Questions: Ryll-Nardzewski Theorem: Syntactic Characterization of Categoricity

Question 1

A complete theory T has infinitely many complete 2-types over the empty set. What does the Ryll-Nardzewski theorem immediately tell you?

• A. T cannot be ω-categorical
• B. T has no countable models
• C. T is categorical in every uncountable cardinality
• D. T has only finitely many 1-types

Answer: (A)

The Ryll-Nardzewski theorem says T is ω-categorical if and only if for each n, S_n(T) is finite. Infinitely many 2-types directly violates this condition for n=2, so T cannot be ω-categorical. The theorem does not say anything about uncountable cardinalities (that requires Morley's theorem), and infinitely many 2-types says nothing about 1-types.

Question 2

The theory DLO (dense linear order without endpoints) is ω-categorical with unique countable model ℚ. For n=3, which best explains why finitely many 3-types exist?

• A. Any 3-tuple of rationals is described, up to automorphism, entirely by the ordering of its elements — there are exactly 6 order types for 3 distinct elements
• B. ℚ has only finitely many elements, so there are finitely many 3-tuples
• C. DLO has no complete types because all its models are infinite
• D. The Stone topology on S₃(DLO) is compact, so it has finitely many points

Answer: (A)

For any n-tuple of distinct rationals, the complete type is fully determined by the linear order among them — which of the n! orderings applies. For n=3 there are 6 permutations, giving at most 6 distinct 3-types (fewer if ties are allowed, but DLO excludes equality). This finite count is exactly why DLO satisfies the Ryll-Nardzewski condition. ℚ is infinite, compactness of the Stone space alone does not imply finiteness (an infinite compact space can still have infinitely many points), and 'no complete types' is wrong.

Question 3

If a complete theory T is ω-categorical, then every complete n-type over ∅ is isolated — that is, there exists a single formula that uniquely determines the entire type.

• T. True
• F. False

Answer: True

This is a key step in the Ryll-Nardzewski theorem. When S_n(T) is finite, the Stone topology on it is discrete, meaning every singleton {p} is open. In the Stone topology, open sets are unions of basic open sets of the form [φ] = {types containing φ}. A singleton being open means p = [φ] for some formula φ, i.e., φ isolates p. Isolated types are exactly those pinned by a single formula — so finite type count implies every type is isolated.

Question 4

A theory that is ω-categorical is expected to also be categorical in nearly every uncountable cardinality, since having mainly finitely many n-types is a property of the theory regardless of cardinality.

• T. True
• F. False

Answer: False

This is false. ω-categoricity is a property specifically about countable models, and the Ryll-Nardzewski theorem characterizes only ω-categoricity. Morley's theorem characterizes categoricity in uncountable cardinalities and requires a different condition (no Vaughtian pairs, totally transcendental). Many ω-categorical theories, including DLO itself, are not categorical in uncountable cardinalities — there are many non-isomorphic uncountable dense linear orders.

Question 5

Why does having only finitely many complete n-types (for each n) force all countable models of T to be isomorphic? Explain the mechanism.

Model answer: When every type is isolated, the omitting types theorem's dual guarantees each type is realized in every model. With finitely many n-types, every countable model realizes exactly the same finite collection of types. A back-and-forth argument then constructs an isomorphism between any two countable models: at each stage, you can always extend a finite partial isomorphism because the type of any tuple is one of finitely many isolated types, all realized on both sides.

The key mechanism is: finite type count → every type isolated → every type realized in every model → back-and-forth succeeds. Isolated types are the bridge: a formula φ that isolates a type p means any model satisfying T contains a tuple realizing p (because T ⊨ ∃x φ(x)), and the type of that tuple is fully determined by φ. The back-and-forth argument exploits this to build an isomorphism step by step.