Notre Dame Journal of Formal Logic
- Notre Dame J. Formal Logic
- Volume 45, Number 3 (2004), 129-146.
On General Boundedness and Dominating Cardinals
Abstract
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
Source
Notre Dame J. Formal Logic, Volume 45, Number 3 (2004), 129-146.
Dates
First available in Project Euclid: 29 October 2004
Permanent link to this document
https://projecteuclid.org/euclid.ndjfl/1099080208
Digital Object Identifier
doi:10.1305/ndjfl/1099080208
Mathematical Reviews number (MathSciNet)
MR2130782
Zentralblatt MATH identifier
1089.03040
Subjects
Primary: 03E10: Ordinal and cardinal numbers
Secondary: 03E35: Consistency and independence results
Keywords
boundedness dominating scale
Citation
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. https://projecteuclid.org/euclid.ndjfl/1099080208