For Allen, much of the joy seems to come in the unsolvability
of the argument itself; for Stillman, it's a question of finding profundity in simple things.
This is because rather than assuming responsibility for its own insight, science falls prey to irrationalism when it turns the alleged unsolvability
of a problem into a higher form of understanding (Zerstorung 93).
After the concepts and theories are introduced, the equivalence of computable partial function and recursive partial function are demonstrated, in part through proofs of the unsolvability
of the halting problem and of the enumeration theorem.
However, we may also, I think, see this very question as already decisive in Turing's development of the definition of a "universal computing machine" and its application to demonstrate the unsolvability
of Hilbert's decision problem in the remarkable "On Computable Numbers, with an Application to the Entscheidungsproblem" written three years earlier, in 1936, and published in 1937.
Subjects covered include the structure theory of various notions of degrees of unsolvability
, algorithmic randomness, reverse mathematics, forcing, large cardinals and inner model theory, with papers on such topics as the strength of some combinatorial principles related to Ramsey's theorem for pairs, absoluteness for universally Baire sets and the uncountable, modaic definability of ordinals, eliminating concepts, rigidity and bi-interpretability in hyperdegrees, fundamental issues of degrees of unsolvability
, a "tt" version of the Posner-Robinson theorem, and prompt simplicity, array computability and cupping.