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Let L be a set of recursive funtions with $L = \{i\in \mathbb{N}|f_i\space is\space surjective\}$ where $i$ is a gödel number of f.Is $L\in RE,\space coRE$?

I can't think of a way to show either of the two.

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Rice-Shapiro proves that $L$ is not RE nor coRE.

Let $F$ be the set of computable surjective functions. $L$ is the the index set associated to $F$.

Assume by contradiction $L$ is RE. By Rice-Shapiro (compactness), since the identity function belongs to $F$, some finite restriction of the identity must belong to $L$ as well. But a finite-domain function can not be surjective -- contradiction.

Assume by contradiction $\bar L$ is RE. The always undefined function $g$ belongs to $\bar F$, so by Rice-Shapiro (monotonicity), $\bar F$ contains any computable extension of $g$, including the identity function, which is surjective. Contradiction.

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  • $\begingroup$ I don't quite understand the second proof. How does Rice-Shapiro state that F^c contains the identity function? $\endgroup$ – Yamahari Jan 10 at 13:02
  • $\begingroup$ @Yamahari $F^c$ contains the always undefined function, so by RS we can extend it in any computable way (say to the identity) and still be inside $F^c$. Pedantically: RS states that the identity belongs to $F^c$ IFF some finite restriction of the identity belongs to $F^c$. The latter is true, so the former must also be true (it's an IFF, not only an implication). $\endgroup$ – chi Jan 10 at 13:58

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