Abstract: With some simple rules Hamkins and Kidder have observed that one can allow Turing machines to run through transfinite segments of time. Obvious questions arise: What is the halting problem for such machines? If such a machine halts, what is on the tape?
Note that we are not claiming that this has any status as a kind of putative physically realisable "supertask" machine, but is simply a mathematical laboratory for such computations.
We may study the mathematical problems of complexity that arise. Further as such machines can read an infinite sequence of digits they are capable of higher type computation on reals, and are capable of deciding surprisingly complicated predicates. There are relationships to proof-theoretic constructions, determinacy of two person games, and mathematical logic through the use of Infinitary Compactness Theorems, and Analytic Boundedness Lemmata.