There are many kinds of limitative results in the sciences, some of which are philosophical. I am interested in examining one kind of limitative result in the neurosciences that is mathematical-a result secured by the Gödel incompleteness theorems. I will view the incompleteness theorems as independence results, develop a connection with independence results in set theory, and then argue that work in the neurosciences (as well as in molecular, systems and synthetic biology) may well avoid these mathematical limitative results. In showing this, I argue that demonstrating that one cannot avoid them is a computational task that is beyond the computational capacities of finitary minds. Along the way, I reformulate three philosophical claims about the nature of consciousness in terms of the Gödel incompleteness theorems and argue that these precise reformulations of the claims can be disarmed.