## Notre Dame Journal of Formal Logic

### On General Boundedness and Dominating Cardinals

J. Donald Monk

#### 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

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

#### References

• Cummings, J., and S. Shelah, "Cardinal invariants above the continuum", Annals of Pure and Applied Logic, vol. 75 (1995), pp. 251–68.
• Jech, T., and K. Prikry, "Cofinality of the partial ordering of functions from $\omega \sb{1}$" into $\omega$ under eventual domination, Mathematical Proceedings of the Cambridge Philosophical Society, vol. 95 (1984), pp. 25–32.
• Szymański, A., "Some remarks on real-valued measurable cardinals", Proceedings of the American Mathematical Society, vol. 104 (1988), pp. 596–602.