In the universe V of all sets, set S is countable — there exists a bijection f: S → ω. Now consider a transitive inner model M ⊆ V that happens to not contain f. What is S's status in M?
AS remains countable in M, because countability is an absolute property preserved in all transitive models
BS may be uncountable in M, because M lacks the witnessing bijection and cannot verify countability
CS does not exist in M, since sets with missing witnesses are removed from inner models
DS is countable in M if and only if S is finite — infinite countable sets lose countability in inner models
Countability is not absolute. To say 'S is countable in M' means: there exists a bijection from S to ω *in M*. If M lacks the bijection f (even though it exists in V), M has no witness to S's countability and will declare S uncountable. This is the standard example of a non-absolute notion: 'x is countable' is Σ₁ (there exists a bijection), and Σ₁ statements are upward absolute (true in M implies true in V) but not downward absolute (true in V does not imply true in M). Forcing exploits this by deliberately adding or removing bijections to change what is 'countable.'
Question 2 Multiple Choice
Which of the following set-theoretic notions is ABSOLUTE for all transitive models of ZF?
Ax is a countable set
Bx is an ordinal
Cx has cardinality ℵ₁
Dx is a real number that codes a well-ordering of ω
'x is an ordinal' is Δ₁ (equivalent to both a Σ₁ and a Π₁ definition), hence absolute for all transitive models. An ordinal is a transitive set well-ordered by membership, and checking this only requires quantifying over elements of x — bounded quantification that transitive models can evaluate locally. 'x is countable' requires an existential quantifier ranging over bijections that may not exist in the model. 'Cardinality ℵ₁' depends on what bijections are available. 'Codes a well-ordering' is more complex and not straightforwardly absolute.
Question 3 True / False
If a formula φ is absolute between transitive models M and N (with M ⊆ N), then φ is expected to be true in both M and N.
TTrue
FFalse
Answer: False
This is one of the key misconceptions about absoluteness. Absolute means the truth value is the SAME in both models — φ holds in M iff it holds in N. A false formula can be absolute: 'x is an ordinal' is absolute, but a specific set x might fail to be an ordinal in both M and N equally. Absoluteness is about model-independence of truth evaluation, not about the truth value being 'true.' A statement like '0 = 1' is trivially absolute (false in all models) even though it is never true.
Question 4 True / False
Shoenfield's absoluteness theorem implies that whether a real number has a certain Σ¹₂ property cannot be made to depend on which forcing extension of ZFC you work in.
TTrue
FFalse
Answer: True
Shoenfield's theorem states that Σ¹₂ (and Π¹₂) statements of second-order arithmetic are absolute between the universe V and any inner model containing all countable ordinals — in particular, between V and any forcing extension. Since Σ¹₂ statements depend only on countable objects and countable ordinals, and forcing extensions agree with the ground model on all countable ordinals, the truth value of any Σ¹₂ statement is unchanged by forcing. This means no Σ¹₂ statement can be shown independent of ZFC using forcing — providing a firm boundary on what independence results are achievable by the main technique of modern set theory.
Question 5 Short Answer
Why is 'x is countable' not absolute between a transitive inner model M and the full universe V, even though countability is a perfectly rigorous mathematical property?
Think about your answer, then reveal below.
Model answer: Countability of x means: there exists a bijection f: x → ω. That bijection must exist *in the model being evaluated*. An inner model M ⊆ V may be 'missing' certain bijections that exist in V — especially after a forcing construction that adds new functions. If M lacks the witnessing bijection for x, M cannot verify x's countability and will declare x uncountable, even though V knows x is countable. The mathematical definition is rigorous, but its truth depends on which functions are available as witnesses, and different models have different function inventories.
This points to the heart of why set theory has genuine model-dependence. Statements with existential quantifiers ranging over all functions (not just functions of elements of x) are sensitive to the ambient universe. 'x is countable' is Σ₁ — one unbounded existential quantifier. Such statements are upward absolute: if M thinks x is countable, V does too (V has at least as many bijections as M). But they are not downward absolute. The classic slogan: 'An uncountable set in M might be countable from the outside' — meaning from V's perspective — is a precise theorem, not a paradox.