Questions: Absolute Formulas and Model-Theoretic Absoluteness

Question 1

Which of the following correctly explains why 'x is a finite set' is absolute between a transitive inner model M and the full universe V?

• A. All properties of sets are absolute because sets are defined entirely by their elements
• B. Finiteness can be expressed using only bounded quantifiers (Δ₀), so evaluating it does not depend on what objects exist outside the set in question
• C. Finiteness is absolute only in models that satisfy the axiom of choice
• D. The formula is not actually absolute — a set finite in M might be infinite in V if V has more natural numbers

Answer: (B)

Finiteness can be expressed as: 'there is no injection from ω into x,' and with some work this reduces to a Δ₀ statement about the elements of x. Δ₀ formulas use only bounded quantifiers (∀y ∈ a, ∃y ∈ a), which range only over elements of specific sets and cannot be affected by objects outside those sets. Whether x is finite is an intrinsic property of x itself, not of the ambient model. Note: option D is tempting but wrong — V doesn't have 'more natural numbers' than M; ordinals and natural numbers are absolute.

Question 2

Consider the claim 'κ is an uncountable cardinal' evaluated in an inner model M versus the full universe V. Why can this fail to be absolute?

• A. Cardinality claims are never absolute between any two models of ZFC
• B. In a larger model V, there may exist bijections between κ and ω that M cannot see, making κ countable in V even if M sees no such bijection
• C. Cardinals are ordinals, and ordinals are absolute, so cardinality claims are always absolute
• D. The claim fails to be absolute only if M does not satisfy the power set axiom

Answer: (B)

Countability is about the existence of an injection from κ into ω. Whether such an injection exists depends on what functions are available in the model. M may contain no bijection between κ and ω, making κ uncountable in M. A larger model V may include additional sets — in particular, additional functions — including a bijection that M cannot see. This is the mechanism of Cohen forcing: you can add a bijection that collapses ℵ₁ of the ground model to a countable set in the extension. Option C is a common but wrong inference: ordinals are absolute, but cardinality (which ordinal is the 'size' of a set) depends on available bijections.

Question 3

An absolute formula should be true in at least one model — a formula that is false in most model cannot be absolute.

• T. True
• F. False

Answer: False

Absoluteness is about truth-preservation between models, not about truth itself. A formula φ is absolute between M and V if for every parameter a in M: M ⊨ φ(a) ↔ V ⊨ φ(a). This bi-conditional is satisfied whether both sides are true or both sides are false. A formula like '0 ≠ 0' is false in every model but trivially absolute between any two, because M ⊨ '0 ≠ 0' is false and V ⊨ '0 ≠ 0' is false, so the equivalence holds. The common misconception confuses absoluteness (a structural relationship between models) with validity (truth in all models).

Question 4

Δ₀ formulas are absolute between transitive models because their bounded quantifiers range over specific sets and cannot be influenced by objects that exist in V but not in M.

• T. True
• F. False

Answer: True

Yes — this is the core absoluteness theorem. A bounded quantifier ∀x ∈ a ranges over the elements of the specific set a, not over the whole model. Since M is a transitive inner model, elements of sets in M are themselves in M (transitivity), so both M and V 'see' the same objects when evaluating ∀x ∈ a. Nothing outside the bounded range can affect the truth value. This is why x ∈ y, x ⊆ y, 'x is an ordinal,' and 'x is a natural number' are all absolute.

Question 5

Explain why cardinality is not absolute between models of set theory, using the notion of what a larger model can 'see' that an inner model cannot.

Model answer: Cardinality is defined via bijections: a set x has cardinality κ if there is a bijection between x and κ. Whether such a bijection exists depends on which functions are in the model. An inner model M may lack a bijection between two sets x and κ — making x 'uncountable' relative to M — while a larger model V contains that bijection, making x countable in V. Models differ not in which sets they contain (inner models contain the same ordinals), but in which functions and relations they include. Cardinality is therefore not an intrinsic property of the set itself, but a relational property that depends on the ambient model's function space.

This is one of the deepest lessons in set theory: 'size' is model-relative. Cohen's forcing constructs models where ℵ₁ collapses to countable by adding a bijection. The inner model had no such bijection — ℵ₁ was genuinely uncountable from its perspective — but the extension does. Neither model is wrong; they simply have different function spaces.