The theorem as stated in my mathematical logic textbook says:
Godel's First Incompleteness Theorem. Let $\Phi$ be consistent and $R$-decidable and suppose $\Phi$ allows representations. Then there is an $S_ {ar}$-sentence $\phi$ such that neither $\Phi\vdash\phi$ nor $\Phi\vdash\neg\phi$.
Is it correct to use the Completeness theorem for first order logic to change the end to:
Then there is an $S_ {ar}$-sentence such that neither $\Phi\vdash\phi$, nor $\Phi\vdash\neg\phi$, and neither $\Phi\vDash\phi$ nor $\Phi\vDash\neg\phi$.
Moreover, I have been thinking about how to understand the first incompleteness theorem more intuitively. I've reached, using informal proof, a conclusion that seems like it must be wrong (since otherwise, I would have heard of it before).
What is wrong with my argument? (or is it correct?)
Firstly, it seems that this follows from the theorem:
Corollary. For every structure $\mathfrak{A}$, there is no consistent and decidable set of first-order formulas $\Phi$ that has as its consequence its first-order theory $Th(\mathfrak{A})$. That is, there is no such $\Phi$ such that $\Phi\vDash Th(\mathfrak{A})$.
Which seems to be equivalent to:
Corollary. There is no structure that has an axiomatizable theory in first-order logic.
my informal argument is: If there were such a structure, then according to the completeness of first order logic, we would be able to derive either $\phi$ or $\neg\phi$ from its axioms, which is a contradiction to the first incompleteness theorem.
Does this make sense?