Maybe I am misunderstanding something, but the problem described in this answer at Math.SE seems to be representable as a program for Infinite Time Turing Machines (under the assumption that any infinite binary string $x \in 2^{\omega}$ can be the input), which leads me to ask the following question:
Which of the two, if any, following propositions is true?
(i) There exists an Infinite Time Turing Machine $M$ such that the existence of at least one real $r$ with the property that $M$ halts (or does not halt) on $r$ necessarily implies that the continuum hypothesis is false (or true);
(ii) There exists an Infinite Time Turing Machine $M$ such that the fact that $M$ halts (or does not halt) on all reals necessarily implies that the continuum hypothesis is false (or true);
(In this question, the term “real” implies an infinite sequence of cells on the input tape of an Infinite Time Turing Machine.)
If both propositions are false, what is the explanation?
Best Answer
Neither is true. There is no link between these questions and the smallest cardinal greater than $\aleph_0$. There might or might not be cardinals between $\aleph_0$ and $\mathfrak c$ but these only talk about what happens at $\mathfrak c$. There might be sets of reals with cardinality strictly between $\aleph_0$ and $\mathfrak c$, but that does not influence what infinite time Turing machines do or do not do.