Given a fixed enumeration of Infinite Time Turing Machines (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:
- if $M_i(x)$ does not halt for any real $x$, then $f(i) = 0$;
- if there exists the largest countable ordinal $alpha$ such that there exists a real $x$ such that $M_i(x)$ halts at time $alpha$ (and there does not exist a real $y neq x$ such that $M_i(y)$ halts at time $beta > alpha$), then $f(i) = alpha$;
- 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”)?
