Exact and reachable cardinals

Question

For any cardinal $\kappa$ we set $2^{<\kappa} = \big|\{A\subseteq \kappa: |A|<\kappa\}\big|$. Let $\kappa$ be an infinite cardinal. We say $\kappa$ is exact if $\kappa = 2^{<\kappa}$, and we say that $\kappa$ is reachable if there is $\lambda < \kappa$ such that $2^\lambda \geq \kappa$.

Note that $\aleph_0$ is exact, but not reachable. Moreover $2^{\aleph_0}$ is reachable, but exactness is not clear to me - it possibly depends on ${\sf CH}$.

For what combinations of "exact" / "reachable" are there examples of cardinals, and/or consistency results?

Answer 1

I'm not sure what kind of answer you're seeking, but here are some elementary things to say about your concepts.

What you would call the non-reachable cardinals are more widely known as the strong limit cardinals, and this is a fundamental cardinal concept in elementary set theory. (A cardinal $\kappa$ is a strong limit cardinal, if $\lambda<\kappa\to 2^\lambda<\kappa$.) So reachable = non-strong-limit. There was an interesting historical shift in cardinal terminology, where at first cardinal notions were formulated in terms of reachability or accessibility, but then later it was realized that actually the negations of those concepts are what is interesting. In this way, one arrives at large cardinals.

Every strong limit cardinal is also what you call exact.

Under the GCH, every infinite cardinal is what you call exact, since this is easy to verify for successor cardinals, and under the GCH every infinite limit cardinal is a strong limit and hence also exact.

Under the GCH, the infinite reachable cardinals are precisely the successor cardinals. In any case, even without the GCH, every successor cardinal is reachable.

A related and often considered property is $\kappa^{<\kappa}=\kappa$, which is not equivalent to $2^{<\kappa}=\kappa$, since singular $\kappa$ never have $\kappa^{<\kappa}=\kappa$, in light of König's theorem that $\kappa^{\text{cof}(\kappa)}>\kappa$.

An inaccessible cardinal is the same thing as a regular strong limit, or a regular non-reachable cardinal in your terminology.