Questions: Inner Models and Relative Consistency Proofs

Question 1

Gödel showed that L ⊨ GCH. A student concludes: 'Therefore GCH must be true, because L is a subclass of the real universe V and inherits its truth.' What is wrong with this reasoning?

• A. It is correct — if GCH holds in L and L ⊆ V, then GCH holds in V
• B. It confuses relative consistency with truth: GCH holds in L, but L is not V, and V may satisfy ¬GCH
• C. It is wrong because L does not actually satisfy GCH
• D. It is wrong because GCH is provable from ZFC, so no model argument is needed

Answer: (B)

Relative consistency is a conditional result, not an absolute one. L ⊆ V means L is a definable class inside V, but L and V are different structures with different truths. Showing L ⊨ GCH proves Con(ZFC) → Con(ZFC + GCH) — GCH cannot be *disproved* from ZFC — but it says nothing about whether GCH holds in V itself. Cohen's forcing later showed V can also satisfy ¬GCH, establishing full independence.

Question 2

Why must an inner model M be transitive in order to function as a genuine model of ZFC?

• A. Transitivity ensures M contains all ordinals of V, giving it sufficient 'height'
• B. Transitivity ensures the ∈-relation inside M agrees with the actual ∈-relation in V, so M's sets are genuine sets rather than artificial simulations
• C. Transitivity ensures M is closed under power sets and unions
• D. Transitivity is a convention, not a logical requirement — non-transitive inner models also work

Answer: (B)

Transitivity means: if a ∈ M and b ∈ a, then b ∈ M. This closure condition ensures that when M 'sees' a membership relation a ∈ b, it is the real ∈ of V — not some artificially restricted version. Without transitivity, M might have sets whose elements fall outside M, making M's ∈-relation a distortion of the true one. Transitive models are honest: their membership relation is the real thing.

Question 3

Showing that L ⊨ GCH proves that ZFC + GCH is consistent relative to ZFC.

• T. True
• F. False

Answer: True

This is exactly what the inner model technique establishes. If ZFC has a model (i.e., if ZFC is consistent), then L — constructed inside any model of ZFC — is a model of ZFC + GCH. So a proof of inconsistency of ZFC + GCH would translate into a proof of inconsistency of ZFC. The result is conditional: Con(ZFC) → Con(ZFC + GCH).

Question 4

A relative consistency proof for a statement φ, using an inner model M where φ holds, establishes that φ is true in the actual set-theoretic universe V.

• T. True
• F. False

Answer: False

A relative consistency proof establishes only that φ cannot be *disproved* from ZFC — not that φ is true. The inner model M is one possible set-theoretic universe satisfying ZFC, but V may be a different universe where φ fails. GCH holds in L (an inner model), but Cohen's forcing constructed models of ZFC where GCH fails, showing GCH is independent of ZFC — neither provable nor disprovable.

Question 5

Why does a relative consistency proof not settle whether a statement like GCH is 'actually true,' and what additional investigation would be required to fully resolve GCH's status?

Model answer: A relative consistency proof shows only that assuming φ leads to no new contradictions — Con(ZFC) → Con(ZFC + φ). It says nothing about truth in V itself, because V might not equal the inner model M used in the proof. To fully resolve GCH's status requires showing both Con(ZFC + GCH) and Con(ZFC + ¬GCH). Gödel's L gives the first; Cohen's forcing gives the second. Together they establish independence: GCH is neither provable nor disprovable from ZFC, so its truth value depends on which additional axioms (if any) one adopts.

Relative consistency answers 'can this coexist with ZFC?' not 'is this true?' Independence is the strongest possible result from this program: it shows ZFC is genuinely underdetermined about GCH. Resolving which value is 'correct' (if the question has an answer) requires stepping outside ZFC to examine large cardinal axioms, inner model theory at the level of L[U], and other foundational commitments — an active research program in set theory today.