Is undecidability of arithmetic a corollary of Tarski undefinability theorem? Arithmetic is undecidable, in other words the set of Godel numbers of theorems of arithmetic is not recursive, and so there is no algorithm/ recursive function to decide if a statement is provable or not.
The Tarski  undefinability theorem for arithmetic states that it is not possible to express by an arithmetic formula the set of Godel numbers of theorems of arithmetic.
But arithmetic formulas contain the set of recursive functions,  so this implies that the above set is not recursive.
Does all this make sense?
 A: To move this off the unanswered queue:
Godel and Tarski are looking at different notions of "arithmetic."

Tarski: $Th(\mathbb{N})$, the set of (Godel numbers of) sentences true in $\mathbb{N}$, is not definable in $\mathbb{N}$.

As you say, we get as a corollary that $Th(\mathbb{N})$ is not recursive. However, that does not immediately yield Godel's theorem:

Godel: The set $Thm(PA)$ of (Godel numbers of) theorems of first-order Peano arithmetic is not recursive. (And this can in fact be strengthened in many ways.)

In fact, $Thm(PA)$ is much less complicated than $Th(\mathbb{N})$; the Turing degrees of these sets are $\bf 0'$ and ${\bf 0^{(\omega)}}$, respectively.
Actually, if anything the implication goes the other way. If $Th(\mathbb{N})$ were definable then we would have $Th(\mathbb{N})\le_T\emptyset^{(n)}$ for some $n$. But we have $\emptyset^{(n+1)}>_T\emptyset^{(n)}$, which yields a contradiction since $\emptyset^{(n+1)}$ is definable in $\mathbb{N}$. See also here. (I've phrased this as a "relativized halting problem" argument, but you could also recast it as a "relativized incompleteness theorem" argument.)

That said, we can use Tarski to prove a version of Godel with some more work; see here, where a slight variant of Tarski's theorem (but via the same proof) can be used to show that $PA$ (or indeed a much weaker theory) is essentially incomplete, and essential incompleteness of $PA$ implies that $Thm(PA)$ is non-recursive.
