• J. D. Hamkins and A. Lewis, “Infinite time Turing machines,” J. Symbolic Logic, vol. 65, iss. 2, pp. 567-604, 2000.  
  Journal MR Citation arχiv

  @article {HamkinsLewis2000:InfiniteTimeTM,
  AUTHOR = {Hamkins, Joel David and Lewis, Andy},
  TITLE = {Infinite time {T}uring machines},
  JOURNAL = {J. Symbolic Logic},
  FJOURNAL = {The Journal of Symbolic Logic},
  VOLUME = {65},
  YEAR = {2000},
  NUMBER = {2},
  PAGES = {567--604},
  ISSN = {0022-4812},
  CODEN = {JSYLA6},
  MRCLASS = {03D10 (03D25 68Q05)},
  MRNUMBER = {1771072 (2001g:03072)},
  MRREVIEWER = {Robert M. Baer},
  DOI = {10.2307/2586556},
  URL = {http://dx.doi.org/10.2307/2586556},
  eprint = {math/9808093}
  }

This is the seminal paper introducing infinite time Turing machines.

We extend in a natural way the operation of Turing machines to infinite ordinal time, and investigate the resulting supertask theory of computability and decidability on the reals. The resulting computability theory leads to a notion of computation on the reals and concepts of decidability and semi-decidability for sets of reals as well as individual reals. Every $\Pi^1_1$ set, for example, is decidable by such machines, and the semi-decidable sets form a portion of the $\Delta^1_2$ sets. Our oracle concept leads to a notion of relative computability for reals and sets of reals and a rich degree structure, stratified by two natural jump operators.