TITLE: Is "A Short Elementary Proof of the Unprovability of the Collatz Conjecture" really as claimed?
QUESTION [9 upvotes]: The paper "A Short Elementary Proof of the Unprovability of the Collatz Conjecture", from a self-proclaimed peer reviewed journal (although the publisher, ScienceDomain International, appears on Beall's list), purports to prove that the Collatz Conjecture is unprovable.
If it's valid, why has it not received more attention? If it’s invalid, what is the flaw, and why has it not been pointed out since initial publication?

REPLY [11 votes]: I endorse Henning Makholm's points about the lack of any consideration of mathematical logic in the paper.
The form of the argument is also seriously flawed.  The author claims to prove (Theorem 3.3) that the Collatz process will take at least $d$ steps to terminate on input $2^{d+1}-1$. (It's not expressed like that, but that's what it says.) Given this, the author lets $d$ tend to $\infty$ and concludes (Corollary 3.4) that the process will never terminate if the binary expansion of its input comprises an unbounded list of $1$s. (Nowhere is it even mentioned how the Collatz process would proceed on such an input.) That the Collatz conjecture is unprovable is deduced from this "corollary" almost as an aside in the concluding paragraph by conflating a number of the form $n=2^u-1$ with one having "an unbounded number of binary digits".