= Solution
Repeat the preceding construction starting at $\alpha_0=\gamma+1$, for any given <ordinal> $\gamma$. A stabilized value or the strictly increasing sequence's supremum is a fixed point at least $\gamma+1$. Thus fixed points occur above every <ordinal>.
If the fixed points formed a set $S$, their union would be an <ordinal> bounding them all. Applying the construction above that bound contradicts it. Hence
$$
\boxed{\{\alpha\in\mathrm{ON}:F(\alpha)=\alpha\}\text{ is a proper class}.}
$$
This is the unboundedness part of the <fixed point of a normal ordinal function> theorem.
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