We show that the halting times of infinite time Turing Machines (considered as ordinals coded by sets of integers) are themselves all capable of being halting outputs of such machines. This gives a clarification of the nature of "supertasks" or infinite time computations. The proof further yields that the class of sets coded by outputs of halting computations coincides with a level of Godel's constructible hierarchy: namely that of L where is the supremum of halting times. A number of other open questions are thereby answered. 1 Introduction: Infinite Time Turing Machines Hamkins and Lewis in [4] give an account of the construction of these machines (first developed by Kidder and Hamkins in 1989) and develop the basic theory of this notion of computability. The reader should refer to this paper for a clear and full account of their basic properties, from which all the results and definitions of this introduction are taken. But to summarise: an infinite time Turing machine has...

We characterise explicitly i, the least ordinal not the length of any eventual output of an Infinite Time Turing machine (halting or otherwise); using this the Infinite Time Turing Degrees are considered, and it is shown how the jump operator coincides with the production of mastercodes for the constructible hierarchy; further that the natural ordinals associated with the jump operator satisfy a Spector criterion, and correspond to the L i -stables. It also implies that the machines devised are "\Sigma 2 Complete" amongst all such other possible machines. It is shown that least upper bounds of an "eventual jump" hierarchy exist on an initial segment. 1 Introduction: Hamkins and Lewis in [4] give a notion of computability degree based on Infinite Time Turing Machines. They define: Definition 1.1 For f; g 2 ! ! write f 1 g /! 9p 2 ! OE g p (0)#f . The right hand side here is to be interpreted in the usual manner for Turing computability, except that computations are allowe...

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In this document I will outline a couple of recent developments, due to Joel Hamkins, Philip Welch and myself, in the theory of infinite-time Turing machines. These results were obtained with the idea of extending the scope of the study of Borel equivalence relations, an area of descriptive set theory. I will introduce the most basic aspects of Borel equivalence relations, and show how infinite-time computation may provide insight into this area. 1 Infinite-time Turing machines We begin by describing a model of transfinite computation called the infinitetime Turing machine (ITTM), invented by Hamkins and Kidder and introduced by Hamkins and Lewis [1]. Like an ordinary Turing machine, an ITTM is a finite state machine with an input tape, an output tape, and a read/write head. The key addition is that when an ITTM reaches a limit ordinal numbered step, the program enters a special limit state, the read/write head is reset to the left, and the value of each cell is set to the limit of the previous values in that cell (or else zero, if the value has flipped unboundedly many times).