# The cofinality of $(\mathbb{N}^\kappa,\le)$ for uncountable $\kappa$?

For a partially ordered set $P$, a set $A\subseteq P$ is cofinal if for each element of $P$ there is a larger element in $A$. The cofinality of $P$, ${\rm cof}(P)$, is the minimal cardinality of a cofinal family in $P$.

Let $\kappa$ be an infinite cardinal number. Consider the set $\mathbb{N}^\kappa$ of all functions from $\kappa$ to $\mathbb{N}$ (equivalently, $\kappa$-sequences of natural numbers), with the pointwise partial order: $f\le g$ if $f(\alpha)\le g(\alpha)$ for all $\alpha<\kappa$.

Let $$f(\kappa):={\rm cof}(\mathbb{N}^\kappa,\le).$$ The cardinal $f(\aleph_0)$ is the well-understood dominating number $\mathfrak{d}$, which is consistently any cardinal of uncountable cofinality that is not larger than the continuum (see Blass).

Basic facts include:

1. For cardinal numbers $\kappa\le\mu$, we have that $f(\kappa)\le f(\mu)$: A projection of a cofinal family in $\mathbb{N}^\mu$ on the first $\kappa$ corrdinates is cofinal in $\mathbb{N}^\kappa$.
2. $\kappa<f(\kappa)$: Given $\{h_\alpha:\alpha<\kappa\}\subseteq\mathbb{N}^\kappa$, define $h(\alpha):=h_\alpha(\alpha)+1$ for all $\alpha<\kappa$.
3. The value of $f(\kappa)$ with "$\le$" replaced by eventual dominance remains the same (see Comfort).

Some deeper results:

1. Jech-Prikry: Assume that $2^{\aleph_0}$ is regular and smaller than $2^{\aleph_1}$. If $f(\aleph_1)=2^{\aleph_0}$, then there is an inner model with a measurable cardinal.
2. If $\kappa^{\aleph_0}=\kappa$, then $f(\kappa)=2^\kappa$ (see Hathaway).

I did not conduct an extensive literature search. I would appreciate your pointing out relevant results that are not mentioned here.

Question: Are there additional cardinals $\kappa$ for which $f(\kappa)$ can be evaluated in ZFC?

(The question also applies to large cadrinals. I do not request that the cardinals provably exist.)

Update. Intially, I also asked whether $\kappa^{\aleph_0}=\kappa$ could be weakened to ${\rm cof}([\kappa]^{\aleph_0},\subseteq)=\kappa$. Todd Eisworth reminds in the comments below that the latter hypothesis is true in ZFC for $\aleph_1$.

Strangely, I couldn't find an explicit reference for the following.

Question. Is it consistent (modulo suitable large cardinals) that $f(\aleph_1)<2^{\aleph_1}$? Assuming that (See Assaf Rinot's comment below), is there an explicit reference for that?

Motivation: The values $f(\kappa)$ pop up whenever I study (usually, jointly with colleagues) the character and other cardinal invariants of topological groups. They are unavoidable.

(Comment: By no means do I mean that the symbol $f(\kappa)$ should be reserved for the function thus defined. This is just a short, temporary name for brevity.)

• If the weakened implication in (2) holds then we would have $f(\aleph_1)=2^{\aleph_1}$ in ZFC. – Todd Eisworth Oct 18 '15 at 18:38
• @ToddEisworth could you provide some details, or a reference? – Boaz Tsaban Oct 18 '15 at 18:58
• A wild guess concerning the consistency of $f(\aleph_1)<2^{\aleph_1}$: have a careful look at Komjath-Shelah model of [F1445]. – saf Oct 18 '15 at 20:40
• Just observe that the cofinality of $([\aleph_1]^{\aleph_0},\subseteq)$ is $\aleph_1$, as witnessed by initial segments. Same holds for all the $\aleph_n$ by induction. – Todd Eisworth Oct 18 '15 at 22:28
• @BoazTsaban: Jech's Chapter 29 mentions that the question of whether $f(\aleph_1)<2^{\aleph_1}$ is consistent is open, and I seem to recall that the question was also mentioned in the original "green version" of his book from 1978. – Todd Eisworth Oct 21 '15 at 22:23