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Your work is wrong. To apply diagonal lemma the definition of phi must be a formula, since you write |- (which is not a formula in PA) I assume you meant it as shorthand for Godels Bew (which is), but you can't existentially quantify Bew like you did in line 3 of the definition.
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Formulas of arithmetic that behave like decision agents
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Knowledge ready for Ankification
Mathematical theorems, axioms and definitions are encodable as flash cards.
It's a terrible idea to try to learn theorems by memorization, if all you want to do is pass math tests fine.. but if you want to understand mathematics it's definitely going to do more harm than good.
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Make sure to wear your rationalist sneakers when you go!