On Computable Numbers, With an Application to the Entscheidungsproblem
Summary: computers could not completely prove mathematical assertions (mathematics could not be completely modelled by computers).
e and π are computable.
1 Church has introduced an idea of “effective calculability”, equivalent to his “computability”.
For the present I shall only say that the justification lies in the fact that the human memory is necessarily limited.
m-configurations: q_1, q_2, ..., q_R ... m-configuration: q_i configuration --- m-configuration and the scanned symbol
3 a-machine is deterministic, c-machine is indeterministic (should make some arbitrary choice; to deal with axiomatic systems). There are two kind of symbols. If first kind is (0, 1), the machine is computing machine. complete configuration: number of the scanned square, the complete sequence of all symbols on the tape, and m-configuration. Machine's step is move.
4 Circular machine: writes down more than a finite number of symbols (otherwise, circle-free). Sequence is computable if it can be computed by a circle-free machine.
Sergey Vartanov, 2007–2020