Morley's Categoricity Theorem

Morley's Categoricity Theorem

In model theory, a branch of mathematical logic, a theory is κ-categorical (or categorical in κ) if it has exactly one model of cardinality κ up to isomorphism. Morley's categoricity theorem is a theorem of Michael D. Morley (1965) which states that if a first-order theory in a countable language is categorical in some uncountable cardinality, then it is categorical in all uncountable cardinalities.

Saharon Shelah (1974) extended Morley's theorem to uncountable languages: if the language has cardinality κ and a theory is categorical in some uncountable cardinal greater than or equal to κ then it is categorical in all cardinalities greater than κ.

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