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]](https://www.semanticscholar.org/paper/ITTMs-with-Feedback-Lubarsky/980d560b8f991113f5d132b4d161a78e712a067c). 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 $\tau\_2$?



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


