Computing uncountably infinite “stuff” is not well defined as stated. So all I can say to if it can “solve undecidable problems” is “Yes, some of them.” Which ones depends on what level of hypercomputer you’ve made, and how high up the arithmetical hierarchy it can take you.
There is a generalized halting problem: no oracle can solve its own halting problem.
Since you mentioned countability, I’ll say I do not know whether any particular type pf hypercomputer would be capable of assigning a specific cardinality (א-n for some n) to the reals.
Specifically, it can compute all of the real number line from 0-1 in finite time. Real numbers are more common than natural numbers. They are also uncountably infinite.
Then if it can compute infinite sets as large as the reals, it can handle any set of cardinality beth-1, but not beth-2 or larger. But because the cardinality of the reals is itself formally undecidable by finite logic systems (or by infinite logic systems of size less than aleph-w), I think this doesn’t give us much specificity about the limits of what that means, or where it falls on the kinds of arithmetical hierarchy schemas finite logic systems enable us to define.
For my own sanity this is about where I stop trying to delve too deep for understanding, and resort to handwaving poetic license:
Great fleas have little fleas upon their backs to bite ’em, And little fleas have lesser fleas, and so ad infinitum. And the great fleas themselves, in turn, have greater fleas to go on; While these again have greater still, and greater still, and so on.
For at the gates that Cantor flung apart (and Hilbert later), Angelic fleas cavort in hosts inordinately greater.
Computing uncountably infinite “stuff” is not well defined as stated. So all I can say to if it can “solve undecidable problems” is “Yes, some of them.” Which ones depends on what level of hypercomputer you’ve made, and how high up the arithmetical hierarchy it can take you.
There is a generalized halting problem: no oracle can solve its own halting problem.
Since you mentioned countability, I’ll say I do not know whether any particular type pf hypercomputer would be capable of assigning a specific cardinality (א-n for some n) to the reals.
Specifically, it can compute all of the real number line from 0-1 in finite time. Real numbers are more common than natural numbers. They are also uncountably infinite.
Then if it can compute infinite sets as large as the reals, it can handle any set of cardinality beth-1, but not beth-2 or larger. But because the cardinality of the reals is itself formally undecidable by finite logic systems (or by infinite logic systems of size less than aleph-w), I think this doesn’t give us much specificity about the limits of what that means, or where it falls on the kinds of arithmetical hierarchy schemas finite logic systems enable us to define.
For my own sanity this is about where I stop trying to delve too deep for understanding, and resort to handwaving poetic license: