Questions: Löwenheim-Skolem Theorems: Overview and Unification

Question 1

A mathematician writes a first-order theory T with the intention that its only model should be the real numbers ℝ (uncountable). What do the Löwenheim-Skolem theorems say about this?

• A. This is achievable if T is sufficiently complex — enough axioms can pin down the reals uniquely
• B. T will also have a countable model satisfying all the same first-order sentences as ℝ
• C. T can have only one model, but its cardinality is undetermined until you specify it
• D. The theorems only apply to theories without constants, so T could still be categorical

Answer: (B)

The downward Löwenheim-Skolem theorem guarantees that if T has an infinite model (like ℝ), it also has a countable elementary submodel — one satisfying exactly the same first-order sentences. No amount of first-order axioms can force uncountability: first-order logic simply cannot express 'this structure has exactly uncountably many elements.' This is why the complete theory of ℝ is not categorical at any infinite cardinality below 2^ℵ₀.

Question 2

Skolem's paradox arises because ZFC (a first-order theory) proves that uncountable sets exist, yet the Löwenheim-Skolem theorem guarantees ZFC has a countable model. How is this resolved?

• A. ZFC is actually inconsistent — no model of ZFC can truly exist
• B. The countable model is not a genuine model of ZFC, only an approximation
• C. Uncountability is not absolute: inside the countable model, there is no internal bijection from ℕ to the 'uncountable' set, even though one exists outside the model
• D. The Löwenheim-Skolem theorem does not apply to ZFC because ZFC has infinitely many axioms

Answer: (C)

Uncountability is a relative concept: a set S is uncountable within a model M when no bijection from ℕ to S exists *inside M*. The countable model of ZFC is countable from the outside, but internally it lacks any bijection between ℕ and its 'uncountable' sets — so from the model's own perspective, those sets are uncountable. This is not a contradiction but an illustration that first-order properties are always interpreted within a model.

Question 3

The downward Löwenheim-Skolem theorem guarantees that any theory with an infinite model has a countable model that is an elementary substructure — satisfying exactly the same first-order sentences.

• T. True
• F. False

Answer: True

This is precisely the downward theorem. 'Elementary substructure' (not just 'substructure') is crucial: every first-order sentence true in the large model remains true in the countable model when the same parameters are used. The construction — closing under witnesses for existential formulas — ensures this elementarity property, not just ordinary substructure.

Question 4

The upward Löwenheim-Skolem theorem shows that first-order logic can express statements that are true in arbitrarily large models, proving that infinite structures come in many distinct sizes.

• T. True
• F. False

Answer: False

The upward theorem shows the opposite: because any infinite model can be expanded to a model of any larger cardinality, first-order logic *cannot* distinguish between infinite cardinalities. It does not show that logic can express size differences — it shows that logic is powerless to enforce them. The existence of models at every infinite cardinality is a limitation, not a capability, of first-order expressiveness.

Question 5

Explain why Skolem's paradox is not actually a paradox, in terms of what 'uncountable' means inside versus outside a model.

Model answer: A set is uncountable within a model when no bijection from ℕ to that set exists *inside the model*. The countable model of ZFC lacks such internal bijections for its 'uncountable' sets — so those sets are genuinely uncountable by the model's own first-order reckoning. The model is countable from the external (meta-level) perspective, but that external bijection does not exist *within* the model. The apparent contradiction dissolves once you recognize that 'countable' and 'uncountable' are model-relative properties, not absolute ones.

This is the deepest lesson of model theory: truth is always relative to a model, and semantic properties like cardinality are evaluated from within. The paradox feels paradoxical because we confuse the external view (the model is countable) with the internal view (the model correctly judges certain sets to be uncountable). First-order logic simply has no way to enforce external cardinality constraints — it is blind to the difference between ℵ₀ and ℵ₁ when viewed from inside a model.