Given the work of Turing and Feferman all arithmetical truths can be isolated through a transfinite progression of theories like $T_0=PA$, $T_{\beta+1}=T_β \ plus \ CON(T_\beta)$ and $T\lambda=\cup T\mu(\mu\prec\lambda)$ - when $\lambda$ is a limit ordinal - through all the recursive ordinals. What is the smallest ordinal $\sigma$ such that $T_\sigma$ proves CON(ZF)? How do such ordinals for arithmetical consistency statements align with proof theoretical ordinals?

Edit: My question does not ask for the proof theoretic ordinal of ZF.

Update: Phillip Welch gives a very readable account of such things as I hint to in comments concerning Feferman's work in an answer to a question here:

Pi1-sentence independent of ZF, ZF+Con(ZF), ZF+Con(ZF)+Con(ZF+Con(ZF)), etc.?

Update 2: My question was badly prepared, as evidenced also by the previous update and the comments in discussion. Noah Schweber kindly suggested that I unaccept his reply until more is clarified concerning my question as related to the Feferman style process I had in mind, and which through a detour into Shoenfield's recursive omega rule (non-constructively) captures all arithmetical truths. I would be surprised if Turing like collapses down to $\omega+1$ could occur in Feferman style processes.

ordinal, rather anordinal notation). $\endgroup$ – Noah Schweber Mar 8 '17 at 21:26