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In mathematical logic, a Gödel logic, sometimes referred to as Dummett logic or Gödel–Dummett logic, is a member of a family of finite- or infinite-valued logics in which the sets of truth values V are closed subsets of the unit interval containing both 0 and 1. Different such sets V in general determine different Gödel logics. The concept is named after Kurt Gödel.

In 1959, Michael Dummett showed that infinite-valued propositional Gödel logic can be axiomatised by adding the axiom schema

${\displaystyle (A\rightarrow B)\lor (B\rightarrow A)}$

to intuitionistic propositional logic.

See also

• Intermediate logic

References

1. ^ von Plato, Jan (2003). "Skolem's Discovery of Gödel-Dummett Logic". Studia Logica. 73 (1): 153–157. doi:10.1023/A:1022997524909.
2. Baaz, Matthias; Preining, Norbert; Zach, Richard (2007-06-01). "First-order Gödel logics". Annals of Pure and Applied Logic. 147 (1): 23–47. arXiv:math/0601147. doi:10.1016/j.apal.2007.03.001. ISSN 0168-0072.
3. Preining, Norbert (2010). "Gödel Logics – A Survey". Logic for Programming, Artificial Intelligence, and Reasoning. Lecture Notes in Computer Science. Vol. 6397. pp. 30–51. doi:10.1007/978-3-642-16242-8_4. ISBN 978-3-642-16241-1. Retrieved 2 March 2022.
4. Dummett, Michael (1959). "A propositional calculus with denumerable matrix". The Journal of Symbolic Logic. 24 (2): 97–106. doi:10.2307/2964753. ISSN 0022-4812. JSTOR 2964753.

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v t e Set theory
Overview	Set (mathematics)
Axioms	Adjunction Choice countable dependent global Constructibility (V=L) Determinacy projective Extensionality Infinity Limitation of size Pairing Power set Regularity Union Martin's axiom Axiom schema replacement specification
Operations	Cartesian product Complement (i.e. set difference) De Morgan's laws Disjoint union Identities Intersection Power set Symmetric difference Union
Concepts Methods	Almost Cardinality Cardinal number (large) Class Constructible universe Continuum hypothesis Diagonal argument Element ordered pair tuple Family Forcing One-to-one correspondence Ordinal number Set-builder notation Transfinite induction Venn diagram
Set types	Amorphous Countable Empty Finite (hereditarily) Filter base subbase Ultrafilter Fuzzy Infinite (Dedekind-infinite) Recursive Singleton Subset · Superset Transitive Uncountable Universal
Theories	Alternative Axiomatic Naive Cantor's theorem Zermelo General Principia Mathematica New Foundations Zermelo–Fraenkel von Neumann–Bernays–Gödel Morse–Kelley Kripke–Platek Tarski–Grothendieck
Paradoxes Problems	Russell's paradox Suslin's problem Burali-Forti paradox
Set theorists	Paul Bernays Georg Cantor Paul Cohen Richard Dedekind Abraham Fraenkel Kurt Gödel Thomas Jech John von Neumann Willard Quine Bertrand Russell Thoralf Skolem Ernst Zermelo

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