Math Conjectures

Shelah’s Categoricity Conjecture for Lω1,ωL_{\omega_1,\omega}

OPENLandmarkConjectureProposed c. 1977 · Standard version

Canonical statement

Let ψ\psi be a sentence of the countable infinitary logic Lω1,ωL_{\omega_1,\omega}, which permits countable conjunctions and disjunctions but only finite strings of quantifiers. If ψ\psi has, up to isomorphism, exactly one model of some cardinality λ≥ℶω1\lambda\ge\beth_{\omega_1}, then it has exactly one model of every cardinality μ≥ℶω1\mu\ge\beth_{\omega_1}, where ℶ0=ℵ0\beth_0=\aleph_0, ℶα+1=2ℶα\beth_{\alpha+1}=2^{\beth_\alpha}, and ℶλ=sup⁡α<λℶα\beth_\lambda=\sup_{\alpha<\lambda}\beth_\alpha at limit ordinals.
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Let \(\psi\) be a sentence of the countable infinitary logic \(L_{\omega_1,\omega}\), which permits countable conjunctions and disjunctions but only finite strings of quantifiers. If \(\psi\) has, up to isomorphism, exactly one model of some cardinality \(\lambda\ge\beth_{\omega_1}\), then it has exactly one model of every cardinality \(\mu\ge\beth_{\omega_1}\), where \(\beth_0=\aleph_0\), \(\beth_{\alpha+1}=2^{\beth_\alpha}\), and \(\beth_\lambda=\sup_{\alpha<\lambda}\beth_\alpha\) at limit ordinals.
LOG-INF-006:[Shelah1971Categoricity][GrossbergVasey2017Universal][Vasey2020Categoricity][Shelah2024E102]
Logic and foundations

Notes

The infinitary logic Lω1,ωL_{\omega_1,\omega} allows countable conjunctions and disjunctions but only finite strings of quantifiers. Shelah's conjecture is an eventual categoricity transfer: if a sentence ψ\psi of this logic has exactly one model up to isomorphism in some cardinality λ≥ℶω1\lambda\ge\beth_{\omega_1}, then it has exactly one model in every cardinality at least ℶω1\beth_{\omega_1} — an infinitary analogue of Morley's categoricity theorem for first-order logic. The program crystallized in Shelah's work of the 1970s, following his study of categoricity in ℵ1\aleph_1 for such sentences [Shelah1971Categoricity]; the date c. 1977c.\,1977 marks the emergence of the standard formulation rather than a single statement.

Most progress is phrased in the framework of abstract elementary classes. Eventual categoricity has been proved for universal classes [GrossbergVasey2017Universal] and for large abstract elementary classes under hypotheses such as amalgamation and tameness, or from large cardinals [Vasey2020Categoricity]; Shelah has continued to develop the categoricity and solvability theory of such classes [Shelah2024E102].

The unrestricted conjecture for Lω1,ωL_{\omega_1,\omega}, free of amalgamation, tameness, universal-class, or large-cardinal assumptions, remains open.

References (4)

The boxed statement is the canonical open formulation — not a stronger variant or a related research program. The status reflects the catalog's last review; do your own literature search before investing serious effort.