Is this set computable? Let be $B$ a Busy Beaver function and set $W=\{\langle M \rangle :\text{$M$ stops in less than $B(10^{1000})$ steps on an empty tape}\}$. Is this set computable?



I'm not sure how to approach this question. I suspect that this set is computable and have tried to see if it is finite, but haven't reached anything.
Your function is computable. Just run the input machine on the empty input for $B(10^{1000})$ steps. Note that $B(10^{1000})$ is just a constant which can be hardcoded into your code.