Editing Proof Theory (section)

== <span style="color: #FFFFFF;">Remembering</span> ==
* '''Proof Theory''' — The study of the structure and properties of mathematical proofs.
* '''Formal Proof''' — A proof written in a precise symbolic language that can be checked by a machine.
* '''Axiom''' — A starting assumption that is accepted without proof.
* '''Inference Rule''' — A rule that allows you to move from one step of a proof to the next (e.g., "If $A$ is true, and $A \to B$ is true, then $B$ is true").
* '''Hilbert's Program''' — A 20th-century goal to provide a complete and consistent foundation for all of mathematics.
* '''Consistency Proof''' — A proof that a system of axioms will never lead to a contradiction.
* '''Natural Deduction''' — A way of writing proofs that mimics "natural" human reasoning.
* '''Sequent Calculus''' — A formal system for proof theory that focuses on "Sequents" (statements of logical implication).
* '''Cut-Elimination''' — A key theorem showing that any proof can be "cleaned up" to remove unnecessary intermediate steps.
* '''Gentzen's Theorem''' — The proof that the consistency of arithmetic can be shown using transfinite induction.
* '''Constructive Proof''' — A proof that shows an object exists by actually showing how to "build" it.
* '''Non-Constructive Proof''' — A proof that shows something must exist (usually by contradiction) without showing what it is.
* '''Structural Induction''' — A method of proof used to show that a property holds for all objects in a formally defined set.
* '''Curry-Howard Correspondence''' — The deep link between "Proofs" in logic and "Programs" in computer science.
</div>

<div style="background-color: #006400; color: #FFFFFF; padding: 20px; border-radius: 8px; margin-bottom: 15px;">