This question is a followup to another one concerning the Hamel basis cardinalities (i.e., dimension qua vector space) of Hilbert spaces. Nik Weaver proved in an answer that they are exactly the cardinals of the form $\lambda^{\aleph_0}$ for some $\lambda\geq 2$. The issue arose in the comments as to whether this is a complete characterization; so as suggested in those comments, I am opening a new question as to whether we can go further than this. Of course it may be a little vague:
Question: Can we give a “better characterization” of the cardinals of the form $\lambda^{\aleph_0}$? Or is there some reason why we can't?
(To point out the obvious, saying that $\kappa$ is of the form $\lambda^{\aleph_0}$ is equivalent to saying $\kappa = \kappa^{\aleph_0}$.)
Here's what I know: first, every $\kappa = \lambda^{\aleph_0}$ must be $\geq\mathfrak{c} := 2^{\aleph_0}$ (trivially) and of uncountable cofinality (because $\newcommand{cf}{\operatorname{cf}}\cf \kappa = \cf \kappa^{\aleph_0} > \aleph_0$ (Jech, Set Theory (Third Millennium Edition), corollary 5.13).
Conversely, if $\kappa$ is $\geq\mathfrak{c}$ and of uncountable cofinality and if the Singular Cardinals Hypothesis (SCH) holds, then $\kappa = \kappa^{\aleph_0}$. This follows from Jech, op. cit., theorem 5.22(ii)(b).
I understand, however, that it is known (by constructions of Magidor and/or Shelah? I don't know and I don't have a precise reference¹) that, under certain large cardinal assumptions, it is consistent that $2^{\aleph_0} = \aleph_1$ but $\aleph_\omega^{\aleph_0} = \aleph_{\omega+2}$: in this case, $\aleph_{\omega+1}^{\aleph_0} = \aleph_\omega^{\aleph_0} = \aleph_{\omega+2}$ so that $\aleph_{\omega+1}$ is not a $\lambda^{\aleph_0}$ despite being $\geq\mathfrak{c}$ and of uncountable cofinality. So “$\geq\mathfrak{c}$ and of uncountable cofinality” does not constitute a characterization of the $\lambda^{\aleph_0}$ in ZFC alone.
But this does not rule out the existence of such a characterization (and, of course, it can't really be ruled out, since $\kappa = \kappa^{\aleph_0}$ is a “characterization” for some definition of “characterization”). Maybe some criterion can be given that involves other things than comparing sizes and cofinalities?
- I got the information from Gitik & Magidor, “The Singular Cardinal Hypothesis Revisited” (p. 243–279 in: Judah, Just & Woodin (eds.), Set Theory of the Continuum (1992)), in the paragraph at the bottom of p. 245, which isn't very clear as to who proved what.
