In the first place, the existence of this direct correspondence between natural numbers and Turing machines shows that the set of all Turing machines is denumerable, and since the set of all partial functions is uncountably infinite, there must certainly be many functions that cannot be computed by Turing machines.
Initially, the direct correspondence between natural numbers and Turing machines indicates that the set of all Turing machines is countable. Since the set of all partial functions is uncountably infinite, there must be numerous functions that cannot be computed by Turing machines.