How did Alonzo Church show that first-order logic is undecidable?

In 1936, Alonzo Church established that first-order logic is undecidable by demonstrating that there is no mechanical procedure (or algorithm) capable of determining, in a finite number of steps, whether any given sentence is logically true.12

While first-order logic is semantically complete—meaning all valid sentences are provable—Church proved that the class of propositions that are not logically true is not axiomatizable. By building upon the development of recursion theory and the formalization of recursive functions, Church and Alan Turing independently showed that an algorithmic test for theoremhood was impossible. This result, often associated with the Church-Turing thesis, confirmed that no purely mechanical device could replace the mathematician in deciding the truth of all statements within the system.3241