Indeed, precisely that is true. The shocking conclusion of Gödel's incompleteness theorem is that the problem that our logical foundations are incomplete is in

Indeed, precisely that is true. The shocking conclusion of Gödel's incompleteness theorem is that the problem that our logical foundations are incomplete is inherently unfixable; any extended system of axioms will have new statements which are true but which it cannot prove. (That said, there are certain sub-areas of mathematics where we can prove that every true result in one of these sub-areas is actually provable from a certain system of axioms adapted to the given sub-area. But there is no way around Gödel's incompleteness phenomenon if one wants to encompass all of mathematics (and even arithmetic with natural numbers suffices to yield the incompleteness phenomenon).)