There is a theorem that states that arithmetic is undecidable: i.e. $Th(\mathcal N)$, the set of all sentences in the standard arithmetic structure $\mathcal N=(\mathbb N,+,\cdot , 0,1)$ where the symbols are interpreted in the standard way, is undecidable.
Godel's first incompleteness theorem states that there does not exist a set of axioms from which we can prove all true arithmetic statements: There does not exist a set of first order formula's $\Phi$ that is consistent and decidable, such that for any first order $(+,\cdot,0,1)$ sentence $\phi$, we have $\Phi\vdash\phi$ or $\Phi \vdash \neg \phi$. By the completeness theorem we could have instead written "$\Phi\models\phi$ or $\Phi \models\neg \phi$"
But since we can construct a first order proof system that is complete, in the sense that all valid sentences can be proven, doesn't this boil down to the same thing?
Isn't godel's incompleteness theorem simply an implication of the undecidability of arithmetic?