Why is there no algorithm for deciding which Diophantine equations have integer solutions?

In the context of metalogic, the absence of an algorithm to determine which Diophantine equations have integer solutions is a consequence of the development of recursion theory and the formalization of computability.1

During the 1930s, logicians such as Alonzo Church, Alan Turing, and Emil Post established precise, mathematical definitions for what constitutes a mechanical or algorithmic procedure. These advancements allowed mathematicians to prove that certain classes of problems are not just difficult, but "mechanically unsolvable" or "absolutely unsolvable"—meaning no finite, repetitive rule can exist to solve them.1

The specific discovery that no such algorithm exists for Diophantine equations was made in 1970. This result provided a negative answer to the 10th problem on the famous list presented by David Hilbert in 1900, confirming that the question of integer solutions for these polynomial equations is undecidable.1