. Further: add(${\cal M})=\min ({\frak b},cov({\cal M}))$, cf(${\cal M})=\max ({\frak d},non({\cal M}))$. I give full proofs of the results establishing the mentioned diagram and references to some of the relevant consistency results.

03E05Combinatorial set theory (logic)
03E10Ordinal and cardinal arithmetic
28A05Classes of sets
03E35Consistency; independence results (set theory)