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Recursion Theorem and c.e. sets

  1. Jan 26, 2012 #1

    jgens

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    Let [itex]\varphi_e[/itex] denote the p.c. function computed by the Turing Machine with code number [itex]e[/itex]. Given [itex]M=\{x : \neg(y<x)[\varphi_y=\varphi_x]\}[/itex] prove that [itex]M[/itex] is infinite and contains no infinite c.e. subset.

    I have already proved that [itex]M[/itex] is infinite. A necessary and sufficient condition to prove that [itex]M[/itex] has no c.e. subset is that given any one-to-one computable function [itex]f[/itex] it follows that [itex]f(\omega) \nsubseteq M[/itex]. One way to demonstrate that this holds is to use the Recursion Theorem to show that there is some index [itex]x_0[/itex] such that [itex]\varphi_{x_0}=\varphi_{f(x_0)}[/itex] but [itex]x_0 < f(x_0)[/itex]. I am having trouble doing this however and was wondering is someone here could help.

    I think that this is the right way to approach the problem since the text suggested we use the Recursion Theorem to tackle it.
     
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