Notre Dame Journal of Formal Logic

On General Boundedness and Dominating Cardinals

J. Donald Monk


For cardinals $\kappa,\lambda,\mu$ we let $\mathfrak{b}_{\kappa,\lambda,\mu}$ be the smallest size of a subset B of $^\lambda\mu$ unbounded in the sense of $\leq_\kappa$; that is, such that there is no function $f\in{}^\lambda\mu$ such that $\{\alpha<\lambda:g(\alpha)>f(\alpha)\}$ has size less than $\kappa$ for all $g\in B$. Similarly for $\mathfrak{d}_{\kappa,\lambda,\mu}$, the general dominating number, which is the smallest size of a subset B of $^\lambda\mu$ such that for every $g\in{}^\lambda\mu$ there is an $f\in B$ such that the above set has size less than $\kappa$. These cardinals are generalizations of the usual ones for $\kappa=\lambda=\mu=\omega$. 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 $\mathfrak{b}^{\uparrow}_{\kappa,\lambda,\mu}$ of $\mathfrak{b}_{\kappa,\lambda,\mu}$ in which B is required to consist of strictly increasing functions. Some of the main results of this paper are: (1) $\mathfrak{b}_{\mu,\mu,{\rm cf}\mu}\leq\mathfrak{b}_{{\rm cf}\mu,{\rm cf}\mu,{\rm cf}\mu}$; (2) for $\lambda\leq\mu$, $\mathfrak{b}^{\uparrow}_{\kappa,\lambda,\mu}$ always exists; (3) if $\mathrm{cf}\lambda= \mathrm{cf}\mu<\lambda\leq\mu$, then $\mathfrak{b}_{{\rm cf}\mu,{\rm cf}\mu,{\rm cf}\mu}= \mathfrak{b}^{\uparrow}_{\lambda,\lambda,\mu}$; (4) $\mathfrak{d}_{\omega,\mu,\mu}=\mathfrak{d}_{1,\mu,\mu}$. For background see Section 1 of the paper. Several open problems are stated.

Article information

Notre Dame J. Formal Logic, Volume 45, Number 3 (2004), 129-146.

First available in Project Euclid: 29 October 2004

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 03E10: Ordinal and cardinal numbers
Secondary: 03E35: Consistency and independence results

boundedness dominating scale


Monk, J. Donald. On General Boundedness and Dominating Cardinals. Notre Dame J. Formal Logic 45 (2004), no. 3, 129--146. doi:10.1305/ndjfl/1099080208.

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