Questions: Categorical Theories and Uniqueness of Models

Question 1

A student argues: 'DLO is ℵ₀-categorical, so by Morley's theorem it must be categorical in all uncountable cardinalities too.' What is wrong with this reasoning?

• A. DLO is not a complete theory, so Morley's theorem doesn't apply
• B. Morley's theorem runs in the other direction: categoricity in some uncountable cardinal implies categoricity in all uncountable cardinals — it says nothing about ℵ₀-categoricity
• C. DLO actually is categorical in all uncountable cardinalities; the student is correct
• D. Morley's theorem only applies to theories with finite models

Answer: (B)

Morley's theorem states: if T is categorical in some *uncountable* cardinality, it is categorical in all uncountable cardinalities. It does not run from ℵ₀ upward. DLO is ℵ₀-categorical (the back-and-forth argument works at ω), but uncountable dense linear orders can differ in cofinality and Dedekind completeness, so DLO has many non-isomorphic uncountable models. Morley's theorem simply does not apply here.

Question 2

A complete theory T in a countable language is known to be categorical in ℵ₃. Which of the following must be true?

• A. T is also ℵ₀-categorical
• B. T is categorical in every uncountable cardinality
• C. T has exactly one model of every infinite cardinality
• D. T is categorical in ℵ₂ but not necessarily ℵ₄

Answer: (B)

By Morley's theorem, categoricity in any uncountable cardinal propagates to all uncountable cardinals. So knowing T is categorical in ℵ₃ immediately gives categoricity in ℵ₁, ℵ₂, ℵ₄, and every other uncountable cardinal. However, this says nothing about ℵ₀: a theory can be categorical in all uncountable cardinals while still having multiple countable models (or none, or one). The theory of algebraically closed fields of fixed characteristic exemplifies this.

Question 3

A κ-categorical theory can have two non-isomorphic models of cardinality κ, provided they satisfy the same sentences.

• T. True
• F. False

Answer: False

This directly contradicts the definition. κ-categoricity means there is exactly *one* model of cardinality κ up to isomorphism — any two models of that cardinality must be isomorphic. Elementary equivalence (satisfying the same sentences) is weaker than isomorphism. Categoricity is precisely the condition that collapses the distinction: at cardinality κ, elementary equivalence implies isomorphism.

Question 4

The theory of algebraically closed fields of characteristic 0 (ACF₀) is categorical in every uncountable cardinality.

• T. True
• F. False

Answer: True

ACF₀ is the canonical example of a Morley-categorical theory. Any two algebraically closed fields of characteristic 0 with the same uncountable cardinality are isomorphic — the cardinality alone determines the transcendence degree over ℚ, and transcendence degree characterizes the field up to isomorphism. This also illustrates Morley's theorem: ACF₀ is categorical in ℵ₁, ℵ₂, and every larger cardinal simultaneously.

Question 5

What does the back-and-forth argument prove about DLO (the theory of dense linear orders without endpoints), and why does it fail at uncountable cardinalities?

Model answer: The back-and-forth argument proves DLO is ℵ₀-categorical: any two countable dense linear orders without endpoints are isomorphic to each other (and to ℚ). At each step, you extend a partial isomorphism by one element, using density and the absence of endpoints to guarantee a matching element exists in the other structure. At uncountable cardinalities, the argument breaks down because uncountable dense linear orders can differ in cofinality (whether they have a cofinal ω-sequence) and whether they are Dedekind complete — structural properties that density and no-endpoints alone do not pin down, producing genuinely non-isomorphic uncountable models.

The key is that the back-and-forth argument is constructive and countable: it builds an isomorphism in countably many steps, matching one element at a time. This works perfectly at ω because density always supplies a matching point. At uncountable cardinalities, however, there are uncountable many points to match and the local density property is no longer enough to control global structure. Cofinality and completeness become independent degrees of freedom, allowing genuinely distinct models.