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Curry's paradox, the Glossary

Curry's paradox is a paradox in which an arbitrary claim F is proved from the mere existence of a sentence C that says of itself "If C, then F".^[1]

34 relations: Axiom schema of specification, China, Combinatory logic, Conditional proof, Conditional sentence, Consistency, Fixed-point combinator, Gödel numbering, Gödel's incompleteness theorems, Germany, Haskell Curry, Implicational propositional calculus, Indicative conditional, Infix notation, Kleene–Rosser paradox, Lambda calculus, Law of identity, Löb's theorem, List of logic symbols, List of paradoxes, Logic, Martin Löb, Modus ponens, Natural deduction, Paradox, Peirce's law, Richard's paradox, Russell's paradox, Set theory, Simply typed lambda calculus, Structural rule, System U, Tautology (logic), Zermelo–Fraenkel set theory.
Mathematical paradoxes
Paradoxes of naive set theory
Self-referential paradoxes

Axiom schema of specification

In many popular versions of axiomatic set theory, the axiom schema of specification, also known as the axiom schema of separation (Aussonderung Axiom), subset axiom or axiom schema of restricted comprehension is an axiom schema.

See Curry's paradox and Axiom schema of specification

China

China, officially the People's Republic of China (PRC), is a country in East Asia.

See Curry's paradox and China

Combinatory logic

Combinatory logic is a notation to eliminate the need for quantified variables in mathematical logic.

See Curry's paradox and Combinatory logic

Conditional proof

A conditional proof is a proof that takes the form of asserting a conditional, and proving that the antecedent of the conditional necessarily leads to the consequent.

See Curry's paradox and Conditional proof

Conditional sentence

Conditional sentences are natural language sentences that express that one thing is contingent on something else, e.g. "If it rains, the picnic will be cancelled." They are so called because the impact of the main clause of the sentence is conditional on the dependent clause.

See Curry's paradox and Conditional sentence

Consistency

In classical deductive logic, a consistent theory is one that does not lead to a logical contradiction.

See Curry's paradox and Consistency

Fixed-point combinator

In combinatory logic for computer science, a fixed-point combinator (or fixpoint combinator), is a higher-order function (i.e. a function which takes a function as argument) that returns some ''fixed point'' (a value that is mapped to itself) of its argument function, if one exists.

See Curry's paradox and Fixed-point combinator

Gödel numbering

In mathematical logic, a Gödel numbering is a function that assigns to each symbol and well-formed formula of some formal language a unique natural number, called its Gödel number. Curry's paradox and Gödel numbering are mathematical logic.

See Curry's paradox and Gödel numbering

Gödel's incompleteness theorems

Gödel's incompleteness theorems are two theorems of mathematical logic that are concerned with the limits of in formal axiomatic theories. Curry's paradox and Gödel's incompleteness theorems are mathematical logic.

See Curry's paradox and Gödel's incompleteness theorems

Germany

Germany, officially the Federal Republic of Germany (FRG), is a country in Central Europe.

See Curry's paradox and Germany

Haskell Curry

Haskell Brooks Curry (September 12, 1900 – September 1, 1982) was an American mathematician and logician.

See Curry's paradox and Haskell Curry

Implicational propositional calculus

In mathematical logic, the implicational propositional calculus is a version of classical propositional calculus that uses only one connective, called implication or conditional.

See Curry's paradox and Implicational propositional calculus

Indicative conditional

In natural languages, an indicative conditional is a conditional sentence such as "If Leona is at home, she isn't in Paris", whose grammatical form restricts it to discussing what could be true.

See Curry's paradox and Indicative conditional

Infix notation

Infix notation is the notation commonly used in arithmetical and logical formulae and statements.

See Curry's paradox and Infix notation

Kleene–Rosser paradox

In mathematics, the Kleene–Rosser paradox is a paradox that shows that certain systems of formal logic are inconsistent, in particular the version of Haskell Curry's combinatory logic introduced in 1930, and Alonzo Church's original lambda calculus, introduced in 1932–1933, both originally intended as systems of formal logic. Curry's paradox and Kleene–Rosser paradox are mathematical paradoxes and self-referential paradoxes.

See Curry's paradox and Kleene–Rosser paradox

Lambda calculus

Lambda calculus (also written as λ-calculus) is a formal system in mathematical logic for expressing computation based on function abstraction and application using variable binding and substitution.

See Curry's paradox and Lambda calculus

Law of identity

In logic, the law of identity states that each thing is identical with itself.

See Curry's paradox and Law of identity

Löb's theorem

In mathematical logic, Löb's theorem states that in Peano arithmetic (PA) (or any formal system including PA), for any formula P, if it is provable in PA that "if P is provable in PA then P is true", then P is provable in PA. Curry's paradox and Löb's theorem are mathematical logic.

See Curry's paradox and Löb's theorem

List of logic symbols

In logic, a set of symbols is commonly used to express logical representation.

See Curry's paradox and List of logic symbols

List of paradoxes

This list includes well known paradoxes, grouped thematically.

See Curry's paradox and List of paradoxes

Logic

Logic is the study of correct reasoning.

See Curry's paradox and Logic

Martin Löb

Martin Hugo Löb (31 March 1921 – 21 August 2006) was a German mathematician.

See Curry's paradox and Martin Löb

Modus ponens

In propositional logic, modus ponens (MP), also known as modus ponendo ponens, implication elimination, or affirming the antecedent, is a deductive argument form and rule of inference.

See Curry's paradox and Modus ponens

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.

See Curry's paradox and Natural deduction

Paradox

A paradox is a logically self-contradictory statement or a statement that runs contrary to one's expectation.

See Curry's paradox and Paradox

Peirce's law

In logic, Peirce's law is named after the philosopher and logician Charles Sanders Peirce. Curry's paradox and Peirce's law are mathematical logic.

See Curry's paradox and Peirce's law

Richard's paradox

In logic, Richard's paradox is a semantical antinomy of set theory and natural language first described by the French mathematician Jules Richard in 1905. Curry's paradox and Richard's paradox are mathematical paradoxes and self-referential paradoxes.

See Curry's paradox and Richard's paradox

Russell's paradox

In mathematical logic, Russell's paradox (also known as Russell's antinomy) is a set-theoretic paradox published by the British philosopher and mathematician Bertrand Russell in 1901. Curry's paradox and Russell's paradox are paradoxes of naive set theory and self-referential paradoxes.

See Curry's paradox and Russell's paradox

Set theory

Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Curry's paradox and set theory are mathematical logic.

See Curry's paradox and Set theory

Simply typed lambda calculus

The simply typed lambda calculus (\lambda^\to), a form of type theory, is a typed interpretation of the lambda calculus with only one type constructor (\to) that builds function types.

See Curry's paradox and Simply typed lambda calculus

Structural rule

In the logical discipline of proof theory, a structural rule is an inference rule of a sequent calculus that does not refer to any logical connective but instead operates on the sequents directly.

See Curry's paradox and Structural rule

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 sorts).

See Curry's paradox and System U

Tautology (logic)

In mathematical logic, a tautology (from ταυτολογία) is a formula or assertion that is true in every possible interpretation. Curry's paradox and tautology (logic) are mathematical logic.

See Curry's paradox and Tautology (logic)

Zermelo–Fraenkel set theory

In set theory, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in the early twentieth century in order to formulate a theory of sets free of paradoxes such as Russell's paradox.

See Curry's paradox and Zermelo–Fraenkel set theory

References

[1] https://en.wikipedia.org/wiki/Curry's_paradox

Also known as Curry paradox, Currys paradox, Löb paradox, Löb's paradox, Loeb paradox, Loeb's paradox.

Curry's paradox, the Glossary

Table of Contents

See also

Mathematical paradoxes

Paradoxes of naive set theory

Self-referential paradoxes

References