Computability and Logic

HISTORICAL REMARKS 239
In the course of this essentially philosophically motivated work, Gödel introduced
the notion of primitive recursive function, and established the arithmetization of syntax
by primitive recursive functions and the representability in formal arithmetic of
primitive recursive functions. But though primitive recursive functions were thus originally
introduced merely as a tool for the proof of the incompleteness theorems, it was
not long before logicians, Gödel himself included, began to wonder how far beyond
the class of primitive recursive functions one had to go before one arrived at a class
of functions that could plausibly be supposed to include all effectively computable
functions. Alonzo Church was the first to publish a definite proposal. A. M. Turing’s
proposal, involving his idealized machines, followed shortly thereafter, and with it
the proof of the existence of a universal machine, another intellectual landmark of
the last century almost on the level of the incompleteness theorems themselves.
Gödel and others went on to show that various other mathematically interesting
statements, besides the consistency statement, are undecidable by P, assuming it to
be consistent, and even by stronger theories, such as are introduced in works on set
theory. In particular, Gödel and Paul Cohen showed that the accepted formal set theory
of their day and ours could not decide an old conjecture of Georg Cantor, the creator of
the theory of enumerable and nonenumberable sets, which Hilbert in 1900 had placed
first on a list of problems for the coming century. The conjecture, called the continuum
hypothesis, was that any nonenumerable set of real numbers is equinumerous with
the whole set of real numbers. Mathematicians would be, according to the results of
Gödel and Cohen, wasting their time attempting to settle this conjecture on the basis
of currently accepted set-theoretic axioms, in the same way people who try to trisect
the angle or square the circle are wasting their time. They must either find some way
to justify adopting new set-theoretic axioms, or else give up on the problem. (Which
they should do is a philosophical question, and like other philosophical questions,
it has been very differently answered by different thinkers. Gödel and Cohen, in
particular, arrayed themselves on opposite sides of the question: Gödel favored the
search for new axioms, while Cohen was for giving up.)