One great problem, three kinds of immersion
From a story to your own conjecture
Problem stories open curiosity. The problem workshop is the next place, where curiosity becomes a testable idea.
- 1 · DiscoverYou are hereMeet the questionLearn when it appeared and why it still holds people’s attention.
- 2 · ChallengeTest one caseMake your first observation in five minutes with a drawing, calculation, or colors.
- 3 · DevelopBuild on ideasPublish an observation and grow it through comparisons, conjectures, and counterexamples.
Continuum Hypothesis — Proven Independent of ZFC
"Is there an infinity between the naturals and the reals?" Conjectured by Cantor in 1878. Gödel’s CH direction (1940) and Cohen’s ¬CH direction (1963) together showed that standard ZFC decides neither side.
Connect a few background ideas before the real wall comes into view.
The starting level measures how easily you can understand and test small cases. It is not the difficulty of a complete proof.
Jump to your first five minutesProved independent by Gödel (1940) + Cohen (1963)
Cohen — Fields Medal 1966
See the puzzle visually
Continuum Hypothesis: there is no infinity between the natural numbers and the real numbers. It was later shown to be neither provable nor disprovable in ZFC. Different axioms produce different mathematical universes.
Problem statement
Does there exist a set X with ℵ₀ < |X| < 2^ℵ₀? (Is there another infinity strictly between countable and continuum?)
The story of this puzzle
Cantor conjectured in 1878 that 2^ℵ₀ = ℵ₁. Hilbert made it his first problem in 1900. Gödel showed in 1940 that CH cannot be refuted from ZFC if ZFC is consistent; Cohen showed in 1963 that CH cannot be proved from ZFC under the same kind of consistency assumption. Together their results establish that ZFC proves neither CH nor ¬CH. This does not make CH truth-valueless; it means ZFC alone does not settle it. Cohen received the Fields Medal in 1966, and later set theory asks what additional axioms may be mathematically natural.
Try it yourself
Try Cantor's diagonal argument. Suppose you list all reals in [0,1]. Walk down the diagonal flipping each digit (1→2, 5→6). The result is a real not in your list. Reals are strictly more numerous than naturals. But how many more? ZFC alone does not decide.
Beyond MathVoyage
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