This is a mischaracterisation of gödels incompleteness theorem.
The set of axioms the theorem is about isn't even mentioned in the article. This is very important. If you pick the empty set as your axioms, most statements will be independent of them.
The important insight is different. One can simulate a computer in the peano axioms.
If you could prove or disprove (starting from a computable set of axioms) every statement about simulated computers you could also solve the halting problem.
Kleenes (the kleene star guy) proof goes that way.
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