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by Konohamaru
2115 days ago
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The Axiom of Choice is not an example of Gödel's true-but-unprovable statements, because Gödel's statements are true in all models (as you said) but AoC is NOT true in all models. The Continuum Hypothesis isn't an example either for the same reason. |
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This is wrong, and I said the exact opposite of this: there are non-standard models of PA in which G_F is false.
There is a fundamental difference between Gödel sentences and AoC, which is that the Gödel sentence is Pi_1, which means its independence implies its truth in the standard model.
I'm just not really a fan of unqualified "true" meaning "true in the standard model"; I think if you're doing a course purely on the incompleteness theorems for an audience without much exposure to mathematical logic, using "true to refer to "truth in the standard model" is not a good idea and is likely to lead to misconceptions.
Perhaps the fact that you think G_F is true in all models is evidence in favour of my claim?