I'm confused about the notion of the cofinality of a cardinal. Since I think the source of the confusion is the Von Neumann cardinal assignment, my first question is:
Question 0. Is there an article or book that purposely distinguishes between a cardinal and its associated initial ordinal?
Anyway, for the duration of this question, lets adopt this distinction. Thus we have two order-isomorphic proper classes, $\mathrm{Ord}$ and $\mathrm{Card}$. And although we have retained the identification $[0,\alpha)=\alpha$ for $\alpha \in \mathrm{Ord}$, we have abandoned it for cardinals. Additionally, given cardinal numbers $\mu$ and $\nu$, lets write $[\mu,\nu)$ for the set of all cardinals $\kappa$ with $\mu \leq \kappa < \nu$.
Furthermore, let $x \mapsto \underline{x}$ denote the unique order isomorphism $\mathrm{Ord}\rightarrow \mathrm{Card}$, so for example: if $\omega$ is the least infinite ordinal, then $\underline{\omega}$ is the least infinite cardinal. Also, let $\eta : \mathrm{Card} \rightarrow \mathrm{Ord}$ denote the proper class function that maps every cardinal number to its initial ordinal. And finally, for every subset $A$ of a well-ordered set, lets write $\mathrm{ord}(A)$ for the unique ordinal that is order-isomorphic to $A$. So in general, we have that $\mathrm{ord}(A) \in \mathrm{Ord}$.
Now my understanding is that the cofinality $\mathrm{cf}(\alpha)$ of an ordinal $\alpha$ is defined as the least ordinal $\beta$ such that there exists a cofinal subset of $\alpha$, call it $B$, such that $\mathrm{ord}(B)=\beta$.
Question 1. Under these definitions, how does one define the cofinality of a cardinal number $\kappa$? Is it:
- The ordinal $\mathrm{cf}\,\mathrm{ord}[0,\kappa)$
- The cardinal $|\mathrm{cf}\,\mathrm{ord}[0,\kappa)|$
- The ordinal $\mathrm{cf}\,\eta({\kappa})$
- The cardinal $|\mathrm{cf}\,\eta({\kappa})|$
- Something else???
I honestly can't work it out.