Truth is not a subset of what is proven but of what is possible.
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Gödel's incompleteness theorems are independent of a notion of truth as commonly understood. There are valid, invalid, and undecidable statements, in a given formal theory. It was only his personal intuition that certain things are true regardless (Platonism).
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The way I understand it, the halting problem implies that there are programs that truly don't halt but for which you won't be able to prove it. However, you can also use a different notion of "true" that means that you can only claim truth for those cases where you can show it.
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