Questions: Independence Results in Set Theory

Question 1

Gödel (1938) proved that CH is consistent with ZFC by:

• A. Showing that CH can be derived as a theorem from the ZFC axioms alone
• B. Constructing a model (the constructible universe L) in which all ZFC axioms and CH both hold, proving ZFC cannot refute CH
• C. Showing that any model of ZFC satisfying ¬CH leads to a contradiction
• D. Proving that the Continuum Hypothesis is equivalent to the Axiom of Choice within ZFC

Answer: (B)

Gödel proved only *half* of independence — that ZFC cannot refute CH. He did this by constructing L (the constructible universe), a specific model in which both ZFC and CH are satisfied. Since L is a model of ZFC + CH, ZFC is consistent with CH; if ZFC could prove ¬CH, then L would be a model satisfying a contradiction. Crucially, Gödel did NOT show ZFC can prove CH — that would require showing CH holds in every model of ZFC. Cohen's forcing (1963) completed the other half by constructing a model where CH fails.

Question 2

A set theorist says: 'The Continuum Hypothesis is independent of ZFC.' What does this mean for the truth value of CH in specific models of ZFC?

• A. CH has no truth value in any model — it is semantically meaningless
• B. CH is true in some models of ZFC and false in others; ZFC alone cannot determine which
• C. CH must be added as a new axiom before it can be assigned a truth value in any model
• D. CH is true in the standard model of ZFC but unprovable from the axioms

Answer: (B)

Independence means ZFC is compatible with both CH and ¬CH — there exist models satisfying each. In Gödel's L, CH holds; in Cohen's forcing extensions, CH fails and 2^ℵ₀ can equal ℵ₂ or larger. So CH has definite truth values in individual models — it is simply not determined by ZFC alone which value it takes. This is very different from meaninglessness. The philosophical question of whether there is a 'real' answer beyond model relativity depends on commitments about mathematical Platonism.

Question 3

Because CH is independent of ZFC, it is neither true nor false in any mathematical sense — it is simply an undecidable sentence with no determinate truth value.

• T. True
• F. False

Answer: False

Independence from ZFC does not strip a statement of its truth value in models. CH is true in Gödel's constructible universe L and false in Cohen's forcing extensions — it has determinate truth values relative to specific models of set theory. What independence shows is that ZFC cannot pin down which universe of sets is 'the' real one. Whether CH has a mind-independent truth value is a separate philosophical question about mathematical Platonism, not a consequence of independence itself.

Question 4

Cohen's forcing method works by constructing a new model of ZFC by extending an existing model with a 'generic' set that was not already in it.

• T. True
• F. False

Answer: True

Forcing adds a new generic set G to a ground model M by specifying a partial order of finite approximations (forcing conditions) that describe G's behavior. A generic filter coherently collects compatible conditions and determines G completely. The extended model M[G] satisfies ZFC, and by choosing the forcing poset carefully, Cohen arranged for M[G] to contain ℵ₂ many subsets of ℵ₀ — making CH fail. Crucially, forcing constructs a new model; it does not modify ZFC's axioms.

Question 5

What are Gödel's and Cohen's respective contributions to proving that CH is independent of ZFC, and why were both needed?

Model answer: Independence requires proving two things: (1) ZFC cannot prove CH, and (2) ZFC cannot refute CH. Gödel proved (2) in 1938 by constructing L, the smallest model of ZFC, in which CH holds — showing ZFC is consistent with CH. Cohen proved (1) in 1963 by inventing forcing: he extended a model of ZFC to a model where 2^ℵ₀ = ℵ₂, making CH false — showing ZFC is consistent with ¬CH. Together, both halves establish that CH is neither provable nor refutable from ZFC; either can be added as an axiom without contradiction.

A common error is attributing the full independence result to Gödel alone. Gödel had only one direction: consistency of CH with ZFC. The other direction — consistency of ¬CH with ZFC — required Cohen's entirely new technique of forcing, which was a major breakthrough. Neither alone establishes independence; you need both models (one satisfying CH, one satisfying ¬CH) to prove the statement is genuinely undecidable from ZFC.