Post's problem for supertasks has both positive and negative solutions
Joel David Hamkins, Andrew Lewis
Abstract
Recently we have introduced a new model of infinite computation by extending the operation of ordinary Turing machines into transfinite ordinal time. In this paper we will show that the infinite time Turing machine analogue of Post's problem, the question whether there are supertask degrees between 0 and the supertask jump 0jump, has in a sense both positive and negative solutions. Namely, in the context of the reals there are no degrees between 0 and 0jump, 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.
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