In 1931, the logician Kurt Gödel showed that Hilbert’s Program will never be achieved i.e. there is no mathematical set that is complete nor consistent. His theorem would.
Kurt Gödel’s Theorem - TED-Ed
So basically,
Every system is incomplete
First Incompleteness Theorem: “Any consistent formal system F within which a certain amount of elementary arithmetic can be carried out is incomplete; i.e., there are statements of the language of F which can neither be proved nor disproved in F.” (Raatikainen 2020)
The way Gödel proved this is using linguistic self-referencing paradoxes. For example, it is a well-known paradox that “this statement is false”. He converted a system’s mathematical functions into number’s, and constructed a sentence that is true if, and only if it is unprovable by the system’s axioms. Therefore, he proved that any such system , would have a Gödel sentence, that cannot be proven.
Every system is inconsistent
Second Incompleteness Theorem: “For any consistent system F within which a certain amount of elementary arithmetic can be carried out, the consistency of F cannot be proved in F itself.” Source: Wikipedia
Gödel states that while a system may be able to prove the consistency of a mathematical statement, it cannot prove the consistency of itself.