How can I prove that this function is computable?

Question

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 $x$.

Answer 1

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.

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(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.)