Set Theory

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25 domain-specific abstractions whose origin domain is Set Theory.

• 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.
• 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 determinacy — A set-theoretic axiom asserting that every infinite two-player perfect-information game on natural numbers has a winning strategy for one player.
• Beth number — A transfinite cardinal sequence beginning at countable infinity and repeatedly applying power set at successors and supremum at limit ordinals.
• 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.
• Continuous function (ordinal theory) — An ordinal-indexed sequence whose value at every limit index equals the supremum of its earlier values, usually considered together with monotonicity in transfinite constructions.
• Diamond principle — A set-theoretic guessing principle asserting a sequence that correctly anticipates every subset of the first uncountable ordinal on a stationary set.
• Generic filter — A forcing filter that meets every dense subset of a partial order belonging to a specified ground model.
• Knaster's condition — A chain condition on a partial order requiring every uncountable subset to contain an uncountable pairwise-compatible or linked subset.
• 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.
• Normal function — An ordinal-valued function that is strictly increasing and continuous at limit ordinals, so its value at a limit is the supremum of all earlier values.
• Ordinal definable set — A set uniquely definable in some rank-initial universe by a first-order formula using finitely many ordinal parameters.
• Partition of a set — A family of nonempty, pairwise disjoint subsets whose union is the whole underlying set, equivalently the classes of an equivalence relation.
• 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.
• 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.
• Set-Theoretic Code — A real coding a hereditarily countable set by a well-founded extensional relation on natural numbers whose Mostowski collapse recovers the set's transitive closure.
• Sierpiński Set — Recognize an uncountable real set whose intersection with every Lebesgue-null set is countable, with existence controlled by additional set-theoretic axioms.
• Square principle — A set-theoretic principle asserting a coherent sequence of short club sets with no single global thread.
• Symmetric difference — The set operation retaining elements that belong to exactly one of two sets.
• Transfinite number — An ordinal or cardinal number larger than every finite number, used to order or measure infinite sets.
• Ω-logic — An infinitary set-theoretic deductive system whose validity is defined through universally Baire sets and generic extensions under large-cardinal assumptions.