Proof theory

rigorous demonstration that a mathematical statement follows from its premises

theorem that a wide class of logical systems cannot be both consistent and complete

In logic and linguistics, a metalanguage is a language used to describe another language, often called the object language. Expressions in a metalanguage are often distinguished from those in the object language by the use of italics, quotation marks, or writing on a separate line. The structure of sentences and phrases in a metalanguage can be described by a metasyntax. For example, to say that the word "noun" can be used as a noun in a sentence, one could write "noun" is a .

branch of mathematical logic

certain type of mistaken proof

sufficient evidence or a sufficient argument for the truth of a proposition

kind of proof calculus

In logic, soundness can refer to either a property of arguments or a property of formal deductive systems.

the direct relationship between computer programs and mathematical proofs

attempt to formalize all of mathematics, based on a finite set of axioms

in logic, rule of inference

fundamental theorem in mathematical logic

property of theories that have computable membership

term in mathematical logic

fundamental concept in metalogic, and the term may be used without qualification with differing meanings depending on the context within mathematical logic

theorem

style of formal logical argumentation

establishment of a theorem using inference from the axioms

theorem

first-order theory of the natural numbers with addition

reduction of first-order mathematical logic to propositional logic

system of formal deduction in logic

Branch of mathematical logic

ordinal-indexed family of rapidly increasing functions: ℕ→ℕ

modal logic

In mathematical logic, a sequent is a very general kind of conditional assertion.

area of research

statement that asserts that a certain premise is true

interpretation of intuitionistic logic

mathematical symbol

large countable ordinal; the proof-theoretic ordinal of arithmetical transfinite recursion

system of arithmetic in proof theiry

In mathematical logic, realizability is a collection of methods in proof theory used to study constructive proofs and extract additional information from them. Formulas from a formal theory are "realized" by objects, known as "realizers", in a way that knowledge of the realizer gives knowledge about the truth of the formula. There are many variations of realizability; exactly which class of formulas is studied and which objects are realizers differ from one variation to another.