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  "body": "\n## Statement\n\nShelah 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:\n\nFor 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* $\\theta \\ge \\mu$.\n\nShelah 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.\n\nSeveral 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.\n",
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        "locator": "Abstract elementary class, section Shelah's categoricity conjecture",
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  "name": "Shelah's categoricity conjecture",
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  "source": "https://github.com/tamnd/math2/blob/main/nodes/TSF/TSFJ71QC4G.md",
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  "url": "https://math2.tamnd.com/wiki/shelahs-categoricity-conjecture/",
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