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  "body": "\n## Summary\n\nMorley rank is an ordinal-valued dimension for definable sets in a first-order structure.\nA definable set has rank at least $\\alpha + 1$ when it contains infinitely many pairwise disjoint definable subsets of rank at least $\\alpha$.\nIn an algebraically closed field, the Morley rank of a constructible set equals its dimension in the sense of algebraic geometry.\nA complete theory in a countable language is $\\omega$-stable exactly when every formula has ordinal Morley rank.\nThe notion is central in [[stable theories|T7JW5QY6KR]] and in Morley's proof of his categoricity theorem.\n\n## Explanations\n\n### L1 human\n\nMorley rank measures the size of a definable set by how many times one can split it into infinitely many smaller pieces.\nA finite nonempty set has rank $0$, since it cannot be split into infinitely many nonempty pieces.\nThe complex line has rank $1$: it splits into infinitely many points, but its definable subsets are finite or cofinite, so it does not split into infinitely many infinite pieces.\nThe complex plane has rank $2$, since it splits into infinitely many disjoint lines.\n",
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