Concept
Is there an infinity between the natural numbers and the reals? Gödel’s 1940 and Cohen’s 1963 complementary relative-consistency results established that CH is independent of ZFC.
Understand it in one breath
"Is there an infinity strictly between |ℕ| and |ℝ|?" Cantor's 1878 conjecture. Gödel’s 1940 relative-consistency result for CH and Cohen’s 1963 result for ¬CH together established that ZFC decides neither CH nor its negation. Later work asks which additional axioms to adopt.
At a glance
Year | Person | Result | Significance |
|---|---|---|---|
1878 | Cantor | Conjecture: no infinity lies strictly between |ℕ| and |ℝ| | The lifelong problem he tried to prove |
1900 | Hilbert | Selected as Problem No. 1 | The first great question of 20th-century mathematics |
1940 | Gödel | CH is consistent with ZFC | Cannot be disproved |
1963 | Paul Cohen | ¬CH is also consistent with ZFC | Cannot be proved either (Cohen received the 1966 Fields Medal for this result) |
Conclusion | Independent statement | CH is undecidable | ZFC does not determine every mathematical truth |
"Mathematics contains choices." CH can receive different answers under different axioms — the shock that there may be more than one mathematical universe.
Key formula
Key moments
Cantor proposes the hypothesis
Is 2^ℵ₀ equal to ℵ₁? Cantor formulated the continuum hypothesis and struggled with it for much of his life.
Number one among Hilbert’s 23 problems
Hilbert placed the continuum hypothesis first on his celebrated list of the most important mathematical problems for the new century.
Gödel — it cannot be disproved
Gödel showed that the continuum hypothesis cannot be disproved from the standard axioms of set theory, assuming those axioms are consistent.
Cohen — it cannot be proved either
Paul Cohen invented forcing to show that the continuum hypothesis cannot be proved from the standard axioms either. Together with Gödel’s 1940 result, this established independence; Cohen’s work was recognized with a Fields Medal.
Modern applications
A central question in set theory and logic, a guide to the limits of formal systems, and a model for studying independence and undecidability.
Beyond MathVoyage
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