User blog comment:Alemagno12/Some more set theory questions/@comment-1605058-20180105165055/@comment-26454151-20180108160448

I believe he meant to say the first 1-inaccessible cardinal.

The (informal) reasoning goes that rules 2 and 3 are all "finite" and "fallback" steps, whereas rule 4 is the only way the values can reach infinities. Rule 2 is invoked when the parameter is a successor cardinal. Rule 3 is invoked when the parameter is a singular limit cardinal. Hence, rule 4 can only be invoked when the parameter is an uncountable regular limit cardinal (weakly inaccessible).

I believe this would result in \(F(I_{\alpha})=\omega \alpha \). From this formula, the fixed points of F are exactly the fixed points of the enumeration of inaccessibles, i.e. the 1-inaccessibles.