Mathematical Types, Functions & Infinity

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Abstractions about mathematical objects organized through types, functions, sets, progressions, continuity, and notions of infinity. They include identity and intersection types, multivalued and piecewise functions, index sets, generic properties, and transformational or classificatory results.

33 abstractions in this family — domain-specific abstractions that sit near one another in structural-signature space (k-means over structural-signature embeddings). Each is shown with its short description.

• Actual and potential infinity — The philosophical distinction between an infinite totality treated as completed and an indefinitely extendable process in which every attained stage remains finite.
• Anamorphism — A coinductive unfold that maps a seed coalgebra uniquely into a final coalgebra, generating a recursive data structure.
• Classification theorem — A theorem enumerating every object of a declared mathematical type up to a stated equivalence, without omission or redundant equivalence classes.
• Container (type theory) — A shape-and-positions representation of strictly positive collection-like type constructors in dependent type theory.
• Dottie number — The unique real fixed point of cosine, satisfying cos x equals x when the angle is measured in radians.
• Fuzzy number — A normalized convex fuzzy subset of the real line, usually with upper-semicontinuous membership and compact support, representing graded compatibility with numerical values.
• Generic property — A mathematical property holding outside a negligible exceptional set under a declared measure-theoretic, topological or algebraic notion of genericity.
• Geometric progression — A sequence in which every term after the first is obtained by multiplying the preceding term by one fixed common ratio.
• Hardy field — A field of germs of real functions at positive infinity that is closed under differentiation.
• Hermite reciprocity — A representation-theoretic isomorphism exchanging degree and symmetric-power indices for binary forms: Sym^m(Symn V) is isomorphic to Sym^n(Symm V) for two-dimensional V.
• Identity type — A type-theoretic proposition whose inhabitants witness equality between two terms of a type.
• Index set — A set whose elements label the members of an indexed family through a declared assignment.
• Intersection type — A type assigned to values satisfying multiple type descriptions simultaneously, written as an intersection such as sigma ∩ tau.
• Law of continuity — Leibniz's heuristic that rules valid across finite cases may be extended consistently to limiting, infinitesimal or infinite cases, provided the resulting transition preserves the relevant relations.
• Mathematical puzzle — A self-contained problem with explicit rules whose solution requires mathematical reasoning, construction or proof and is pursued primarily as a recreational challenge.
• Multivalued function — A relation assigning each input a set of possible outputs, represented by a subset of the Cartesian product and often equipped with nonempty-value or regularity conditions.
• Number line — A spatial representation of an ordered number system on a directed line, with a chosen origin, unit interval, and monotone coordinate correspondence.
• Osculant — An invariant that vanishes when hypersurfaces have contact of unusually high order at a common point.
• Perpetuant — A stable irreducible covariant in classical invariant theory whose degree–weight space persists once the binary form’s degree exceeds the covariant weight.
• Piecewise function — A function specified by different formulas or rules on declared subsets that partition or cover its domain with consistent treatment of their boundaries.
• Proportionality (mathematics) — A relation in which corresponding quantities maintain a constant ratio, or under inverse proportionality a constant product.
• Pseudoreflection — A finite-order nonidentity linear automorphism whose fixed subspace is a hyperplane.
• Radon–Nikodym set — The convex range of a vector measure formed from several agents' nonatomic valuations of measurable cake pieces, representing all simultaneously attainable value vectors.
• Real-valued function — A function whose codomain is the real numbers.
• Rising sun lemma — A one-dimensional decomposition lemma partitioning the points below a later higher value of a continuous function into disjoint intervals with controlled endpoints.
• Set inversion — The problem of characterizing all inputs whose image under a function lies in a specified target set.
• Sign (mathematics) — A positive, negative, or zero classification attached to a real quantity, and by extension a binary orientation or parity factor represented by plus or minus one in typed mathematical structures.
• Single-crossing condition — An ordered-comparison property under which the difference between two functions changes sign at most once and in a fixed direction.
• Symmetric difference — The set operation retaining elements that belong to exactly one of two sets.
• Unit type — A type with exactly one inhabitant up to equality, carrying no information beyond successful presence or completion.
• Unitary method — A proportional-reasoning technique that first finds the value of one unit and then scales to the requested number of units.
• Universe (mathematics) — A contextually fixed collection large enough to contain every object and construction under consideration while controlling size, paradox, and quantification in mathematical foundations.
• Vague set — A set-valued uncertainty model assigning separate lower evidence for membership and lower evidence against membership, leaving an explicit hesitation interval.