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Natural functions outside $\sf PA$?
Can theories stronger than $\sf PA$ manage to define functions from the naturals to the naturals, that $\sf PA$ cannot? If so, what are examples of those functions?
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Is there a definable model of PA whose domain is a proper class and whose complete theory is not definable?
Assume ZFC. Is there a formula of $\mathcal{L}_\in$ (without parameters) defining a model $\mathcal{M}$ of PA whose domain is a proper class but the complete theory of that model is not definable by ...
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