TITLE: How can I prove that this function is computable?
QUESTION [1 upvotes]: Is the following function
$$
g(x) = \begin{cases}
    1 & \text{if $\exists n \geq 0:\ \varphi_n(x) \downarrow$}\\
    \uparrow & \text{otherwise}
  \end{cases}
$$
computable?
Please note that $\varphi_n(x) \downarrow$ means that the function with index $i$ halts on input $x$.

REPLY [0 votes]: Quick loophole:
If $\varphi_i$ means something even remotely like what it looks like it should, then your function is the same function as
$$ g(x) = 1 $$
which is very easy to compute.

(If $\varphi_i$ is something so unconventional that this doesn't work, you'll probably want to look up dovetailing. The Wikipedia article I link to is perhaps not particularly approachable, but a bit of determined googling may turn up something better.)
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