From what I remember "things you can compute" splits into "things you can mathematically define" and "things for which you can construct a Turing machine which computes them".
If we define "x is computable" as "there exists a Turing machine T(x) which takes n as input and produces n-th digit of x" then there are numbers which are defineable but not computable.
If we define "x is computable" as "there exists a Turing machine T(x) which takes n as input and produces n-th digit of x" then there are numbers which are defineable but not computable.