User blog comment:Alemagno12/Some more set theory questions/@comment-1605058-20180102220818

Q3: The answer is \(2^{\aleph_0}\) - by transfinite induction you can show the number of such tuples of depth smaller than \(\alpha\) is at most \(2^{\aleph_0}\) for all \(\alpha<\omega_1\) (using \((2^{\aleph_0})^{\aleph_0}=2^{\aleph_0}\)), so there are at most \(2^{\aleph_0}\cdot\aleph_1=2^{\aleph_0}\) ones of depth smaller than \(\omega_1\). The lower bound is easier.

Q4: Note that for limit ordinals \(\alpha\), \(F(\alpha)\) always has cofinality at least \(\alpha\) (since \(F(\beta)\geq\beta\)). Hence if \(\Delta=F(\Delta)\), then \(\Delta\) is a limit cardinal equal to its cofinality, i.e. it's weakly inaccessible. Conversely, it's not hard to show that weakly inaccessible cardinals are fixed points.