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In Theorem $3.21$, we showed that a language is Turing-recognizable iff some enumerator enumerates it. Why didn’t we use the following simpler algorithm for the forward direction of the proof? As before, $s_{1}, s_{2},\dots$ is a list of all strings in $\Sigma^{\ast}.$
$E = "$ Ignore the input.

1. Repeat the following for $i = 1,2,3,\dots.$
2. Run $M$ on $s_{i}.$
3. If it accepts, print out $s_{i}."$
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