Suppose I start with the set of computable real numbers between 0 and 1. Now these are countable.
So I follow Cantor's diagonalization argument to construct a real number B between 0 and 1 outside this set.
I enumerate the countable real numbers S1, S2,S3... and I construct B to have it's nth digit after the decimal place be one plus the nth digit of SN if it's 0-8, and 0 if it's 9.
So following this process have I constructed a non-computable real number? Because it feels like what I did above was a computation (I followed an algorithmic process)... How can I compute a non-computable real number? Was the above not a computation... ie couldn't a computer follow this process to create a number outside the countable real numbers... ie: compute the first digit of the first countable number... add 1 to it to calculate the first digit of B... then go on to the second digit of the second countable number etc.
Where is my error? Thanks.