When Kurt Gödel published his famous Incompleteness Theorems in 1931, it shook the foundations of mathematical logic: He refuted that all axioms that can be set up as a possible basis are inevitably incomplete in order to prove all statements about numbers - and destroyed that Hilbert's dream to prove the consistency of mathematical theory.

The introduction of Gödel numbers (the unambiguous mapping of formulas to natural numbers) and diagonalization (the replacement of free variables in functions with their respective Gödel numbers) are two central concepts that Gödel introduces in his proof. The decisive proof in which Gödel combines these concepts can be written down as follows:

$$P(p) \, \text{wahr} \Leftrightarrow p \in \, \overline{B}^* \Leftrightarrow d(p) \in \overline{B} \Leftrightarrow d(p) \notin B \Leftrightarrow g(P(p)) \notin B \Leftrightarrow P(p) \, \text{unbeweisbar}$$

Now since \(P(p)\) cannot be false (otherwise it would be provable and therefore true), \(P(p)\) must be true and therefore not provable. Thus there is always a true proposition in a language (with any choice of axioms) which cannot be proved, where \(g\) is the dodecision, \(p\) is the dodecode number of the predicate \(P\), which characterizes the complementary archetype \(\overline{B}^*\) of \(B\) (the set of all dodecode numbers of all provable propositions) under the diagonal function \(d\).

For further reading, we recommend Gödel's 1931 publication and the article by Stepan Parunashvili, which is well worth reading: in addition to the incompleteness theorem, Gödel has made other groundbreaking achievements, including the irrefutability of Cantor's continuum hypothesis and the ontological proof of God in the language of modal logic.