"Capacity" concept in Shannon Theorem
The formula you wrote corresponds to the error-free capacity. Namely, Shannon is telling us that $C$ is the maximum (theoretical) limit one can achieve.

Shannon managed to prove that if the information rate on the channel being used is less than $C$, then it is theoretically possible to find a coding system for the signal such that the transmission will have an arbitrarilly small probability of error. Unfortunately, the theorem does not tell us how to find that code, just that it exists.

That's why there is no reference to error rate in the formula for capacity. The formula itself is telling you the maximum information rate you can have with an arbitrary small amount of errors, and that value corresponds to $C$.

Sidenote: remember that the formula is only valid for white Gaussian noise.