User blog comment:Edwin Shade/Turing Machines and a Problem by Vel! Posed Half a Decade Ago/@comment-1605058-20171010173654

As you most certainly know, continuum is the cardinality of the set of real numbers. But this is not the same thing as \(\aleph_1\). \(\aleph_1\) is, by definition, the smallest cardinal which is larger than \(\aleph_0\), the cardinal of countable sets. Saying that continuum is \(\aleph_1\), i.e. there are no cardinalities strictly between \(\aleph_0\) and continuum, is a statement known as the continuum hypothesis and it is known to be independent of the axioms of Zermelo–Fraenkel set theory. So while within the axioms it is possible for continuum to be equal to \(\aleph_1\), saying this as if it was the definition is plain wrong.

Additionally, the complex plane also has cardinality continuum (the very same continuum as the real line).

Now we come to a more serious thing, namely your machines. You have hardly explained how the operate - you've only stated that their tape space is the set of real numbers. However, there is much more you have to explain. For example, does your machine operate in steps? If it does, then how can it move in one step (it can't move to the "next real number")? Also, if the number of steps is greater than \(\omega\) (as you claim), how does the machine behave after the first \(\omega\) steps, where does the read-write head end up and how does the tape look like? If the machine doesn't work in steps but rather works somehow in a continuous manner, you also have to explain how movement rules and what not look like. Everything the same applies to the complex-number-machine.

I am not trying to discourage you, quite the opposite - I would love to see this idea work out into something interesting and, hopefully, very powerful. Let me also point your attention to infinite time Turing machines, the idea of which bears some resemblance to yours; perhaps reading up on this will help you flesh out your ideas.