Cichoń’s diagram. (English) Zbl 0559.03029

Fremlin, D. H.

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 \(\omega_ 1\) and the continuum. If \({\mathcal I}\) is an ideal of sets such that \(\cup {\mathcal I}=X\not\in {\mathcal I}\), write add(\({\mathcal I})=\min \{\#({\mathcal E}):\) \({\mathcal E}\subseteq {\mathcal I}\), \(\cup {\mathcal E}\not\in {\mathcal I}\}\); non(\({\mathcal I})=\min \{\#(A):\) \(A\subseteq X\), \(A\not\in {\mathcal I}\}\); cov(\({\mathcal I})=\min \{\#({\mathcal E}):\) \({\mathcal E}\subseteq {\mathcal I}\), \(\cup {\mathcal E}=X\}\); cf(\({\mathcal I})=\min \{\#({\mathcal E}):\) \({\mathcal E}\subseteq {\mathcal I}\), \({\mathcal E}\) is cofinal with \({\mathcal I}\}\). Write \({\mathcal N}\) for the ideal of Lebesgue negligible subsets of \({\mathbb{R}}\), \({\mathcal M}\) for the ideal of meagre subsets of \({\mathbb{R}}\), \({\mathcal K}\) for the \(\sigma\)-ideal of subsets of \({\mathbb{N}}^{{\mathbb{N}}}\) generated by the compact sets. We find that non(\({\mathcal K})=add({\mathcal K})\) and that cov(\({\mathcal K})=cf({\mathcal K})\); call these cardinals \({\mathfrak b}\) and \({\mathfrak d}\) respectively. All what is known about the cardinals associated with \({\mathcal N}\), \({\mathcal M}\) and \({\mathcal K}\) is captured by a diagram, developed by J. Cichoń, A. Kamburelis and J. Pawlikowski [Proc. Am. Math. Soc. 94, 142-146 (1985)]. In detail, the following is known: \(\omega_ 1\leq add({\mathcal N})\leq add({\mathcal M})\leq {\mathfrak b}\leq {\mathfrak d}\leq cf({\mathcal M})\leq cf({\mathcal N})\leq {\mathfrak c}\), but cov(\({\mathcal N})>{\mathfrak b}\) and cov(\({\mathcal N})<{\mathfrak b}\) are both relatively consistent with ZFC. Further: add(\({\mathcal M})=\min ({\mathfrak b},cov({\mathcal M}))\), cf(\({\mathcal M})=\max ({\mathfrak d},non({\mathcal M}))\). I give full proofs of the results establishing the mentioned diagram and references to some of the relevant consistency results.

MSC:

03E05	Other combinatorial set theory
03E10	Ordinal and cardinal numbers
28A05	Classes of sets (Borel fields, \(\sigma\)-rings, etc.), measurable sets, Suslin sets, analytic sets
03E35	Consistency and independence results