Infinite Sets & Large Cardinals

← Back to Domain-Specific Families

Abstractions about transfinite cardinality, large cardinals, infinite orders, trees, measures, determinacy, and structured subsets of infinite domains.

11 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.

• 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.
• Beth number — A transfinite cardinal sequence beginning at countable infinity and repeatedly applying power set at successors and supremum at limit ordinals.
• Kurepa tree — An uncountable tree of height omega-one with countable levels and at least omega-two many cofinal branches.
• 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.
• 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.
• Transfinite number — An ordinal or cardinal number larger than every finite number, used to order or measure infinite sets.
• 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.