Exercise 20, § 6

(a) For a cardinal $\alpha$ to be regular (Exercise 17) it is necessary that for every cardinal $\beta\neq0$ we should have

(Use Exercise 19 and consider separately the cases (i) $\beta$ is finite, (ii) $\aleph_0\leq\beta<\alpha$, (iii) $\beta\geq\alpha$; also use Exercise 3 of § 3.). The generalized continuum hypothesis implies that the above condition is also sufficient.

(b) Show that, if a cardinal $\alpha$ is such that $\alpha^m=\alpha$ for every cardinal $m$ such that $0<m<\alpha$, then $\alpha$ is regular (use Exercise 3 of § 3).

(c) Show that the proposition “for every regular cardinal $\alpha$ and every cardinal $m$ such that $0<m<\alpha$, we have $\alpha=\alpha^m$” is equivalent to the generalized continuum hypothesis (use (a)).