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1931 – incompleteness theorems

## 1931 – incompleteness theorems

In 1931, Kurt Godel published two theorems of mathematical logic that have become known as the incompleteness theorems. Prior to these theorems, many mathematicians were trying to prove that all of mathematics could someday be encoded in a complete set of axioms.

First incompleteness theorem:

Any effectively generated theory capable of expressing elementary arithmetic cannot be both consistent and complete. In particular, for any consistent, effectively generated formal theory that proves certain basic arithmetic truths, there is an arithmetical statement that is true,[1] but not provable in the theory.

Translation – It’s impossible to find a complete system of mathematical axioms that can prove all mathematical truths with no inconsistencies. There will always be some statements that can’t be proven either true or false without going outside of the system of axioms, which introduces a larger system with it’s own unprovable statements.

Second incompleteness theorem:

For any formal recursively enumerable (i.e., effectively generated) theory T including basic arithmetical truths and also certain truths about formal provability, T includes a statement of its own consistency if and only if T is inconsistent.

Translation: If a system of axioms can be proven to be complete and consistent entirely from within itself, then it is inconsistent. This implies that any such system will always be incomplete.

PRECURSOR:
1862 – Hilbert Hilberts second problem