However I'm not convinced that problems which require multiple /infinite/ tapes can *realistically* be mapped onto a single /infinite/ tape... at least without requiring infinite time for the housekeeping in addition to the finite time required to execute the algorithm.
Leaving any housekeeping aside that may be necessary to carry out the mapping for real, as counter intuitive as it sounds, it is possible to map an infinite number of infinities onto a single infinity.
For instance, take the infinity of natural numbers 1, 2, 3, ....
That infinity contains an infinite number of infinities, for instance, the infinity 2, 4, 6, 8... (enumerate by 2), and 3, 6, 9, 12, ... (enumerate by 3) and so on... there is an infinite number of enumerations in the natural numbers infinity BUT...
The infinity of _irrational_ numbers is "larger" than the infinity of natural numbers because it is not possible to establish a 1 to 1 relationship between all irrational numbers and natural numbers (there will always be unmapped irrational numbers)
Infinities are real mind benders...
