Hilbert's Tenth Problem and nonrecursive Diophantine sets In her paper Defining Integers, Alexandra Shlapentokh presents the following as an immediate corollary of the undecidability of Hilbert's Tenth Problem --- that is, the language $\{p : p$ is a polynomial with integer coefficients with at least an integer root $\}$:


Corollary 1.5. There are undecidable Diophantine subsets of $\mathbb{Z}$


I'm not sure this is a consequence of the undecidability of HTP, so maybe I misread, and what the author is saying is that this is a consequence of the correspondence between recognizable sets and Diophantine sets.

Question: Is Corollary 1.5 a direct consequence of the undecidability of HTP?

Question (rephrased): Suppose every Diophantine set is decidable; does this imply that HTP is decidable?
Looking at the paper, I believe that what Shlapentokh is saying is that "there is an undecidable Diophantine subset of $\mathbb Z$" is an immediate consequence of "there is an r.e. set that is not recursive" and the MRDP theorem, where by the "MRDP theorem" is meant "A set is Diophantine if and only if it is r.e."  Nothing deeper than that.