Editing Mathematical logic (section)

== Proof theory and constructive mathematics ==
{{Main|Proof theory}}
'''[[Proof theory]]''' is the study of formal proofs in various logical deduction systems. These proofs are represented as formal mathematical objects, facilitating their analysis by mathematical techniques.  Several deduction systems are commonly considered, including [[Hilbert-style deduction system]]s, systems of [[natural deduction]], and the [[sequent calculus]] developed by Gentzen.

The study of '''constructive mathematics''', in the context of mathematical logic, includes the study of systems in non-classical logic such as intuitionistic logic, as well as the study of [[Impredicativity|predicative]] systems.  An early proponent of predicativism was [[Hermann Weyl]], who showed it is possible to develop a large part of real analysis using only predicative methods.{{sfn|Weyl|1918}}

Because proofs are entirely finitary, whereas truth in a structure is not, it is common for work in constructive mathematics to emphasize provability.   The relationship between provability in classical (or nonconstructive) systems and provability in intuitionistic (or constructive, respectively) systems is of particular interest.  Results such as the [[Gödel–Gentzen negative translation]] show that it is possible to embed (or ''[[Logic translation|translate]]'') classical logic into intuitionistic logic, allowing some properties about intuitionistic proofs to be transferred back to classical proofs.

Recent developments in proof theory include the study of [[proof mining]] by [[Ulrich Kohlenbach]] and the study of [[proof-theoretic ordinal]]s by [[Michael Rathjen]].