I realize this is an old question, but in case this is still of interest: if I correctly understand that you are talking about the ramified analytical hierarchy, namely the one in which each next level consists of sets of natural numbers definable over the previous in second-order number theory (as opposed to full set theory which leads to the constructible hierarchy, as Andreas Blass points out in his answer), then the relevant paper here seems to be Boyd, Hensel & Putnam, “A Recursion-Theoretic Characterization of the Ramified Analytical Hierarchy”, Trans. Amer. Math. Soc.141 (1969) 37–62. The ordinal at which this hierarchy collapses, usually written $β_0$ after that paper and known as the “ordinal of ramified analysis”. It is greater than the Church-Kleene ordinal $ω_1^{CK}$, but less than the true $ω_1$ (or even the ordinal $ω_1^L$ that is $ω_1$ in the constructible universe $L$); it is the smallest ordinal $β$ such that $L_β$ models ZFC minus powerset, and the ramified analytical hierarchy coincides with the constructible hierarchy on $\mathscr{P}(ω)$ up to that point. See also Marek & Srebrny, “Gaps in the Constructible Universe”, Ann. Math. Logic6 (1974) 359–394.
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