[{"n": "Set (mathematics)", "u": "/e/sets-and-logic/set-mathematics/", "s": "A set is a collection of different things, typically mathematical objects."}, {"n": "Set theory", "u": "/e/sets-and-logic/set-theory/", "s": "The branch of mathematical logic that studies sets, founded by Georg Cantor in the 1870s."}, {"n": "Union (set theory)", "u": "/e/sets-and-logic/union-set-theory/", "s": "Set operation combining all elements from given sets."}, {"n": "Subset", "u": "/e/sets-and-logic/subset/", "s": "A set whose elements all belong to another set."}, {"n": "Zermelo–Fraenkel set theory", "u": "/e/sets-and-logic/zermelo-fraenkel-set-theory/", "s": "Zermelo–Fraenkel set theory is an axiomatic system that formulates a theory of sets free of paradoxes such as "}, {"n": "Sheffer stroke", "u": "/e/sets-and-logic/sheffer-stroke/", "s": "The logical NAND operation, also known as the Sheffer stroke, is functionally complete and fundamental to digi"}, {"n": "Signature (logic)", "u": "/e/sets-and-logic/signature-logic/", "s": "Formal description of non-logical symbols in a language."}, {"n": "Structure (mathematical logic)", "u": "/e/sets-and-logic/structure-mathematical-logic/", "s": "A structure is a set with finitary operations and relations, used to define the semantics of first-order logic"}, {"n": "Tarski's axioms", "u": "/e/sets-and-logic/tarski-s-axioms/", "s": "A first-order axiom system for Euclidean geometry, complete and decidable."}, {"n": "Tautology (logic)", "u": "/e/sets-and-logic/tautology-logic/", "s": "A formula true under all interpretations, central to propositional logic."}, {"n": "Theory (mathematical logic)", "u": "/e/sets-and-logic/theory-mathematical-logic/", "s": "A set of sentences in a formal language, closed under logical consequence."}, {"n": "Type (model theory)", "u": "/e/sets-and-logic/type-model-theory/", "s": "A set of formulas describing possible element behavior in a structure."}, {"n": "Type theory", "u": "/e/sets-and-logic/type-theory/", "s": "A formal system classifying expressions by types, used in logic, programming, and mathematics."}, {"n": "Ultraproduct", "u": "/e/sets-and-logic/ultraproduct/", "s": "Quotient of a direct product modulo an ultrafilter."}, {"n": "Universal quantification", "u": "/e/sets-and-logic/universal-quantification/", "s": "A logical constant interpreted as \"for all\" or \"for every\", expressing that a predicate holds for every member"}, {"n": "Universe (mathematics)", "u": "/e/sets-and-logic/universe-mathematics/", "s": "A universe is a collection that contains all entities one wishes to consider in a given situation."}, {"n": "Well-formed formula", "u": "/e/sets-and-logic/well-formed-formula/", "s": "A finite sequence of symbols from a given alphabet, constructed following the defined grammar of a formal lang"}, {"n": "Von Neumann universe", "u": "/e/sets-and-logic/von-neumann-universe/", "s": "Class of hereditary well-founded sets in ZFC set theory."}, {"n": "Zorn's lemma", "u": "/e/sets-and-logic/zorn-s-lemma/", "s": "A set theory lemma equivalent to the axiom of choice."}, {"n": "Transfinite induction", "u": "/e/sets-and-logic/transfinite-induction/", "s": "Extension of mathematical induction to ordinal numbers."}, {"n": "Transfinite number", "u": "/e/sets-and-logic/transfinite-number/", "s": "Numbers larger than all finite numbers, including transfinite cardinals and ordinals."}, {"n": "Well-ordering theorem", "u": "/e/sets-and-logic/well-ordering-theorem/", "s": "Every set can be well-ordered, equivalent to axiom of choice."}, {"n": "Skolem's paradox", "u": "/e/sets-and-logic/skolem-s-paradox/", "s": "A countable model can contain an uncountable set."}, {"n": "Skolem normal form", "u": "/e/sets-and-logic/skolem-normal-form/", "s": "A prenex normal form with only universal quantifiers."}, {"n": "Cantor's theorem", "u": "/e/sets-and-logic/cantor-s-theorem/", "s": "No set can be mapped onto its power set."}, {"n": "Axiom of choice", "u": "/e/sets-and-logic/axiom-of-choice/", "s": "Axiom asserting existence of a choice function for any collection of nonempty sets."}, {"n": "Aleph number", "u": "/e/sets-and-logic/aleph-number/", "s": "Aleph numbers measure sizes of infinite sets."}, {"n": "Cardinal number", "u": "/e/sets-and-logic/cardinal-number/", "s": "A number measuring the size of a set."}, {"n": "Ordinal number", "u": "/e/sets-and-logic/ordinal-number/", "s": "Ordinals extend counting to infinite well-ordered sets."}, {"n": "Empty set", "u": "/e/sets-and-logic/empty-set/", "s": "The unique set containing no elements."}, {"n": "Power set", "u": "/e/sets-and-logic/power-set/", "s": "The set of all subsets of a given set."}, {"n": "Russell's paradox", "u": "/e/sets-and-logic/russell-s-paradox/", "s": "A set-theoretic paradox showing that unrestricted comprehension leads to contradictions."}]