a consistent mathematical system is also complete are complementary to Russell’s discovery of the paradox in set theory, which suggests that an attempt to make a formal system complete will introduce an inconsistency.

Turing presumably imposed the restrictions of finite machine states and finite steps in order to ensure that Turing machines would be close analogues of the recursive equations which were the standard at that time for computing proofs. These restrictions on the Turing machine were necessary conditions for proof, but as his results showed, they did not amount to sufficient conditions. In making these assumptions he effectively restricted the machines to producing time-free results. This was entirely reasonable given that his goal was to formalize a procedure for producing mathematical results which were themselves timeless. Nevertheless he found the restrictions to be somewhat problematic, as did Church and G[ö]{}del, because they meant that a Turing machine could not capture the generation of the new concepts and procedures that flowed from the exercise of mathematical intuition by mathematicians. There was much informal discussion of this issue, including possibilities for circumventing the limitations. Turing suggested that no single system of logic could include all methods of proof, and so a 