How did the unsolvability of Diophantine equations answer Hilbert’s tenth problem?

Hilbert’s tenth problem asked for a mechanical procedure—an algorithm—to determine whether any given Diophantine equation has integer solutions.1

In 1970, this problem was answered in the negative. Through the development of recursion theory and the formalization of what constitutes a "mechanical" or "computable" procedure, mathematicians proved that no such algorithm can exist. This discovery demonstrated that the problem is "absolutely unsolvable," meaning it is impossible to construct a rule of repetitive procedure that works for all such polynomial equations.1

This result is a significant milestone in metalogic, as it provides a concrete example of a problem that is mechanically undecidable. It highlights the limitations of formal systems and confirms that there are mathematical questions for which no general, finite method of resolution can be found.1