= Hilbert–Bernays paradox
{wiki=Hilbert–Bernays_paradox}
The Hilbert–Bernays paradox is a philosophical and logical issue related to the foundations of mathematics and formal systems, particularly concerning the relationship between provability and truth. The paradox arises in the context of formal systems and the principles that govern them. It highlights a potential clash between two different forms of reasoning: syntactic (formal proofs) and semantic (truth in models). Specifically, the paradox involves certain statements that can be proven within a formal system but that also have implications about their own provability.
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