1 5 A pr 1 99 8 Ordinal Computers
نویسنده
چکیده
Can a computer which runs for time ω 2 compute more than one which runs for time ω? No. Not, at least, for the infinite computer we describe. Our computer gets more powerful when the set of its steps gets larger. We prove that they theory of second order arithmetic cannot be decided by computers running to countable time. Our motivation is to build a computer that will store and manipulate surreal numbers. Hackelroad [1] and Lurie [4] examined at least two ways to compute surreals in finite time, and shown the difficulty of building a field of surreals in which x > y, x = y + z, and x = y × z are decidable. Likewise for reals. In the recursive reals algebra but not order is decidable. And so it seems that the question of whether the theory of either field under +, ×, < is decidable, ought to refer to decidability by some class of computers that can compute more than finite-time Turing Machines. If we're going to talk about whether a computer can decide facts about numbers, then let's have a computer that can construct all the numbers we want to talk about and decide the algebra and order relations. Computers running to time ℵ 1 can compute all reals, and to ordinal time can compute all surreals. Now, what facts about such numbers can ordinal computers decide?
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