Departure question
Test the floor beneath mathematicsPort 5 of 8Meet paradox, incompleteness, and independence in attempts to build a foundation for mathematics from sets.
This voyage is an editorial path for understanding, not a claim of direct historical influence or sole invention.
"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.
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.
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.
Ports in time
Follow the scenes to see problems, notation, standards of proof, and applications changing across different times and places.
Is 2^ℵ₀ equal to ℵ₁? Cantor formulated the continuum hypothesis and struggled with it for much of his life.
No reliable place is given, so time continues without an invented pin
Continue through the world of this yearHilbert placed the continuum hypothesis first on his celebrated list of the most important mathematical problems for the new century.
No reliable place is given, so time continues without an invented pin
Continue through the world of this yearGödel showed that the continuum hypothesis cannot be disproved from the standard axioms of set theory, assuming those axioms are consistent.
No reliable place is given, so time continues without an invented pin
Continue through the world of this yearPaul 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.
No reliable place is given, so time continues without an invented pin
Continue through the world of this yearA central question in set theory and logic, a guide to the limits of formal systems, and a model for studying independence and undecidability.
Curated sources and problems. Bring one discovery back from OEIS, Project Euler, MathOverflow, or arXiv.
No concept belongs to one person
These are not inventor credits. They are different ports: opening a problem, sharpening a language, or carrying it into another world.
Number lenses
These numbers are editorial lenses for the voyage, not required prerequisites.
Meet paradox, incompleteness, and independence in attempts to build a foundation for mathematics from sets.
Open the number voyage
Meet paradox, incompleteness, and independence in attempts to build a foundation for mathematics from sets.
Open the number voyage
Meet paradox, incompleteness, and independence in attempts to build a foundation for mathematics from sets.
Open the number voyage
Concept genealogy
Concepts arriving from before
Current port
Continuum Hypothesis
Concepts opened from here
No direct successor port is curated yet.
Only direct editorial links are shown; this is not a complete learning order or historical influence line.
