Why is there no algorithm that can decide every theorem of first-order logic?

First-order logic is semantically complete, meaning every valid sentence is a theorem. However, it is not decidable, a result established independently by Alonzo Church and Alan Turing in 1936.123

The reason there is no algorithm to decide every theorem of first-order logic is that while the set of logical truths is axiomatizable, the set of propositions that are not logically true is not. Consequently, no mechanical procedure exists that can determine, in a finite number of steps, whether an arbitrary sentence is a logical truth or whether its negation is satisfiable.2

This undecidability, known as the Church-Turing theorem, demonstrates that a purely mechanical device cannot replace the work of mathematicians in identifying all theorems. This finding, alongside Gödels incompleteness theorems and Alfred Tarski’s work on the undefinability of truth, fundamentally limits the scope of algorithmic automation in formal systems.32