Morley's Categoricity Theorem states that if a first-order theory in a countable language has exactly one model, up to isomorphism, of some uncountable cardinality, then it has exactly one model of every uncountable cardinality. Named for Michael D. Morley, it was a foundational result launching the modern field of model theory, and its later generalization to uncountable languages by Saharon Shelah became a central motivation for stability theory.
Facts
StatementIf a first-order theory in a countable language is categorical in some uncountable cardinality, then it is categorical in all uncountable cardinalities. 1 Classification
Statement FormCharacterization Theorem 1 Connections
Has Statement Form
In Branch
Source Categorical theory
Sources
1. Categorical theory
Lead section, model theory paragraph
if a first-order theory in a countable language is categorical in some uncountable cardinality, then it is categorical in all uncountable cardinalities.
In Branch: Logic and Foundations, Lead sentence
In mathematical logic, a theory is categorical if it has exactly one model (up to isomorphism).
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