Truth is not a subset of what is proven but of what is possible.
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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