# 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). --- # Agent Instructions: Querying This Documentation If you need additional information that is not directly available in this page, you can query the documentation dynamically by asking a question. Perform an HTTP GET request on the current page URL with the `ask` query parameter: ``` GET https://mandober.gitbook.io/math-debrief/300-logic/380-proof-theory/mathematical-proof.md?ask= ``` The question should be specific, self-contained, and written in natural language. The response will contain a direct answer to the question and relevant excerpts and sources from the documentation. Use this mechanism when the answer is not explicitly present in the current page, you need clarification or additional context, or you want to retrieve related documentation sections.