Questions: Countable Model Existence and Representation

Question 1

A theory T in a countable language has a model of cardinality ℵ₁. Which of the following must be true?

• A. T also has a countable model, by the downward Löwenheim–Skolem theorem
• B. T has no countable model — ℵ₁ is the minimal model size
• C. Whether T has a countable model depends on whether T is complete
• D. T has a countable model only if it is ω-categorical

Answer: (A)

The downward Löwenheim–Skolem theorem says that any infinite model (in a countable language) has an elementary substructure with a countable domain. So a model of size ℵ₁ immediately implies the existence of a countable elementary substructure, which is itself a model of T. The key insight is that for theories in countable languages, having *any* infinite model is equivalent to having a countable model.

Question 2

ZFC has a countable model M. Inside M, the sentence 'the real numbers are uncountable' holds. How is this possible without contradiction?

• A. M must be mistaken — ZFC actually proves the reals are countable
• B. Inside M, no bijection between M's 'reals' and ω exists as an element of M, even though such a bijection exists from outside M — uncountability is model-relative
• C. M is not a genuine model of ZFC; it only satisfies a weakened version
• D. The Löwenheim–Skolem theorem does not apply to ZFC because ZFC is too strong

Answer: (B)

This is Skolem's paradox. M satisfies the ZFC sentence 'there is no bijection between ℝ and ω' — because inside M, that bijection does not exist as an *element of M*. From outside M, we can see that M itself is countable and construct such a bijection externally, but M cannot 'see' it. Uncountability is not an absolute property; it is defined relative to what bijections exist within a model. This is why the paradox is a philosophical puzzle rather than a genuine contradiction.

Question 3

If a first-order theory T in a countable language is consistent, it necessarily has a countable model.

• T. True
• F. False

Answer: True

This is the content of the downward Löwenheim–Skolem theorem combined with the completeness theorem. Consistency means T has some model (by the completeness theorem, consistency implies satisfiability). Any infinite model in a countable language contains a countable elementary substructure. And if T is only satisfied by finite models, then those finite models are themselves countable (finite sets are countable). Either way, a countable model exists.

Question 4

A theory T in a countable language that has no uncountable models also has no countable models.

• T. True
• F. False

Answer: False

This reverses the logical relationship. Having no uncountable models says nothing about countable models — these are independent. A theory can have only countable models (ω-categorical theories like the theory of dense linear orders without endpoints have exactly one countable model and no uncountable models up to isomorphism). The implication runs the other direction: having no countable model means no model at all (consistency is equivalent to having a countable model).

Question 5

Explain Skolem's paradox: how can a countable model M satisfy the sentence 'the real numbers are uncountable'?

Model answer: Uncountability is relative to the model. M satisfies 'ℝ is uncountable' because, inside M, there is no bijection between M's version of ℝ and ω that exists as an element of M. ZFC proves 'no bijection between ℝ and ω exists,' meaning no such bijection is an element of any model of ZFC. But from outside M, we can see that M itself is countable, so such a bijection exists externally — M just can't 'see' it. The apparent contradiction dissolves because 'uncountable' means 'no bijection to ω exists within this model,' not 'no bijection exists anywhere.'

The resolution to Skolem's paradox hinges on the distinction between internal and external perspective. 'Uncountable' in first-order logic means there is no bijection to ω that the model can access. A countable model of ZFC contains only countably many elements, but among those elements is no bijection between M's reals and M's ω — so M correctly satisfies the uncountability sentence. This shows that mathematical properties like cardinality are not absolute; they depend on which functions exist in the ambient model.