Mathematical proof

https://leanprover.github.io/logic_and_proof/introduction.html#mathematical-proof

• The concept of proof is formalized in mathematical logic, where it is expressed in a formal instead of a natural language.

• A formal proof is a sequence of formulas, written in a formal language, within a formal system of mathematical logic, that starts from a set of assumptions and then applies a rule of inference (from its establshed set of inference rules) deriving subsequent formulas, one after the other, with each one being a logical consequence of the preceding ones. This form makes the concept of proof amenable to analysis and that's exactly the central subject of proof theory.

• Proof theory studies formal proofs and their properties, the most famous and surprising result of which is that, almost all, axiomatic systems can generate undecidable theorems that are not provable within that system (Gödel's incompleteness theorem).