The recent Arxiv paper "Learning the undecidable from networked systems" attempts to construct a network of $N$ Turing machines$^1$ that can solve the Halting problem for any program of size $O(\log N)$. From this, they claim in the conclusion that
This result shows that there is at least one fixed algorithm that can be distributedly run on networked randomly generated universal Turing machines, so that the entire algorithmic network can compute a function in the Turing degree $0'$, if the population/network size in large enough.
However, each network solves the Halting problem only for programs of bounded size, a restriction that makes the problem computable (For finitely many programs, there exists a look-up table that tells whether each of those programs halts). So, 'large enough' here has to mean some sort of infinity, and I do not really see how their construction extends to networks of infinite size. Am I misunderstanding their claim?
If (as I suspect), this network indeed does not solve an uncomputable problem, I also wonder whether this result is relevant to the notion of computability at all?
$^1$: Apparently, a halting oracle is involved as well, see section 4.2 in the paper. The authors even claim
... the oracle is only necessary to deal with the non-halting computations.
(emphasis mine) This confuses me greatly, because isn't the fact that there are non-halting computations the entire point? Also, if we are using an oracle for the halting problem anyway, why not use the oracle directly? However, even if we assume that this oracle is not nessecary, I still do not see why the main claim would hold.