Which ordinal is larger, the supremum of ordinals writable by iterated Infinite Time Turing Machines or the smallest $\Sigma_2^1$-reflecting ordinal?

Question

The ordinal $\tau_1$ corresponds to $\lambda^{\textit{it}}$ (the supremum of all ordinals writable by iterated ITTMs) — see Definition 3.1 in the paper “ITTMs with Feedback” [Robert S. Lubarsky]. According to Theorem 3.4 in the paper, the ordinal $\zeta^{\textit{it}}$ (the supremum of the ordinals eventually writable by iterated ITTMs) is the least $\kappa$ which is $\kappa$-extendible, and $\lambda^{\textit{it}}$ is the smallest $\Sigma_1$ substructure of $\zeta^{\textit{it}}$.

The ordinal $\tau_2$ corresponds to #2.25 in “A zoo of ordinals” [David A. Madore]. It is defined as the smallest $\sigma$ such that $L_{\sigma} \preceq_1 L$, or equivalently $L_{\sigma} \preceq_1 L_{\omega_1}$. This is also the smallest $\Sigma_2^1$-reflecting ordinal.

Which ordinal is larger, $\tau_1$ or $\tau_2$?

Answer 1

The quickest answer is to say $\zeta^{it}$ is the smaller because it can be computed inside any transitive $ZF^-$ model; if $L_\gamma$ is the least such model, then $\gamma < \sigma$ - the latter ordinal also being the smallest $\Sigma^1_2$-reflecting ordinal. (Indeed it can be computed inside the least $L_\tau$ where $\tau$ is $\Sigma_2$-non-projectible. Such a $\tau$ is a $\tau$-extendible limit of $\kappa$-extendibles etc. $ \Pi^1_3$-$CA$ is enough to prove the existence of the ordinal $\zeta ^{it}$. So $ZF^-$ was overkill.) So $\tau_1$ is less than $\tau_2$ in the nomenclature of the question.