- 1901: Bertrand Russell publishes Russell’s Paradox, which shows that mathematics contains contradictions, and is therefore undecidable and inconsistent. This leads to the foundational crisis of mathematics.
- 1920s: Hilbert’s Program is developed in an attempt to solve this crisis. Hilbert theorises that using a certain set of axioms, all mathematics may be formalised and be methodically shown to be complete, consistent and decidable.
- 1931: Gödel’s Incompleteness Theorem proves that no formal system can be complete nor consistent, as there are some undecidable axioms, and the consistency of the system can never be proven within the system. Therefore it disproves Hilbert’s Program
- 1936: Church-Turing Thesis is developed in order to determine what is defined by ‘methodical’. Alan Turing defines computability as a program that can be run on a Turing Machine. Alonzo Church defines methodical as a way to represent the problem via lambda calculus. Turing realises both formalisms are identical.
- 1936: Turing also proves Hilbert’s Halting Problem is undecidable, via Turing Machines.