The Recursion Theorem and Infinite Sequences

Arnold W. Miller

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arXiv · 0801.2097

The Recursion Theorem and Infinite Sequences

Arnold W. Miller

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Abstract

In this paper we use the Recursion Theorem to show the existence of various infinite sequences and sets. Our main result is that there is an increasing sequence e_0, e_1, e_2 .. such that W_{e_n}={e_{n+1}} for every n. Similarly, we prove that there exists an increasing sequence such that W_{e_n}={e_{n+1},e_{n+2},...} for every n. We call a nonempty computably enumerable set A self-constructing if W_e=A for every e in A. We show that every nonempty computable enumerable set which is disjoint from an infinite computable set is one-one equivalent to a self-constructing set

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BibTeX RIS

Arnold W. Miller. 2008-01-14. The Recursion Theorem and Infinite Sequences. https://arxiv.org/abs/0801.2097

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