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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?

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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.

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