---
tag: TSFJ71QC4G
kind: statement
role: proposition
name: Shelah's categoricity conjecture
sources:
  - wikipedia: en:27344508@r1373726617
    locator: Abstract elementary class, section Shelah's categoricity conjecture
    note: adopted
    mention: wikipedia://en/27344508#R0H6
    sections: [Statement]
inventory: [V0RAQZ304J]
---

## Statement

Shelah introduced AECs to provide a uniform framework in which to generalize first-order classification theory. Classification theory started with Morley's categoricity theorem, so it is natural to ask whether a similar result holds in AECs. This is **Shelah's eventual categoricity conjecture**. It states that there should be a Hanf number for categoricity:

For every AEC *K* there should be a cardinal $\mu$ depending only on $\operatorname{LS}(K)$ such that if *K* is categorical in *some* $\lambda \geq \mu$ (i.e. *K* has exactly one (up to isomorphism) model of size $\lambda$), then *K* is categorical in $\theta$ for *all* $\theta \ge \mu$.

Shelah also has several stronger conjectures: The threshold cardinal for categoricity is the Hanf number of pseudoelementary classes in a language of cardinality LS(K). More specifically when the class is in a [[countable|01TQB88H4C]] language and axiomatizable by an $L_{\omega_1,\omega}$ sentence the threshold number for categoricity is $\beth_{\omega_1}$. This conjecture dates back to 1976.

Several approximations have been published (see for example the results section below), assuming set-theoretic assumptions (such as the existence of large cardinals or variations of the generalized continuum hypothesis), or model-theoretic assumptions (such as amalgamation or tameness). As of 2014, the original conjecture remains open.