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Here by «algorithm» I mean a (halting) Turing machine with finite alphabet and memory. Is it possible to obtain by purely existential (i.e. non-constructive) means the existence of an algorithm ...

Call a real $x$ limit computable iff there is a Turing machine $T$ such that, for any $i\in\omega$, there is $t(i)\in\omega$ such that the $i$th entry on the tape is not changed after time $t(i)$ and ...

Otherwise, if all the elements in a set can be represented by a at most n symbols (finite Kolmogorov complexity), I could count them by creating a n dimensional pairing function. Or atleast, that is ...