I think that this is a reasonable response and I think the responses in this thread exaggerate the consequences of Godel's Incompleteness Theorems. But you per

I think that this is a reasonable response and I think the responses in this thread exaggerate the consequences of Godel's Incompleteness Theorems. But you perhaps may find it interesting to consider what could be meant when someone says "there are true statements that are not provable" in a way that doesn't exaggerate the results of the Incompleteness Theorems. I don't actually support this line of thinking, but I find it interesting to engage with. So here goes: Whatever you might say about more abstract mathematical structures, there seems to be something indelibly real about the natural numbers. I can verify simple facts about them by physically counting things. While definitionally perhaps it might make sense to create a formal system that says 5 + 6 = 8, I can go and count apples and realize that nope actually 5 + 6 = 11. Now luckily statements of this form, that don't use any quantification (i.e. the words "for all," "there exists," or their equivalents such as "always," or "eventually"), are entirely decided by our usual axioms about the natural numbers. But it seems like, while perhaps a tad less concrete than statements such as 1 + 1 = 2, there are other statements involving quantification that still have some "one-sided" concrete aspect to them. For example if I claim there exists a natural number with some arithmetical property and there indeed is such a number, simply by physically counting up I will eventually come across such a number. However, if I'm wrong, then I will never know (this is what is meant by one-sided). Likewise if I claim a property is true of every natural number, I will eventually have experimental proof if I'm wrong, but I will never know for certain if I'm right (at least by physically counting things). Godel's Incompleteness Theorem says there will always be such quantified statements that any given formal system cannot decide. Yet they seem to have definite truth values in our physical universe! Either my physical act of counting will come to an end or it won't. This seems like a very real consequence. So sure you can claim that these statements are different than theorems because either themselves or their negation can be added to our pre-existing axioms about the natural numbers without any conflict and that might be true in theory, but we would like to preserve the property that our axioms about the natural numbers correctly reflect how natural numbers act in our universe, and it seems, at least at a first glance, that Godel's Incompleteness Theorem disallows that, or at the very least that the act of trying to find these axioms requires extra-mathematical justification.