Has any work been done on generalizing any characteristics of the cardinality of the continuum?

The bounding number $\mathfrak{b}$ and dominating number $\mathfrak{d}$ could be easily generalized for ordinals $\alpha$, as follows:

$$\forall f,g\in\alpha^\alpha(f\leq_{\alpha}g\Leftrightarrow|\{\beta\in\alpha:f(\beta)>g(\beta)\}|<|\alpha|)$$ $$\mathfrak{b}_\alpha=\min\{|F|:F\subseteq\alpha^\alpha\land\forall f\in\alpha^\alpha\exists g\in F(g\not\leq_\alpha f)\}$$ $$\mathfrak{d}_\alpha=\min\{|F|:F\subseteq\alpha^\alpha\land\forall f\in\alpha^\alpha\exists g\in F(f\leq_\alpha g)\}$$

A few points to be made at this point:

  • $\mathfrak{b}_\alpha\leq\max\{|F|:F\subseteq\alpha^{\alpha}\}=|\alpha^\alpha|=2^{|\alpha|}$

  • For cardinals $\kappa,\mathfrak{b}_\kappa>\kappa$ (this is shown by generalized diagonalization)

  • The last two points combined make $\mathfrak{b}_\kappa=\kappa^+$ if $\kappa^+=2^\kappa$ (or if GCH is assumed)

  • $\mathfrak{b}_\omega$ is clearly, in this case, $\mathfrak{b}$, and the same is true with $\mathfrak{d}_\omega$.

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    $\begingroup$ Yes, this is actually a very active area. $\endgroup$ – Andrés E. Caicedo Oct 3 '17 at 14:36
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    $\begingroup$ The introduction to "Sheva-Sheva-Sheva: Large Creatures" by Roslanowski and Shelah contains a lot of references. To the best of my knowledge, a good and reasonably comprehensive survey of these matters is an "open exposition problem" in Set Theory... The paper is available as number 777 at Shelah's archive: shelah.logic.at/files/777.pdf $\endgroup$ – Todd Eisworth Oct 3 '17 at 14:38
  • $\begingroup$ I did not know that Shelah had an easily accessible paper archive. Thank you. Are there any other famous set theory mathematicians with similar archives? $\endgroup$ – Zetapology Oct 4 '17 at 4:21
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    $\begingroup$ See also this paper by S.-D. Friedman. $\endgroup$ – Noah Schweber Oct 12 '17 at 17:49

Don Monk has a paper describing generalized $\mathfrak b$ and $\mathfrak d$ as you describe. In his paper he further generalizes to $\mathfrak b_{\kappa,\lambda,\mu}$ (and analogously for $\mathfrak d$), considering families of functions in ${}^\lambda\mu$ with $\lambda, \mu,$ and $\kappa$ all possibly distinct. I think in his notation the numbers you describe are $\mathfrak b_{\alpha, \alpha, \alpha}$ and $\mathfrak d_{\alpha, \alpha, \alpha}$

Generalized ${\frak b}$ and ${\frak d}$. Notre Dame J. Formal Logic 45 (2004), no. 3, 129--146.


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