> You can represent any finite series of real numbers with a series of computations performed on some other finite series of real numbers.

That statement caught my eye. It's either trivially true or quite clearly wrong, depending on how you mean it.

In the literal meaning it's true. Given any finite set of real numbers, I can easily produce a different set (like taking the original set and adding a number which wasn't in there like one plus the largest or so) from which you can trivially produce the original set computationally.

But if you mean you give me both sets then that can't be true. For example if you give me a single real number as set A and the empty set as set B then I can't create a program which generates set A from set B. Your real number in set A could encode anything.