Truth is not a subset of what is proven but of what is possible.
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Replying to @shobith
That depends on your definition of truth. In constructive mathematics, that is true, but for classical logic, Gödel has shown that some true things are not provable.
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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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I think I was trying to say that we shouldn't use the term "truth" if there are multiple definitions for it or if we don't want to confuse the philosophers.
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How do you suggest we talk about whether something is true without using a suitable notion of truth? (Validity usually means that an expression, statement or term is well-formed, not that it is true.)
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I mean the same thing as a theorem, i.e. a statement that can be mechanically derived from the axioms and rules of the system in finite time. The point is that it's all mechanical.
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My understanding is that model theory introduces a notion of truth: a statement can be true in one model and false in another, while being undecidable in the host system. Maybe I misunderstand it. I haven't found very clear explanations of these things.
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There is a veritable zoo of notions of truth: https://plato.stanford.edu/entries/truth/
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