OAI.CHObstruction.main

The theorem states that, assuming the continuum hypothesis CH (2^ℵ₀ = ℵ₁, in universe u), there is a relational signature given by a family L(n) of n-ary relation symbols for n ∈ ℕ, with countably many symbols in total, and a class K of structures whose carriers are ZFSets, equipped with a designated strong-substructure relation, such that K is an abstract elementary class (AEC), meaning the strong relation is reflexive, transitive and antisymmetric, relates only members of K, implies the substructure relation, is invariant under isomorphism, satisfies coherence, and has unions of nonempty continuous chains that are least upper bounds, and K has a downward Löwenheim–Skolem bound; K has Löwenheim–Skolem number exactly ℵ₀, meaning ℵ₀ is the least cardinal θ that bounds the signature size and lets every subset A of a member of K lie in a strong substructure in K of cardinality at most |A|+θ. Moreover, with endpoint = ℶ_{ω₂} and hanf(ℵ₀) = ℶ_{(2^ℵ₀)⁺ as an ordinal}, the theorem asserts endpoint = hanf(ℵ₀), that K has two non-isomorphic members of cardinality endpoint, and that for every cardinal μ at least tailThreshold = ℶ at the ordinal of (2^ℵ₁)⁺, K is categorical in μ, meaning it has a member of cardinality μ and any two such members are isomorphic.