Exercise 22, § 6

A cardinal $\aleph_\alpha$ is said to be inaccessible if the ordinal $\omega_\alpha$ is inaccessible (Exercise 16 (b)). We have then $\omega_\alpha=\alpha$ if $\omega_\alpha\neq\omega_0$. A cardinal $\alpha$ is said to be strongly inaccessible if it is inaccessible and dominant (Exercise 21).

(a) The generalized continuum hypothesis implies that every inaccessible cardinal is strongly inaccessible.

(b) For a cardinal $\mathfrak{a} \geqslant 3$ to be strongly inaccessible it is necessary and sufficient that, for each family $(\mathfrak{a}_\iota)_{\iota \in I}$ of cardinals such that

for all $\iota \in I$, we should have $\prod_{\iota \in I} \mathfrak{a}_\iota < \mathfrak{a}$.

(c) For an infinite cardinal $\mathfrak{a}$ to be strongly inaccessible it is necessary and sufficient that it should be dominant (Exercise 21) and that it should satisfy one of the following two conditions : (i) $\mathfrak{a}^{\mathfrak{b}} = \mathfrak{a}$ for every cardinal $\mathfrak{b}$ such that $0 < \mathfrak{b} < \mathfrak{a}$; (ii) $\mathfrak{a}^{\mathfrak{b}} = \mathfrak{a}.2^{\mathfrak{b}}$ for every cardinal $\mathfrak{b} > 0$. (Use Exercises 20 and 21.)