Abstract. The infinite time Turing machine analogue of Post's problem, the question whether there are supertask degrees between 0 and the supertask jump 0 ▽ , has in a sense both positive and negative solutions. Namely, in the context of the reals there are no degrees between 0 and 0 ▽ , but in the context of sets of reals, there are; indeed, there are incomparable semi-decidable supertask degrees. Both arguments employ a kind of transfinite-injury construction which generalizes canonically to oracles. The supertask model of infinitary computation extends the operation of an ordinary Turing machine into transfinite ordinal time. The theory, of which we gave a full introductory account in [HamLew], leads naturally to notions of computability and semi-decidability for reals and sets of reals, as well as an oracle concept and two jump operators: the lightface ▽ and boldface jumps. Left open in [HamLew] was the supertask analogue of Post's problem, the question whether there are semidecidable degrees between 0 and the jump 0 ▽ , or indeed, whether there are any degrees between 0 and 0 ▽ at all. Here, we prove that the answer is in a sense both positive and negative.
They showed in REF that in this model, • for sets of reals the answer is "no" just like in the classical discrete case.
Post's problem for supertasks has both positive and negative solutions
Arch. Math. Log.
Mathematics Computer Science