Can generalized Turing machines compute all reals? Super-recursive algorithms are theoretical super-recursive algorithms are a generalization of ordinary algorithms that are more powerful, that is, compute more than Turing machines.

In this entry it is said:


  A symbol sequence is computable in the limit if there is a finite, possibly non-halting program on a universal Turing machine that incrementally outputs every symbol of the sequence. This includes the dyadic expansion of π but still excludes most of the real numbers, because most cannot be described by a finite program. Traditional Turing machines with a write-only output tape cannot edit their previous outputs; generalized Turing machines, according to Jürgen Schmidhuber, can edit their output tape as well as their work tape


Does that mean that, theoretically, generalized Turing machines, cannot compute all reals? Can generalized Turing machines do (fully) hypercomputation (theoretically)?

(I know that there is strong evidence against hypercomputation. I'm asking this only from a theoretical perspective)
Any machine model in which a machine can be described by a string over a fixed alphabet can only compute countably many things. Since there are uncountably many real numbers, all of these machine models fail to compute almost all real numbers.

In fact, the alphabet need not be fixed. It is enough that the alphabet is taken from some countable set of finite alphabets, for example $\{1,\ldots,n\}$ for every $n$.