{"/e/sets-and-logic/union-set-theory/": {"t": "Union (set theory)", "s": "Set operation combining all elements from given sets."}, "/e/sets-and-logic/empty-set/": {"t": "Empty set", "s": "The unique set containing no elements."}, "/e/sets-and-logic/set-theory/": {"t": "Set theory", "s": "The branch of mathematical logic that studies sets, founded by Georg Cantor in the 1870s."}, "/e/sets-and-logic/zermelo-fraenkel-set-theory/": {"t": "Zermelo–Fraenkel set theory", "s": "Zermelo–Fraenkel set theory is an axiomatic system that formulates a theory of sets free of paradoxes such as Russell's paradox."}, "/e/sets-and-logic/axiom-of-choice/": {"t": "Axiom of choice", "s": "Axiom asserting existence of a choice function for any collection of nonempty sets."}, "https://codexery.com/historical-figures/e/notable/georg-cantor/": {"t": "Georg Cantor", "s": "Founder of set theory and transfinite numbers"}, "/e/sets-and-logic/cardinal-number/": {"t": "Cardinal number", "s": "A number measuring the size of a set."}, "/e/sets-and-logic/russell-s-paradox/": {"t": "Russell's paradox", "s": "A set-theoretic paradox showing that unrestricted comprehension leads to contradictions."}, "https://codexery.com/historical-figures/e/notable/david-hilbert/": {"t": "David Hilbert", "s": "German mathematician and philosopher of mathematics"}, "https://codexery.com/discovery-and-invention-controversies/e/discovery-and-invention-controversies/controversy-over-cantor-s-theory/": {"t": "Controversy over Cantor's theory", "s": "Cantor's set theory was controversial but later largely accepted."}, "https://codexery.com/set-theory-logic/e/set-theory-and-logic/naive-set-theory/": {"t": "Naive set theory", "s": "Informal set theory foundational to modern mathematics."}, "https://codexery.com/set-theory-and-foundations/e/set-theory-and-foundations/axiom-of-extensionality/": {"t": "Axiom of extensionality", "s": "Axiom defining set equality by identical members."}, "https://codexery.com/combinatorics/e/combinatorics/bijection/": {"t": "Bijection", "s": "A function pairing each element of two sets exactly once.", "i": "https://codexery.com/combinatorics/img/combinatorics/bijection.webp", "c": "Schapel · Public domain"}, "https://codexery.com/geometry-and-shapes/e/geometry-and-shapes/space-mathematics/": {"t": "Space (mathematics)", "s": "A set with structure defining relationships among its elements."}, "https://codexery.com/mathematicians/e/mathematicians/ernst-zermelo/": {"t": "Ernst Zermelo", "s": "German logician and mathematician who axiomatized set theory.", "i": "https://codexery.com/mathematicians/img/mathematicians/ernst-zermelo.webp", "c": "via Wikipedia: Ernst Zermelo · see source"}, "/e/sets-and-logic/universe-mathematics/": {"t": "Universe (mathematics)", "s": "A universe is a collection that contains all entities one wishes to consider in a given situation."}, "https://codexery.com/set-theory-logic/e/set-theory-and-logic/operation-mathematics/": {"t": "Operation (mathematics)", "s": "A function combining elements of a set into another element."}, "https://codexery.com/graphs-and-data-visualization/e/graphs-and-data-visualization/venn-diagram/": {"t": "Venn diagram", "s": "A diagram style showing logical relations between sets, popularized by John Venn.", "i": "https://codexery.com/graphs-and-data-visualization/img/graphs-and-data-visualization/venn-diagram.webp", "c": "ASpiegler · CC BY-SA 4.0"}, "/e/sets-and-logic/structure-mathematical-logic/": {"t": "Structure (mathematical logic)", "s": "A structure is a set with finitary operations and relations, used to define the semantics of first-order logic."}, "/e/sets-and-logic/theory-mathematical-logic/": {"t": "Theory (mathematical logic)", "s": "A set of sentences in a formal language, closed under logical consequence."}, "/e/sets-and-logic/set-mathematics/": {"t": "Set (mathematics)", "s": "A set is a collection of different things, typically mathematical objects."}, "https://codexery.com/european-scientific-pioneers/e/european-scientific-pioneers/richard-dedekind/": {"t": "Richard Dedekind", "s": "German mathematician who defined real numbers via Dedekind cuts."}, "https://codexery.com/set-theory-and-foundations/e/set-theory-and-foundations/axiomatic-system/": {"t": "Axiomatic system", "s": "Axiomatic systems: the deductive backbone of mathematics and logic."}, "https://codexery.com/computing-digital/e/computing-and-digital/computer-science/": {"t": "Computer science", "s": "Study of computation, information, and automation spanning theory to application."}, "https://codexery.com/set-theory-and-foundations/e/set-theory-and-foundations/large-cardinal/": {"t": "Large cardinal", "s": "Large cardinal axioms extend ZFC and are linearly ordered by consistency strength."}, "https://codexery.com/historical-figures/e/notable/carl-friedrich-gauss/": {"t": "Carl Friedrich Gauss", "s": "German mathematician, astronomer, geodesist, and physicist"}, "https://codexery.com/scientists-and-mathematicians/e/scientists-and-mathematicians/bernhard-riemann/": {"t": "Bernhard Riemann", "s": "German mathematician who revolutionized analysis, geometry, and number theory"}, "https://codexery.com/topology-and-abstract-spaces/e/topology-and-abstract-spaces/topology/": {"t": "Topology", "s": "Study of properties preserved under continuous deformations."}, "https://codexery.com/geometry-and-topology/e/geometry-and-topology/manifold/": {"t": "Manifold", "s": "A topological space locally resembling Euclidean space near each point."}, "https://codexery.com/set-theory-and-foundations/e/set-theory-and-foundations/class-set-theory/": {"t": "Class (set theory)", "s": "A class is a collection of mathematical objects defined by a property, differing from sets to avoid paradoxes."}, "https://codexery.com/set-theory-logic/e/set-theory-and-logic/partition-of-a-set/": {"t": "Partition of a set", "s": "Grouping elements into non-empty, disjoint subsets that cover the whole set."}, "https://codexery.com/set-theory-and-foundations/e/set-theory-and-foundations/cardinality/": {"t": "Cardinality", "s": "Fundamental concept defining the size of sets."}, "https://codexery.com/set-theory-and-foundations/e/set-theory-and-foundations/cantor-s-diagonal-argument/": {"t": "Cantor's diagonal argument", "s": "Proof that some infinities are larger than others."}, "/e/sets-and-logic/power-set/": {"t": "Power set", "s": "The set of all subsets of a given set."}, "https://codexery.com/set-theory-and-foundations/e/set-theory-and-foundations/axiom-of-union/": {"t": "Axiom of union", "s": "The axiom that unpacks sets of sets."}, "https://codexery.com/set-theory-and-foundations/e/set-theory-and-foundations/axiom-of-pairing/": {"t": "Axiom of pairing", "s": "Any two objects have a pair."}, "https://codexery.com/set-theory-and-foundations/e/set-theory-and-foundations/complement-set-theory/": {"t": "Complement (set theory)", "s": "Set of elements not in a given set."}, "https://codexery.com/set-theory-logic/e/set-theory-and-logic/reflexive-relation/": {"t": "Reflexive relation", "s": "A binary relation where every element relates to itself."}, "https://codexery.com/geometry-and-topology/e/geometry-and-topology/cartesian-product/": {"t": "Cartesian product", "s": "Set of all ordered pairs from two sets."}, "/e/sets-and-logic/transfinite-number/": {"t": "Transfinite number", "s": "Numbers larger than all finite numbers, including transfinite cardinals and ordinals."}, "/e/sets-and-logic/von-neumann-universe/": {"t": "Von Neumann universe", "s": "Class of hereditary well-founded sets in ZFC set theory."}, "/e/sets-and-logic/aleph-number/": {"t": "Aleph number", "s": "Aleph numbers measure sizes of infinite sets."}, "https://codexery.com/set-theory-and-foundations/e/set-theory-and-foundations/axiom-of-regularity/": {"t": "Axiom of regularity", "s": "Axiom ensuring no set contains itself and no infinite descending chains."}, "https://codexery.com/mathematicians/e/mathematicians/john-von-neumann/": {"t": "John von Neumann", "s": "Mathematician who shaped quantum physics, computing, and nuclear strategy."}, "/e/sets-and-logic/well-formed-formula/": {"t": "Well-formed formula", "s": "A finite sequence of symbols from a given alphabet, constructed following the defined grammar of a formal language."}, "https://codexery.com/polish-inventions/e/polish-inventions/polish-notation/": {"t": "Polish notation", "s": "Prefix notation invented by Jan Łukasiewicz in 1924."}, "https://codexery.com/mathematical-logic-and-computation/e/mathematical-logic-and-computation/truth-table/": {"t": "Truth table", "s": "A tabular method for evaluating logical expressions."}, "https://codexery.com/historical-figures/e/notable/charles-sanders-peirce/": {"t": "Charles Sanders Peirce", "s": "American scientist, mathematician, logician, and philosopher; father of pragmatism"}, "https://codexery.com/philosophers-thinkers/e/philosophers-and-thinkers/alfred-north-whitehead/": {"t": "Alfred North Whitehead", "s": "English mathematician and philosopher who founded process philosophy."}, "https://codexery.com/historical-figures/e/notable/bertrand-russell/": {"t": "Bertrand Russell", "s": "English philosopher, logician, and social reformer"}, "https://codexery.com/set-theory-logic/e/set-theory-and-logic/propositional-logic/": {"t": "Propositional logic", "s": "Branch of classical logic dealing with propositions and logical connectives."}, "https://codexery.com/classical-philosophy-rhetoric/e/classical-philosophy-and-rhetoric/principia-mathematica/": {"t": "Principia Mathematica", "s": "Three-volume work on foundations of mathematics by Whitehead and Russell."}, "https://codexery.com/geometry-and-topology/e/geometry-and-topology/affine-transformation/": {"t": "Affine transformation", "s": "A transformation preserving lines and parallelism in affine spaces."}, "/e/sets-and-logic/type-model-theory/": {"t": "Type (model theory)", "s": "A set of formulas describing possible element behavior in a structure."}, "/e/sets-and-logic/type-theory/": {"t": "Type theory", "s": "A formal system classifying expressions by types, used in logic, programming, and mathematics."}, "https://codexery.com/historical-figures/e/scientists/willard-van-orman-quine/": {"t": "Willard Van Orman Quine", "s": "American logician and philosopher who developed naturalized epistemology"}, "/e/sets-and-logic/signature-logic/": {"t": "Signature (logic)", "s": "Formal description of non-logical symbols in a language."}, "https://codexery.com/computing-digital/e/computing-and-digital/database/": {"t": "Database", "s": "Organized data store managed by a database management system."}, "https://codexery.com/greek-inventions/e/greek-inventions/euclidean-geometry/": {"t": "Euclidean geometry", "s": "Ancient Greek mathematician who systematized geometry in Elements."}, "https://codexery.com/mathematical-logic-and-computation/e/mathematical-logic-and-computation/decidability-logic/": {"t": "Decidability (logic)", "s": "Decidability concerns effective methods for logical membership."}, "/e/sets-and-logic/universal-quantification/": {"t": "Universal quantification", "s": "A logical constant interpreted as \"for all\" or \"for every\", expressing that a predicate holds for every member of a domain."}, "https://codexery.com/mathematical-logic-and-computation/e/mathematical-logic-and-computation/algorithm/": {"t": "Algorithm", "s": "A finite sequence of logical instructions for solving problems."}, "https://codexery.com/geometry-and-topology/e/geometry-and-topology/affine-geometry/": {"t": "Affine geometry", "s": "Geometry without distance or angle, focusing on parallelism."}, "https://codexery.com/geometry-and-topology/e/geometry-and-topology/metric-space/": {"t": "Metric space", "s": "A set with a distance function between its points."}, "https://codexery.com/geometry-and-shapes/e/geometry-and-shapes/three-dimensional-space/": {"t": "Three-dimensional space", "s": "Three coordinates define a point in space."}, "https://codexery.com/geometry-and-topology/e/geometry-and-topology/congruence-geometry/": {"t": "Congruence (geometry)", "s": "Two figures are congruent if they have identical shape and size."}, "https://codexery.com/geometry-and-shapes/e/geometry-and-shapes/solid-geometry/": {"t": "Solid geometry", "s": "Geometry of three-dimensional Euclidean space and solid figures."}, "/e/sets-and-logic/subset/": {"t": "Subset", "s": "A set whose elements all belong to another set."}, "https://codexery.com/philosophical-concepts/e/philosophical-concepts/contradiction/": {"t": "Contradiction", "s": "A proposition unconditionally false, central to logical proof."}, "https://codexery.com/philosophical-traditions/e/philosophical-traditions/contingency-philosophy/": {"t": "Contingency (philosophy)", "s": "A statement that is neither necessary nor impossible."}, "https://codexery.com/set-theory-logic/e/set-theory-and-logic/quantifier-logic/": {"t": "Quantifier (logic)", "s": "Operators specifying how many individuals satisfy a property."}, "https://codexery.com/linguistic-concepts-and-phenomena/e/linguistic-concepts-and-phenomena/pejorative/": {"t": "Pejorative", "s": "A term expressing negative or disrespectful connotation."}, "https://codexery.com/historical-figures/e/scientists/immanuel-kant/": {"t": "Immanuel Kant", "s": "Central thinker of the Enlightenment, known for transcendental idealism and the categorical imperative."}, "https://codexery.com/mathematicians/e/mathematicians/gottlob-frege/": {"t": "Gottlob Frege", "s": "German philosopher, logician, and mathematician; father of analytic philosophy.", "i": "https://codexery.com/mathematicians/img/mathematicians/gottlob-frege.webp", "c": "via Wikipedia: Gottlob Frege · see source"}, "https://codexery.com/philosophy-theory/e/philosophy-and-theory/tractatus-logico-philosophicus/": {"t": "Tractatus Logico-Philosophicus", "s": "A philosophical work on language, reality, and the limits of science.", "i": "https://codexery.com/philosophy-theory/img/philosophy-and-theory/tractatus-logico-philosophicus.webp", "c": "via Wikipedia: Tractatus Logico-Philosophicus · see source"}, "https://codexery.com/mathematicians/e/mathematicians/henri-poincar/": {"t": "Henri Poincaré", "s": "French polymath and last universalist mathematician.", "i": "https://codexery.com/mathematicians/img/mathematicians/henri-poincar.webp", "c": "via Wikipedia: Henri Poincaré · see source"}, "https://codexery.com/scientists-and-mathematicians/e/scientists-and-mathematicians/kurt-g-del/": {"t": "Kurt Gödel", "s": "Logician who proved limits of formal axiomatic systems."}, "https://codexery.com/set-theory-logic/e/set-theory-and-logic/recursion/": {"t": "Recursion", "s": "A process defined by reference to a simpler version of itself."}, "https://codexery.com/set-theory-logic/e/set-theory-and-logic/rule-of-inference/": {"t": "Rule of inference", "s": "Norms for deriving conclusions from premises in deductive logic."}, "https://codexery.com/philosophical-concepts-and-theories/e/philosophical-concepts-and-theories/logical-consequence/": {"t": "Logical consequence", "s": "A fundamental concept in logic describing valid inference."}, "https://codexery.com/set-theory-logic/e/set-theory-and-logic/principle-of-explosion/": {"t": "Principle of explosion", "s": "From a contradiction, any proposition can be inferred."}, "/e/sets-and-logic/ultraproduct/": {"t": "Ultraproduct", "s": "Quotient of a direct product modulo an ultrafilter."}, "/e/sets-and-logic/ordinal-number/": {"t": "Ordinal number", "s": "Ordinals extend counting to infinite well-ordered sets."}, "https://codexery.com/geometry-and-topology/e/geometry-and-topology/topological-space/": {"t": "Topological space", "s": "A set with a structure defining closeness without distance."}, "https://codexery.com/geometry-and-topology/e/geometry-and-topology/compact-space/": {"t": "Compact space", "s": "Compactness makes a space behave like a finite set."}, "https://codexery.com/set-theory-logic/e/set-theory-and-logic/quantifier-elimination/": {"t": "Quantifier elimination", "s": "Quantifier elimination is a concept of simplification used in mathematical logic, model theory, and theoretical computer science."}, "https://codexery.com/algebra/e/algebra/polynomial-ring/": {"t": "Polynomial ring", "s": "Ring formed from polynomials with coefficients in another ring."}, "https://codexery.com/complexity-theory/e/complexity-theory/theoretical-computer-science/": {"t": "Theoretical computer science", "s": "Study of abstract and mathematical foundations of computation."}, "https://codexery.com/computing-software/e/computing-and-software/programming-language/": {"t": "Programming language", "s": "Engineered languages for expressing computer programs."}, "https://codexery.com/computer-scientists-and-programmers/e/computer-scientists-and-programmers/alonzo-church/": {"t": "Alonzo Church", "s": "Founder of computer science and inventor of lambda calculus."}, "https://codexery.com/geometry-and-topology/e/geometry-and-topology/algebraic-topology/": {"t": "Algebraic topology", "s": "Branch of mathematics using algebra to study topological spaces."}, "https://codexery.com/computing-digital/e/computing-and-digital/compiler/": {"t": "Compiler", "s": "Software that translates high-level code into low-level machine code."}, "https://codexery.com/communication-and-media/e/communication-and-media/natural-language/": {"t": "Natural language", "s": "A language emerging organically in a human community."}, "https://codexery.com/archaeologists-and-anthropologists/e/archaeologists-and-anthropologists/gregory-bateson/": {"t": "Gregory Bateson", "s": "English anthropologist and cyberneticist known for double-bind theory and systems thinking."}, "https://codexery.com/set-theory-logic/e/set-theory-and-logic/reductio-ad-absurdum/": {"t": "Reductio ad absurdum", "s": "Proof by contradiction: one of a mathematician's finest weapons."}, "/e/sets-and-logic/sheffer-stroke/": {"t": "Sheffer stroke", "s": "The logical NAND operation, also known as the Sheffer stroke, is functionally complete and fundamental to digital electronics."}, "https://codexery.com/set-theory-logic/e/set-theory-and-logic/new-foundations/": {"t": "New Foundations", "s": "A non-well-founded set theory conceived by Quine."}, "https://codexery.com/mathematicians/e/mathematicians/hermann-weyl/": {"t": "Hermann Weyl", "s": "Universal mathematician who bridged pure mathematics and theoretical physics.", "i": "https://codexery.com/mathematicians/img/mathematicians/hermann-weyl.webp", "c": "via Wikipedia: Hermann Weyl · see source"}, "/e/sets-and-logic/transfinite-induction/": {"t": "Transfinite induction", "s": "Extension of mathematical induction to ordinal numbers."}, "https://codexery.com/astronomy-space/e/astronomy-and-space/observable-universe/": {"t": "Observable universe", "s": "The portion of the universe visible from Earth."}, "https://codexery.com/set-theory-and-foundations/e/set-theory-and-foundations/axiom-of-infinity/": {"t": "Axiom of infinity", "s": "Axiom guaranteeing existence of an infinite set."}, "/e/sets-and-logic/well-ordering-theorem/": {"t": "Well-ordering theorem", "s": "Every set can be well-ordered, equivalent to axiom of choice."}, "https://codexery.com/algebra/e/algebra/ring-mathematics/": {"t": "Ring (mathematics)", "s": "Algebraic structure with two binary operations, generalizing integer arithmetic."}, "https://codexery.com/mathematics-and-cryptography/e/mathematics-and-cryptography/wac-aw-sierpi-ski/": {"t": "Wacław Sierpiński", "s": "Mathematician mentioned in the Mathemalchemy installation."}, "https://codexery.com/set-theory-and-foundations/e/set-theory-and-foundations/cardinality-of-the-continuum/": {"t": "Cardinality of the continuum", "s": "The cardinality of the set of real numbers."}, "/e/sets-and-logic/zorn-s-lemma/": {"t": "Zorn's lemma", "s": "A set theory lemma equivalent to the axiom of choice."}, "https://codexery.com/mathematicians/e/mathematicians/felix-hausdorff/": {"t": "Felix Hausdorff", "s": "Founder of modern topology and pioneer of set theory", "i": "https://codexery.com/mathematicians/img/mathematicians/felix-hausdorff.webp", "c": "via Wikipedia: Felix Hausdorff · see source"}, "/e/sets-and-logic/cantor-s-theorem/": {"t": "Cantor's theorem", "s": "No set can be mapped onto its power set."}, "https://codexery.com/historical-figures/e/scientists/hilary-putnam/": {"t": "Hilary Putnam", "s": "American philosopher who changed positions through rigorous self-critique."}, "https://codexery.com/set-theory-and-foundations/e/set-theory-and-foundations/axiom-schema-of-specification/": {"t": "Axiom schema of specification", "s": "Axiom schema that asserts any definable subclass of a set is itself a set."}, "https://codexery.com/set-theory-and-foundations/e/set-theory-and-foundations/axiom-of-determinacy/": {"t": "Axiom of determinacy", "s": "Axiom asserting every infinite two-player game of a certain type is determined."}, "https://codexery.com/geometry-and-topology/e/geometry-and-topology/basis-linear-algebra/": {"t": "Basis (linear algebra)", "s": "A basis is a linearly independent spanning set of a vector space."}, "https://codexery.com/complexity-theory/e/complexity-theory/p-versus-np-problem/": {"t": "P versus NP problem", "s": "A major unsolved problem in theoretical computer science."}, "https://codexery.com/set-theory-and-foundations/e/set-theory-and-foundations/axiom-of-constructibility/": {"t": "Axiom of constructibility", "s": "Axiom asserting all sets are constructible, resolving many independent questions."}, "https://codexery.com/set-theory-and-foundations/e/set-theory-and-foundations/axiom-of-global-choice/": {"t": "Axiom of global choice", "s": "A stronger variant of the axiom of choice for proper classes."}, "https://codexery.com/set-theory-and-foundations/e/set-theory-and-foundations/axiom-of-limitation-of-size/": {"t": "Axiom of limitation of size", "s": "Axiom identifying sets as classes smaller than the universe."}, "https://codexery.com/combinatorics/e/combinatorics/combinatorics/": {"t": "Combinatorics", "s": "Mathematics of counting, arrangement, and optimization in finite systems.", "i": "https://codexery.com/combinatorics/img/combinatorics/combinatorics.webp", "c": "Dom walden · CC BY-SA 4.0"}, "https://codexery.com/writing-systems/e/writing-systems/hebrew-script/": {"t": "Hebrew script", "s": "A 22-letter, right-to-left abjad for Hebrew and other Jewish languages.", "i": "https://codexery.com/writing-systems/img/writing-systems/hebrew-script.webp", "c": "Treehill · CC BY-SA 4.0"}, "https://codexery.com/writing-systems/e/writing-systems/greek-alphabet/": {"t": "Greek alphabet", "s": "Earliest alphabet to systematically write vowels and consonants.", "i": "https://codexery.com/writing-systems/img/writing-systems/greek-alphabet.webp", "c": "Unknown author · Public domain"}, "https://codexery.com/set-theory-and-foundations/e/set-theory-and-foundations/axiom-schema-of-replacement/": {"t": "Axiom schema of replacement", "s": "Axiom schema ensuring definable images of sets are sets."}, "https://codexery.com/combinatorics/e/combinatorics/permutation/": {"t": "Permutation", "s": "A bijection rearranging a set's elements into a new order."}, "https://codexery.com/geometry-and-topology/e/geometry-and-topology/closure-mathematics/": {"t": "Closure (mathematics)", "s": "Closure is the smallest superset closed under given operations."}, "https://codexery.com/chess-terminology/e/chess-terminology/chess-opening/": {"t": "Chess opening", "s": "The opening is the initial phase of a chess game."}, "https://codexery.com/set-theory-and-foundations/e/set-theory-and-foundations/axiom-of-power-set/": {"t": "Axiom of power set", "s": "Axiom ensuring every set has a power set."}, "https://codexery.com/combinatorics/e/combinatorics/binomial-theorem/": {"t": "Binomial theorem", "s": "Describes the expansion of powers of a binomial.", "i": "https://codexery.com/combinatorics/img/combinatorics/binomial-theorem.webp", "c": "Cmglee · CC BY-SA 3.0"}, "https://codexery.com/historical-figures/e/notable/edmund-husserl/": {"t": "Edmund Husserl", "s": "Austrian and German philosopher who founded phenomenology."}}