In this question, the notation $P^x(\alpha)$ denotes a situation where a particular OTM-program $P$ performs a computation on input $x$ with an ordinal parameter $\alpha$, assuming that $x$ is written on the initial segment of length $\omega$ (the smallest limit ordinal) of the tape of $P$ at time $0$. That is, $x$ is the input for $P$ written in cells indexed by finite ordinals $(0, 1, 2, \ldots)$ before the start of computation, yet all cells indexed by all ordinals greater than or equal to $\omega$ are initially blank, except one cell indexed by $\alpha$ (this cell is marked by a non-zero symbol.)

Let $\beta$ denote *the smallest* ordinal such that for *any* pair of an OTM-program $P$ and a real $x$ (that is, $P$ quantifies over *all* programs and $x$ quantifies over *all* reals) exactly one of the following statements is true:

There does not exist an (uncountable or countable) ordinal $\alpha$ such that $P^x(\alpha)$ halts;

If there exists at least one (uncountable or countable) ordinal $\alpha$ such that $P^x(\alpha)$ halts, then, assuming that $\alpha_0$ is the smallest such ordinal, $\alpha_0 < \beta.$

How large is $\beta$?

allprograms,allinputs andallordinal parameters (assuming that the parameter isminimal, as is written in the question). $\endgroup$6more comments