A countable set M is constructed so that M ≺ V (the full set-theoretic universe). Inside M, the statement 'ℝ is uncountable' is true — M ⊨ ¬∃ bijection from ω to ℝ^M. From V's perspective, M itself is countable. How is this possible?
AIt is a contradiction — if M is countable then M cannot satisfy the uncountability of ℝ
BThe bijection between ω and M's reals exists in V but not in M, so uncountability is not absolute across models
CM uses a different logic where 'uncountable' means something weaker than in V
DThe Löwenheim-Skolem theorem guarantees that M is actually uncountable despite appearances
This is Skolem's Paradox, and option B is its resolution. 'Uncountable' means 'no bijection to ω exists in this model.' M lacks the bijection that witnesses countability — that bijection exists in V but not as an element of M — so from M's internal perspective its reals are uncountable. The lesson: uncountability is not an absolute property. It depends on which functions the model contains. Option A is the naive reaction; option C misunderstands that M uses ordinary first-order logic.
Question 2 Multiple Choice
What is the key difference between a formula being 'absolute' for a transitive model and a model being an 'elementary submodel'?
AThey are the same: absoluteness and elementarity both require all formulas to have the same truth value
BAbsoluteness applies only to Δ₀ (bounded) formulas; elementary submodels preserve all first-order formulas simultaneously
CElementary submodels are stronger because they must be transitive, while absolute formulas hold in non-transitive models
DAbsoluteness is a property of a single formula; elementary submodels only guarantee preservation of atomic formulas
Absoluteness is formula-specific: a Δ₀ formula has the same truth value in any transitive model as in V, but higher-complexity formulas (Σ₁, Π₁, etc.) may not. An elementary submodel M ≺ V preserves every first-order formula simultaneously — even Σ_n formulas for all n. The price is that M need not be transitive. Option C reverses the relationship: transitive models support absoluteness, but elementary submodels need not be transitive.
Question 3 True / False
If M ≺ V, then M and V agree on which sets are countable — a set is countable in M if and only if it is countable in V.
TTrue
FFalse
Answer: False
False. This is precisely what Skolem's Paradox denies. A set X can be countable in V (there exists a bijection f: ω → X in V) while being 'uncountable' from M's perspective, simply because f is not an element of M. Countability is not absolute — it depends on which functions are present in the model. M ≺ V preserves all first-order sentences, but 'X is countable' quantifies over functions, and the relevant functions may not be in M.
Question 4 True / False
Every ZFC axiom holds in M if M ≺ V, because each ZFC axiom is a first-order sentence true in V.
TTrue
FFalse
Answer: True
True. The ZFC axioms are first-order sentences (e.g., the Axiom of Extensionality, Pairing, Union, Power Set schema, etc.). Since M ≺ V means M and V agree on the truth of every first-order sentence with parameters from M, and the axioms are sentences with no free variables (or universally quantified), they are all true in M. This is why elementary submodels are useful for constructing models of ZFC: you get ZFC for free from elementarity.
Question 5 Short Answer
Explain why a countable elementary submodel M of V can satisfy 'there are uncountably many reals,' and what this reveals about the concept of uncountability in set theory.
Think about your answer, then reveal below.
Model answer: M satisfies 'ℝ is uncountable' because uncountability means 'no bijection from ω to ℝ exists in this model.' The bijection that witnesses M's countability from V's perspective is not itself an element of M — it exists in V but outside M. Since M cannot 'see' this bijection, it correctly (by its own lights) concludes its reals are uncountable. This reveals that uncountability is model-relative, not absolute: it depends on which functions exist within the model, not on any intrinsic cardinality property.
Skolem's Paradox is a feature, not a bug. It shows that 'uncountable' is not an intrinsic property of a set but a relational one: X is uncountable relative to a model M iff no bijection f: ω → X is an element of M. This is why set theorists distinguish between 'internally uncountable' (no such bijection in M) and 'externally countable' (such a bijection exists in V). The same insight drives forcing: by carefully choosing what functions a model contains, you can make it satisfy radically different cardinality claims.