Set-Theoretic & Order Structures

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Abstractions about the foundational architecture of infinite sets and ordered structures, spanning ordinal and cardinal arithmetic (aleph and beth numbers, admissible sets), order-theoretic completeness (lattices, ideals, posets), and set-theoretic forcing and combinatorics (diamond and square principles, Aronszajn trees, proper forcing axiom).

53 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.

• Additively indecomposable ordinal — A nonzero ordinal alpha that cannot be reached or exceeded by adding two smaller ordinals, equivalently an ordinal of the form omega raised to an ordinal power.
• Admissible set — A transitive set whose membership structure satisfies Kripke–Platek set theory.
• Aleph number — A member of the transfinite sequence of well-ordered infinite cardinalities, indexed by ordinals with aleph-null as the size of the natural numbers.
• Amoeba order — An order on cardinal characteristics that compares how strongly families of measure-small sets can cover or absorb other small sets under a declared null-ideal convention.
• Aronszajn line — A linear order of cardinality aleph-one containing neither an omega-one or reverse-omega-one suborder nor an uncountable real-type suborder.
• Aronszajn tree — A tree of height ω₁ whose levels and branches are all countable, generalized to κ-trees with levels and branches smaller than κ.
• Axiom of infinity — Assert in Zermelo–Fraenkel set theory that an inductive set exists—one containing the empty set and closed under the successor x mapped to x union singleton x—thereby supplying a set from which omega and the natural-number sequence can be isolated.
• Beth number — A transfinite cardinal sequence beginning at countable infinity and repeatedly applying power set at successors and supremum at limit ordinals.
• Binary relation — Represent which ordered pairs from two declared sets stand in a relation by selecting a subset of their Cartesian product, enabling converse, composition, closure, and relational properties.
• Biordered set — An abstract set of idempotent-like elements equipped with compatible left and right quasiorders and partial basic products that axiomatize the idempotent structure of a semigroup.
• Bounded complete poset — A partially ordered set in which every subset having an upper bound also has a least upper bound, expressing completeness for mutually consistent collections.
• Club principle — A set-theoretic guessing principle asserting a sequence of cofinal subsets that is fully contained in every unbounded set at some indexed stage.
• Complement (set theory) — The set of elements in a declared universe that are not members of a selected set, or the elements of one set left after removing another.
• Complete Heyting algebra — Combine arbitrary joins and meets with Heyting implication, equivalently requiring finite meets to distribute over arbitrary joins, to form the algebraic objects called frames.
• Complete lattice — A partially ordered set in which every subset, including the empty set, has both a supremum and an infimum.
• Completely distributive lattice — A complete lattice in which arbitrary meets distribute over arbitrary joins according to the choice-function identity, equivalently satisfying the self-dual complete distributivity law.
• Dedekind cut — Represent a boundary in a linear order by a downward-closed lower part with no greatest element, constructing order completion and the real numbers from rational cuts.
• Diaconescu's theorem — The constructive-logic result that a sufficiently strong axiom of choice entails the law of excluded middle.
• Diamond principle — A set-theoretic guessing principle asserting a sequence that correctly anticipates every subset of the first uncountable ordinal on a stationary set.
• Distributivity (order theory) — A family of order-theoretic laws governing how infima and suprema interact, from finite lattice distributivity to complete and infinite distributive variants.
• Finite set — A set equipotent with the natural numbers below some n, equivalently one whose elements can be completely counted and assigned a natural-number cardinality.
• Frink ideal — A subset I of a partially ordered set such that every common lower bound of the common upper bounds of each finite subset of I also belongs to I.
• Ideal (order theory) — A nonempty directed lower set of a partially ordered set, equivalently in a lattice a lower set closed under finite joins.
• Idempotent Relation — A binary relation on one set is idempotent when composing it with itself yields exactly the same related pairs.
• Join and meet — The least upper bound and greatest lower bound, respectively, of a subset in a partially ordered set when those bounds exist.
• Knaster's condition — A chain condition on a partial order requiring every uncountable subset to contain an uncountable pairwise-compatible or linked subset.
• Kripke–Platek set theory — A weak axiomatic set theory centered on bounded separation and collection, used to formalize admissible sets and the predicative or recursion-theoretic fragment of set theory.
• Kurepa tree — An uncountable tree of height omega-one with countable levels and at least omega-two many cofinal branches.
• L(R) — The smallest transitive inner model of ZF containing every ordinal and every real, constructed by iterating definability from the real numbers and used to study determinacy under large-cardinal assumptions.
• Monad (nonstandard analysis) — The set of hyperreal points infinitesimally close to a given hyperreal point, with a finite point's monad containing exactly one real standard part.
• Non-well-founded set theory — Study axiomatic set universes in which membership may contain infinite descent or cycles because Foundation is omitted or replaced by a declared anti-foundation principle, with graph decoration and bisimulation specifying which circular presentations denote equal sets.
• Normal measure — A kappa-complete nonprincipal ultrafilter on a measurable cardinal that is closed under diagonal intersections, equivalently makes every regressive function constant on a measure-one set.
• Ordinal definable set — A set uniquely definable in some rank-initial universe by a first-order formula using finitely many ordinal parameters.
• Partially ordered set — A set equipped with a reflexive, antisymmetric and transitive binary relation whose elements need not all be comparable.
• Pocket set theory — Pocket set theory (PST) is an alternative set theory in which there are only two infinite cardinal numbers, ℵ 0 (aleph-naught, the cardinality of the set of all natural numbers) and c (the cardinality of the continuum).
• Proper forcing axiom — A strong set-theoretic forcing axiom asserting that for any proper partial order and any family of ℵ₁ dense sets, a filter meeting every one of them exists.
• Property of Baire — The property of a set that it differs from some open set by a meager set.
• Ramified forcing — Cohen's original forcing construction, which adds a generic object while building names through a hierarchy ramified by formula complexity and constructibility assumptions.
• Reflection principle — A set-theoretic principle asserting that any specified finite collection of truths about the universe of sets already holds in some set-sized rank-initial structure, with stronger variants serving as large-cardinal axioms.
• Sperner property of a partially ordered set — The property that a graded poset’s largest antichain has the same size as its largest rank level.
• Square principle — A set-theoretic principle asserting a coherent sequence of short club sets with no single global thread.
• Strong antichain — An antichain whose distinct elements have no common lower bound, or dually no common upper bound, in the ambient poset.
• Strong measure zero set — A set coverable, for every positive length sequence, by intervals whose respective lengths are bounded by that sequence.
• Supercompact cardinal — A large cardinal kappa for which, at every target scale lambda, there is an elementary embedding with critical point kappa into an inner model closed under lambda-length sequences.
• Syndetic set — A subset of a semigroup—canonically the natural numbers—with uniformly bounded gaps or finitely many left translates covering the carrier.
• Tarski–Grothendieck set theory — An axiomatic set theory extending ZFC with an axiom that places every set inside a Grothendieck-style universe, thereby implying unbounded inaccessible cardinals.
• Transfinite number — An ordinal or cardinal number larger than every finite number, used to order or measure infinite sets.
• Universal set — A set intended to contain every object admitted by a theory, including every set in the relevant universe and potentially itself.
• Well-quasi-ordering — A quasi-order in which every infinite sequence contains an earlier element below a later one, equivalently having neither infinite descending chains nor infinite antichains.
• Whitehead problem — The question whether every abelian group A with Ext-one of A and the integers equal to zero must be free, a statement independent of ZFC.
• XYZ inequality — A correlation inequality constraining relative ordering probabilities for three incomparable elements in a finite partially ordered set.
• Η set — An eta-alpha set is a dense linear order in which every two less-than-aleph-alpha-sized subsets separated left from right have an interpolating element.
• Ω-logic — An infinitary set-theoretic deductive system whose validity is defined through universally Baire sets and generic extensions under large-cardinal assumptions.