Confusion about cofinality

Question

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:

1. The ordinal $\mathrm{cf}\,\mathrm{ord}[0,\kappa)$
2. The cardinal $|\mathrm{cf}\,\mathrm{ord}[0,\kappa)|$
3. The ordinal $\mathrm{cf}\,\eta({\kappa})$
4. The cardinal $|\mathrm{cf}\,\eta({\kappa})|$
5. Something else???

I honestly can't work it out.

Answer 1

One can formulate two forms of cofinalities:

1. Ordinal cofinality, which is the least order type of an unbounded set.
2. Cardinal cofinality, which is the least cardinality of a partition that all its parts are smaller then the original cardinal.

Some facts that are useful to know:

• Both versions have the property that the resulting ordinal is always a regular cardinal. That is, the cofinality of the cofinality is the original cofinality.

• If $\delta$ is an initial ordinal, i.e. a cardinal, then both cofinalities are equal.

We are not interested not usually interested in the cofinality of $\text{ord}[0,\kappa)$, because it will usually not coincide with either the cofinalities above. For example, $\text{ord}[0,\omega_1)=\omega+1$, and its cofinality as an ordinal is $1$. That is very uninteresting, and it doesn't tell us much. Whereas knowing that $\omega_1$ is a regular cardinal tells us plenty.

And as for your question $0$. If the same context uses both ordinal and cardinal arithmetics, then there is benefit to using $\aleph_\alpha$ instead of $\omega_\alpha$, to discern the cardinal and ordinal arithmetics. But in many cases people use $\kappa,\lambda,\mu,\nu$ for cardinals and $\alpha,\beta,\gamma,\delta$ for ordinals so it's possible to understand from the context of the letters, Shelah is a prime example and many of his papers begin with this. (Of course they always say whether or not a letter denotes a cardinal or an ordinal.)

Answer 2

Here is an ordinal-free way of defining cofinality of a cardinal.

The cofinality of a cardinal $\kappa$ is the least cardinal $\lambda$ for which there exists a set $X$ of cardinality $\lambda$ such that every member of $X$ is a set of cardinality $< \kappa$ and $\coprod_{x \in X} x$ has cardinality equal to $\kappa$.

A slightly cleaner definition is available if you permit cardinal arithmetic:

The cofinality of a cardinal $\kappa$ is the least cardinal $\lambda$ for which there exists a set $X$ of cardinality $\lambda$ and a function $f : X \to [0, \kappa)$ such that $\sum_{x \in X} f (x) = \kappa$.

Of course, these both compute the same answer as the standard definition (exercise!), but have the advantage of making sense in contexts where the well-ordering principle is not available.