# Category: Proof theory

Theories of iterated inductive definitions
In set theory and logic, Buchholz's ID hierarchy is a hierarchy of subsystems of first-order arithmetic. The systems/theories are referred to as "the formal theories of ν-times iterated inductive defi
Deep inference
Deep inference names a general idea in structural proof theory that breaks with the classical sequent calculus by generalising the notion of structure to permit inference to occur in contexts of high
Mathematical fallacy
In mathematics, certain kinds of mistaken proof are often exhibited, and sometimes collected, as illustrations of a concept called mathematical fallacy. There is a distinction between a simple mistake
Resolution (logic)
In mathematical logic and automated theorem proving, resolution is a rule of inference leading to a refutation complete theorem-proving technique for sentences in propositional logic and first-order l
Proof (truth)
A proof is sufficient evidence or a sufficient argument for the truth of a proposition. The concept applies in a variety of disciplines,with both the nature of the evidence or justification and the cr
Disjunction and existence properties
In mathematical logic, the disjunction and existence properties are the "hallmarks" of constructive theories such as Heyting arithmetic and constructive set theories (Rathjen 2005).
Resolution proof reduction via local context rewriting
In proof theory, an area of mathematical logic, resolution proof reduction via local context rewriting is a technique for resolution proof reduction via local context rewriting. This proof compression
Setoid
In mathematics, a setoid (X, ~) is a set (or type) X equipped with an equivalence relation ~. A setoid may also be called E-set, Bishop set, or extensional set. Setoids are studied especially in proof
Cirquent calculus
Cirquent calculus is a proof calculus that manipulates graph-style constructs termed cirquents, as opposed to the traditional tree-style objects such as formulas or sequents. Cirquents come in a varie
Reverse mathematics
Reverse mathematics is a program in mathematical logic that seeks to determine which axioms are required to prove theorems of mathematics. Its defining method can briefly be described as "going backwa
Dialectica interpretation
In proof theory, the Dialectica interpretation is a proof interpretation of intuitionistic arithmetic (Heyting arithmetic) into a finite type extension of primitive recursive arithmetic, the so-called
Hilbert system
In logic, especially mathematical logic, a Hilbert system, sometimes called Hilbert calculus, Hilbert-style deductive system or Hilbert–Ackermann system, is a type of system of formal deduction attrib
System U
In mathematical logic, System U and System U− are pure type systems, i.e. special forms of a typed lambda calculus with an arbitrary number of sorts, axioms and rules (or dependencies between the sort
Bounded quantifier
In the study of formal theories in mathematical logic, bounded quantifiers (a.k.a. restricted quantifiers) are often included in a formal language in addition to the standard quantifiers "∀" and "∃".
Veblen function
In mathematics, the Veblen functions are a hierarchy of normal functions (continuous strictly increasing functions from ordinals to ordinals), introduced by Oswald Veblen in . If φ0 is any normal func
Independence (mathematical logic)
In mathematical logic, independence is the unprovability of a sentence from other sentences. A sentence σ is independent of a given first-order theory T if T neither proves nor refutes σ; that is, it
Focused proof
In mathematical logic, focused proofs are a family of analytic proofs that arise through goal-directed proof-search, and are a topic of study in structural proof theory and reductive logic. They form
LowerUnits
In proof compression LowerUnits (LU) is an algorithm used to compress propositional logic resolution proofs. The main idea of LowerUnits is to exploit the following fact: Theorem: Let be a potentially
Large countable ordinal
In the mathematical discipline of set theory, there are many ways of describing specific countable ordinals. The smallest ones can be usefully and non-circularly expressed in terms of their Cantor nor
Curry–Howard correspondence
In programming language theory and proof theory, the Curry–Howard correspondence (also known as the Curry–Howard isomorphism or equivalence, or the proofs-as-programs and propositions- or formulae-as-
Friedman translation
In mathematical logic, the Friedman translation is a certain transformation of intuitionistic formulas. Among other things it can be used to show that the Π02-theorems of various first-order theories
Non-surveyable proof
In the philosophy of mathematics, a non-surveyable proof is a mathematical proof that is considered infeasible for a human mathematician to verify and so of controversial validity. The term was coined
Structural proof theory
In mathematical logic, structural proof theory is the subdiscipline of proof theory that studies proof calculi that support a notion of analytic proof, a kind of proof whose semantic properties are ex
Takeuti–Feferman–Buchholz ordinal
In the mathematical fields of set theory and proof theory, the Takeuti–Feferman–Buchholz ordinal (TFBO) is a large countable ordinal, which acts as the limit of the range of Buchholz's psi function an
Deduction theorem
In mathematical logic, a deduction theorem is a metatheorem that justifies doing conditional proofs—to prove an implication A → B, assume A as an hypothesis and then proceed to derive B—in systems tha
Peano–Russell notation
In mathematical logic, Peano–Russell notation was Bertrand Russell's application of Giuseppe Peano's logical notation to the logical notions of Frege and was used in the writing of Principia Mathemati
Resolution proof compression by splitting
In mathematical logic, proof compression by splitting is an algorithm that operates as a post-process on resolution proofs. It was proposed by Scott Cotton in his paper "Two Techniques for Minimizing
Epsilon calculus
Hilbert's epsilon calculus is an extension of a formal language by the epsilon operator, where the epsilon operator substitutes for quantifiers in that language as a method leading to a proof of consi
Gödel's completeness theorem
Gödel's completeness theorem is a fundamental theorem in mathematical logic that establishes a correspondence between semantic truth and syntactic provability in first-order logic. The completeness th
Paraconsistent mathematics
Paraconsistent mathematics, sometimes called inconsistent mathematics, represents an attempt to develop the classical infrastructure of mathematics (e.g. analysis) based on a foundation of paraconsist
Gentzen's consistency proof
Gentzen's consistency proof is a result of proof theory in mathematical logic, published by Gerhard Gentzen in 1936. It shows that the Peano axioms of first-order arithmetic do not contain a contradic
Weak interpretability
In mathematical logic, weak interpretability is a notion of translation of logical theories, introduced together with interpretability by Alfred Tarski in 1953. Let T and S be formal theories. Slightl
Geometry of interaction
The Geometry of Interaction (GoI) was introduced by Jean-Yves Girard shortly after his work on linear logic. In linear logic, proofs can be seen as various kinds of networks as opposed to the flat tre
Soundness
In logic, more precisely in deductive reasoning, an argument is sound if it is both valid in form and its premises are true. Soundness also has a related meaning in mathematical logic, wherein logical
Self-verifying theories
Self-verifying theories are consistent first-order systems of arithmetic, much weaker than Peano arithmetic, that are capable of proving their own consistency. Dan Willard was the first to investigate
Metalanguage
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 obje
Reverse Mathematics: Proofs from the Inside Out
Reverse Mathematics: Proofs from the Inside Out is a book by John Stillwell on reverse mathematics, the process of examining proofs in mathematics to determine which axioms are required by the proof.
Pure type system
In the branches of mathematical logic known as proof theory and type theory, a pure type system (PTS), previously known as a generalized type system (GTS), is a form of typed lambda calculus that allo
Gödel's incompleteness theorems
Gödel's incompleteness theorems are two theorems of mathematical logic that are concerned with the limits of provability in formal axiomatic theories. These results, published by Kurt Gödel in 1931, a
Judgment (mathematical logic)
In mathematical logic, a judgment (or judgement) or assertion is a statement or enunciation in a metalanguage. For example, typical judgments in first-order logic would be that a string is a well-form
Completeness (logic)
In mathematical logic and metalogic, a formal system is called complete with respect to a particular property if every formula having the property can be derived using that system, i.e. is one of its
Interpretability
In mathematical logic, interpretability is a relation between formal theories that expresses the possibility of interpreting or translating one into the other.
Redundant proof
In mathematical logic, a redundant proof is a proof that has a subset that is a shorter proof of the same result. In other words, a proof is redundant if it has more proof steps than are actually nece
Extension by definitions
In mathematical logic, more specifically in the proof theory of first-order theories, extensions by definitions formalize the introduction of new symbols by means of a definition. For example, it is c
Nested sequent calculus
In structural proof theory, the nested sequent calculus is a reformulation of the sequent calculus to allow deep inference.
Proof procedure
In logic, and in particular proof theory, a proof procedure for a given logic is a systematic method for producing proofs in some proof calculus of (provable) statements.
Realizability
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 "reali
Büchi arithmetic
Büchi arithmetic of base k is the first-order theory of the natural numbers with addition and the function which is defined as the largest power of k dividing x, named in honor of the Swiss mathematic
Fast-growing hierarchy
In computability theory, computational complexity theory and proof theory, a fast-growing hierarchy (also called an extended Grzegorczyk hierarchy) is an ordinal-indexed family of rapidly increasing f
Proof calculus
In mathematical logic, a proof calculus or a proof system is built to prove statements.
Hypersequent
In mathematical logic, the hypersequent framework is an extension of the proof-theoretical framework of sequent calculi used in structural proof theory to provide analytic calculi for logics that are
Ordinal notation
In mathematical logic and set theory, an ordinal notation is a partial function mapping the set of all finite sequences of symbols, themselves members of a finite alphabet, to a countable set of ordin
Double-negation translation
In proof theory, a discipline within mathematical logic, double-negation translation, sometimes called negative translation, is a general approach for embedding classical logic into intuitionistic log
Turnstile (symbol)
In mathematical logic and computer science the symbol has taken the name turnstile because of its resemblance to a typical turnstile if viewed from above. It is also referred to as tee and is often re
Gödel's speed-up theorem
In mathematics, Gödel's speed-up theorem, proved by Gödel, shows that there are theorems whose proofs can be drastically shortened by working in more powerful axiomatic systems. Kurt Gödel showed how
Natural deduction
In logic and proof theory, natural deduction is a kind of proof calculus in which logical reasoning is expressed by inference rules closely related to the "natural" way of reasoning. This contrasts wi
Primitive recursive functional
In mathematical logic, the primitive recursive functionals are a generalization of primitive recursive functions into higher type theory. They consist of a collection of functions in all pure finite t
Japaridze's polymodal logic
Japaridze's polymodal logic (GLP) is a system of provability logic with infinitely many provability modalities. This system has played an important role in some applications of provability algebras in
Proof-theoretic semantics
Proof-theoretic semantics is an approach to the semantics of logic that attempts to locate the meaning of propositions and logical connectives not in terms of interpretations, as in Tarskian approache
Tolerant sequence
In mathematical logic, a tolerant sequence is a sequence ,..., of formal theories such that there are consistent extensions ,..., of these theories with each interpretable in . Tolerance naturally gen
Ordinal analysis
In proof theory, ordinal analysis assigns ordinals (often large countable ordinals) to mathematical theories as a measure of their strength. If theories have the same proof-theoretic ordinal they are
Feferman–Schütte ordinal
In mathematics, the Feferman–Schütte ordinal Γ0 is a large countable ordinal.It is the proof-theoretic ordinal of several mathematical theories, such as arithmetical transfinite recursion.It is named
Sequent calculus
In mathematical logic, sequent calculus is a style of formal logical argumentation in which every line of a proof is a conditional tautology (called a sequent by Gerhard Gentzen) instead of an uncondi
Proof theory
Proof theory is a major branch of mathematical logic that represents proofs as formal mathematical objects, facilitating their analysis by mathematical techniques. Proofs are typically presented as in
Presburger arithmetic
Presburger arithmetic is the first-order theory of the natural numbers with addition, named in honor of Mojżesz Presburger, who introduced it in 1929. The signature of Presburger arithmetic contains o
Proof net
In proof theory, proof nets are a geometrical method of representing proofs thateliminates two forms of bureaucracy that differentiate proofs: (A) irrelevant syntactical features of regular proof calc
Formal proof
In logic and mathematics, a formal proof or derivation is a finite sequence of sentences (called well-formed formulas in the case of a formal language), each of which is an axiom, an assumption, or fo
Buchholz's ordinal
In mathematics, ψ0(Ωω), widely known as Buchholz's ordinal, is a large countable ordinal that is used to measure the proof-theoretic strength of some mathematical systems. In particular, it is the pro
Completeness of atomic initial sequents
In sequent calculus, the completeness of atomic initial sequents states that initial sequents A ⊢ A (where A is an arbitrary formula) can be derived from only atomic initial sequents p ⊢ p (where p is
Lambda-mu calculus
In mathematical logic and computer science, the lambda-mu calculus is an extension of the lambda calculus introduced by M. Parigot. It introduces two new operators: the μ operator (which is completely
VIPER microprocessor
VIPER is a 32-bit microprocessor design created by Royal Signals and Radar Establishment (RSRE) in the 1980s, intended to be used in safety-critical systems such as avionics. It was the first commerci
Extension by new constant and function names
In mathematical logic, a theory can be extended withnew constants or function names under certain conditions with assurance that the extension will introduceno contradiction. Extension by definitions
Slow-growing hierarchy
In computability theory, computational complexity theory and proof theory, the slow-growing hierarchy is an ordinal-indexed family of slowly increasing functions gα: N → N (where N is the set of natur
Hilbert's program
In mathematics, Hilbert's program, formulated by German mathematician David Hilbert in the early part of the 20th century, was a proposed solution to the foundational crisis of mathematics, when early
Elementary function arithmetic
In proof theory, a branch of mathematical logic, elementary function arithmetic (EFA), also called elementary arithmetic and exponential function arithmetic, is the system of arithmetic with the usual
Bachmann–Howard ordinal
In mathematics, the Bachmann–Howard ordinal (or Howard ordinal) is a large countable ordinal.It is the proof-theoretic ordinal of several mathematical theories, such as Kripke–Platek set theory (with
Proof mining
In proof theory, a branch of mathematical logic, proof mining (or proof unwinding) is a research program that studies or analyzes formalized proofs, especially in analysis, to obtain explicit bounds,
Analytic proof
In mathematics, an analytic proof is a proof of a theorem in analysis that only makes use of methods from analysis, and which does not predominantly make use of algebraic or geometrical methods. The t
Herbrand's theorem
Herbrand's theorem is a fundamental result of mathematical logic obtained by Jacques Herbrand (1930). It essentially allows a certain kind of reduction of first-order logic to propositional logic. Alt
Slicing the Truth
Slicing the Truth: On the Computability Theoretic and Reverse Mathematical Analysis of Combinatorial Principles is a book on reverse mathematics in combinatorics, the study of the axioms needed to pro
Structural rule
In proof theory, a structural rule is an inference rule that does not refer to any logical connective, but instead operates on the judgment or sequents directly. Structural rules often mimic intended
Provability logic
Provability logic is a modal logic, in which the box (or "necessity") operator is interpreted as 'it is provable that'. The point is to capture the notion of a proof predicate of a reasonably rich for
Sequent
In mathematical logic, a sequent is a very general kind of conditional assertion. A sequent may have any number m of condition formulas Ai (called "antecedents") and any number n of asserted formulas
Decidability (logic)
In logic, a true/false decision problem is decidable if there exists an effective method for deriving the correct answer. Zeroth-order logic (propositional logic) is decidable, whereas first-order and
Conservative extension
In mathematical logic, a conservative extension is a supertheory of a theory which is often convenient for proving theorems, but proves no new theorems about the language of the original theory. Simil
Hardy hierarchy
In computability theory, computational complexity theory and proof theory, the Hardy hierarchy, named after G. H. Hardy, is a hierarchy of sets of numerical functions generated from an ordinal-indexed
Consistency
In classical deductive logic, a consistent theory is one that does not lead to a logical contradiction. The lack of contradiction can be defined in either semantic or syntactic terms. The semantic def
Takeuti's conjecture
In mathematics, Takeuti's conjecture is the conjecture of Gaisi Takeuti that a sequent formalisation of second-order logic has cut-elimination (Takeuti 1953). It was settled positively: * By Tait, us
Ω-consistent theory
In mathematical logic, an ω-consistent (or omega-consistent, also called numerically segregative) theory is a theory (collection of sentences) that is not only (syntactically) consistent (that is, doe
Cut-elimination theorem
The cut-elimination theorem (or Gentzen's Hauptsatz) is the central result establishing the significance of the sequent calculus. It was originally proved by Gerhard Gentzen in his landmark 1934 paper
Decidable sublanguages of set theory
In mathematical logic, various sublanguages of set theory are decidable. These include: * Sets with Monotone, Additive, and Multiplicative Functions. * Sets with restricted quantifiers.
Proof compression
In proof theory, an area of mathematical logic, proof compression is the problem of algorithmically compressing formal proofs. The developed algorithms can be used to improve the proofs generated by a
Nonrecursive ordinal
In mathematics, particularly set theory, non-recursive ordinals are large countable ordinals greater than all the recursive ordinals, and therefore can not be expressed using ordinal collapsing functi
Original proof of Gödel's completeness theorem
The proof of Gödel's completeness theorem given by Kurt Gödel in his doctoral dissertation of 1929 (and a shorter version of the proof, published as an article in 1930, titled "The completeness of the