I have a problem in my logic course which I can't get my head around:(adsbygoogle = window.adsbygoogle || []).push({});

I have to show that there is a well formed formula [itex]\mathcal{A}(x_1)[/itex] in the formal first order system for arithmetics, [itex]\mathcal{N}[/itex], with precisely one free variable [itex]x_1[/itex], such that [itex]\mathcal{A}(0^{(n)})[/itex] is a theorem in [itex]\mathcal{N}[/itex] for all [itex]n\in D_N[/itex], but where [itex]\forall x_1\mathcal{A}(x_1)[/itex] is not a theorem in [itex]\mathcal{N}[/itex]. Here [itex]D_N[/itex] denotes the set of natural numbers.

My initial idea was to use the statement and proof of Gödel incompleteness theorem, but I get stuck in a bit of a circle argument with the ω-consistency, so perhaps my idea of using this theorem is all wrong.

Edit: If it isn't obvious from the context, it is fair to assume that [itex]\mathcal{N}[/itex] is consistent.

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# An application of Gödel's incompleteness theorem?

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