Notre Dame Journal of Formal Logic 45 (3):129-146 (2004)

Abstract
For cardinals we let be the smallest size of a subset B of unbounded in the sense of ; that is, such that there is no function such that has size less than for all . Similarly for , the general dominating number, which is the smallest size of a subset B of such that for every there is an such that the above set has size less than . These cardinals are generalizations of the usual ones for . When all three are the same regular cardinal, the relationships between them have been completely described by Cummings and Shelah. We also consider some variants of the functions, following van Douwen, in particular the version of in which B is required to consist of strictly increasing functions. Some of the main results of this paper are: (1) ; (2) for , always exists; (3) if , then ; (4) . For background see Section 1 of the paper. Several open problems are stated
Keywords boundedness   dominating   scale
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DOI 10.1305/ndjfl/1099080208
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Cardinal Invariants Above the Continuum.James Cummings & Saharon Shelah - 1995 - Annals of Pure and Applied Logic 75 (3):251-268.

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