6 On Computable Numbers, with an Application to the Entscheidungsproblem (1936)

"Entscheidungsproblem" is German for "decision problem," and with the definite article refers to the problem of determining whether an arbitrary mathematical proposition is provable. In 1928, the logical foundations of mathematics had been firmly enough established that David Hilbert and his student Wilhelm Ackermann could stipulate that the axiomatic system in question was the predicate calculus (referred to here as the "functional calculus K"). The hunt was on to find a procedure that would, in effect, answer all mathematical questions and complete Hilbert's program. Various researchers offered decision procedures for fragments of the predicate calculus, but there was at first no serious work toward an impossibility proof—in part because the notion of a "decision procedure" was merely intuitive. Imagining that no procedure might exist required that the class of all procedures be circumscribed.