Questions: Indiscernible Sequences and Morley's Categoricity Theorem

Question 1

A complete theory T (in a countable language) is categorical in ℵ₁ — it has exactly one model of cardinality ℵ₁ up to isomorphism. What does Morley's theorem imply?

• A. T is categorical in all infinite cardinalities, including ℵ₀ (countably infinite models are also unique up to isomorphism)
• B. T is categorical in all uncountable cardinalities, but may still have multiple non-isomorphic countably infinite models
• C. T is categorical in ℵ₂ but the theorem gives no information about larger cardinalities without further argument
• D. Categoricity in one uncountable cardinal gives no information about categoricity at other cardinalities

Answer: (B)

Morley's theorem states that categoricity in any one uncountable cardinality implies categoricity in all uncountable cardinalities. However, it says nothing about the countable case — a theory can be categorical in all uncountable cardinalities while having many non-isomorphic countably infinite models (or even none). The theorem establishes a strong propagation across uncountable sizes but does not bridge the gap to ℵ₀.

Question 2

Why are indiscernible sequences central to the proof of Morley's categoricity theorem?

• A. They allow compactness-based constructions of models of any prescribed uncountable size
• B. They serve as 'coordinates' that uniformly determine a model's structure: in a categorical theory, indiscernible sequences built from an ω-stable type system characterize models up to isomorphism at each uncountable cardinality
• C. They provide a canonical well-ordering of every uncountable model, establishing its cardinality
• D. They replace Ramsey-theoretic arguments with purely algebraic ones, simplifying the proof

Answer: (B)

The proof proceeds roughly as: (1) a categorical theory is ω-stable, so types over countable sets are countable; (2) ω-stability enables the construction of infinite indiscernible sequences in models of any uncountable size; (3) these sequences act as 'coordinates' — the theory of the sequence (an indiscernible type) uniquely determines the model up to isomorphism. Without indiscernibles, there is no systematic way to compare or uniquely characterize models of different uncountable sizes.

Question 3

An indiscernible sequence is one where any two finite subsequences of the same length satisfy exactly the same first-order formulas, making the individual elements 'interchangeable' from the theory's perspective.

• T. True
• F. False

Answer: True

This is the defining property of indiscernibles: for any two k-element subsequences (a_{i₁}, ..., a_{iₖ}) and (a_{j₁}, ..., a_{jₖ}) with i₁ < ... < iₖ and j₁ < ... < jₖ, they satisfy the same formulas. The elements have no distinguishing first-order properties — they are 'generic' instances of the same type. This uniformity is what makes them useful as structural coordinates.

Question 4

Morley's categoricity theorem states that if a theory T is categorical in some uncountable cardinality, then T has exactly one model in nearly every infinite cardinality, including the countably infinite case.

• T. True
• F. False

Answer: False

This is the most common misstatement of Morley's theorem. The theorem applies only to uncountable cardinalities: categorical in one uncountable cardinal implies categorical in all uncountable cardinals. The countably infinite case (ℵ₀) is not covered. A theory can be uncountably categorical (e.g., the theory of algebraically closed fields of characteristic 0) while having many non-isomorphic countably infinite models, or no countable models, or a unique countable model — each case is possible independently.

Question 5

Why is Morley's categoricity theorem considered surprising, and what does it reveal about the internal structure of categorical theories?

Model answer: The theorem is surprising because models of different uncountable cardinalities can differ dramatically in general theories — there is no obvious reason why 'uniqueness at ℵ₁' should force 'uniqueness at ℵ₂, ℵ₃, ...' The theorem reveals that categorical theories possess an unusually rigid, uniform structure: they must be ω-stable (types over countable parameter sets are countable), their models can be built from indiscernible sequences in a controlled way, and those sequences uniquely determine each model up to isomorphism. Categoricity at any uncountable size is evidence of deep structural simplicity that propagates to all larger sizes, launching the classification-theoretic program of studying how many models a theory can have.

Morley's theorem was the opening move of Shelah's classification theory, which eventually characterized exactly how many models a theory can have at each cardinality. The key insight is that tame structure (few models) is globally uniform, while wild structure (many models) is locally variable.