Questions: Continuum Hypothesis and Independence from ZFC

Question 1

Someone claims: 'Cohen's proof showed that ZFC can prove the Continuum Hypothesis.' What is wrong with this claim?

• A. Cohen showed ZFC can disprove CH, not prove it
• B. Cohen showed ¬CH is consistent with ZFC, meaning ZFC cannot prove CH
• C. Cohen showed CH is true in every model of ZFC
• D. Cohen proved CH using large cardinal axioms, not forcing

Answer: (B)

Cohen's forcing construction built a model of ZFC in which CH *fails* — showing ¬CH is consistent with ZFC. This means ZFC cannot prove CH (if it could, CH would hold in all models). Combined with Gödel's result (ZFC cannot disprove CH), the two results together establish *independence*: ZFC neither proves nor disproves CH.

Question 2

What did Gödel's constructible universe L establish about the Continuum Hypothesis?

• A. CH is true in all possible set-theoretic universes
• B. CH is equivalent to the Axiom of Choice
• C. ZFC can prove CH, establishing it as a theorem
• D. ZFC cannot disprove CH — CH is consistent with ZFC

Answer: (D)

Gödel showed that L is a model of ZFC in which CH is true. This means you cannot derive a contradiction from ZFC + CH: if ZFC is consistent, so is ZFC + CH. Crucially, this establishes only that CH cannot be *disproved* from ZFC — not that it is provable. Cohen's later result (¬CH is also consistent) completed the independence proof.

Question 3

The independence of CH from ZFC means that both ZFC + CH and ZFC + ¬CH are consistent (assuming ZFC itself is consistent).

• T. True
• F. False

Answer: True

This is exactly what independence means: Gödel showed ZFC + CH is consistent (so ¬CH is not provable from ZFC), and Cohen showed ZFC + ¬CH is consistent (so CH is not provable from ZFC). Together, these results place CH beyond the reach of ZFC in either direction.

Question 4

The independence of CH from ZFC settles that the Continuum Hypothesis has no definite truth value.

• T. True
• F. False

Answer: False

Independence is a statement about *provability from axioms*, not about truth. A Platonist would say CH is either true or false in the actual universe of sets — we simply cannot determine which from ZFC. A pluralist might say there are many valid set-theoretic universes, some satisfying CH and some not. Independence leaves the philosophical question open but does not resolve it by declaring CH truth-valueless.

Question 5

Why does establishing that CH is independent of ZFC not settle the question of whether CH is 'really' true?

Model answer: Independence shows that ZFC is silent on CH — the axioms neither entail CH nor entail ¬CH. But 'true' depends on what we take as the intended universe of sets. ZFC is not the only possible axiomatic standard: adding large cardinal axioms, Martin's Axiom, or other principles can decide CH one way or another. Some set theorists believe there is a unique correct set-theoretic universe in which CH is determinately true or false; others accept a plurality of universes. Independence rules out a ZFC-proof, but leaves room for richer frameworks that do settle the question.

The key distinction is provability versus truth. Gödel's incompleteness theorems already showed that no consistent system strong enough to express arithmetic can prove all truths about its own subject matter. CH is another instance: the question of its truth lives beyond the reach of ZFC, but not necessarily beyond all mathematical investigation.