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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 ω 1 and the continuum. If is an ideal of sets such that =X, write add()=min{#(): , }; non()=min{#(A): AX, A}; cov()=min{#(): , =X}; cf()=min{#(): , is cofinal with }. Write 𝒩 for the ideal of Lebesgue negligible subsets of , for the ideal of meagre subsets of , 𝒦 for the σ-ideal of subsets of generated by the compact sets. We find that non(𝒦)=add(𝒦) and that cov(𝒦)=cf(𝒦); call these cardinals 𝔟 and 𝔡 respectively. All what is known about the cardinals associated with 𝒩, and 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: ω 1 add(𝒩)add()𝔟𝔡cf()cf(𝒩)𝔠, but cov(𝒩)>𝔟 and cov(𝒩)<𝔟 are both relatively consistent with ZFC. Further: add()=min(𝔟,cov()), cf()=max(𝔡,non()). I give full proofs of the results establishing the mentioned diagram and references to some of the relevant consistency results.
MSC:
03E05Combinatorial set theory (logic)
03E10Ordinal and cardinal arithmetic
28A05Classes of sets
03E35Consistency; independence results (set theory)