Given a fixed enumeration of [Infinite Time Turing Machines](https://arxiv.org/abs/math/9808093) (ITTMs), let $M_i(x)$ denote a computation of an $i$-th ITTM, assuming that the input $x$ is a _real_ (an _infinite_ binary sequence).  
  
Then the function $f(i)$ (input: natural number, output: countable ordinal) is defined as follows:  
  
1. if $M_i(x)$ does not halt for any real $x$, then $f(i) = 0$;  
2. if there exists a countable ordinal $\alpha$ such that there exists a real $x$ such that $M_i(x)$ halts at time $\alpha$, but there does not exist a real $y$ such that $M_i(x)$ halts at time $\beta > \alpha$, then $f(i) = \alpha$; 
3. if both (1) and (2) are false (that is, if for any countable ordinal $\alpha$ there exists a real $x$ such that $M_i(x)$ halts at time $\beta > \alpha$), then $f(i) = 0$.  
  
The ordinal $\tau$ is defined as the supremum of the infinite set $\{f(0), f(1), f(2), \ldots \}$. Question: how large is $\tau$? In particular, what is $\tau$ in comparison with the least $\Sigma_1$-stable ordinal (mentioned in the Definition 3.1 in the paper [“Recognizable sets and Woodin cardinals: Computation beyond the constructible universe”](https://arxiv.org/abs/1512.06101))?