Is there anything that can not be computed?
Yes. There exists a function f: N ->N that is not computable.
(1) Consider all Java programs of the form:
static void main(String [] args)
TokenReader in = new TokenReader(System.in);
/* Let y be f(x), for some f: N->N. */
All such programs are enumerable; e.g., they can be listed from smallest to largest in alphabetical within each given size.
(2) All functions f: N->N that are computed by such programs are enumerable, e.g., just list the functions computed by the above programs and eliminate duplicates.
(3) (Cantor) It is impossible to enumerate all functions f: N->N; i.e. any conceivable list
naturals programs
0 first function f0
1 second function f1
2 third function f2
etc. etc.
necessarily omits some function, e.g., f(i) = fi(i)+1.
(4) So the list given in (2) must be incomplete. Q.E.D.