Departure question
Test the floor beneath mathematicsPort 2 of 8Where two intuitions collide
Infinity was long treated as an unending process; once handled as sets, different sizes and undecidable questions appeared.
This voyage is an editorial path for understanding, not a claim of direct historical influence or sole invention.
"Infinity itself comes in different sizes." Cantor's diagonal argument shows that the reals have greater cardinality than the naturals, while rationals and algebraic numbers remain countable. His work met criticism, including from Kronecker, but also gained important support during his lifetime; his recurring illness cannot be reduced to one mathematical dispute.
Set | Size (cardinality) | A one-to-one correspondence with ℕ? |
|---|---|---|
Even numbers | ℵ₀ | ✓ (n ↔ 2n) |
Integers ℤ | ℵ₀ | ✓ (same size as the natural numbers) |
Rational numbers ℚ | ℵ₀ | ✓ (diagonal enumeration) |
Algebraic numbers | ℵ₀ | ✓ |
Real numbers ℝ | 2^ℵ₀ (= 𝔠) | ✗ (larger, by the diagonal argument) |
Power set of ℝ, 𝒫(ℝ) | 2^𝔠 | ✗ (larger still) |
An infinite hierarchy of infinities. Each step brings a genuinely larger infinity — the paradise and torment Cantor discovered.
A family of mathematical ideas that distinguish potential processes, infinite sets, and different infinite cardinalities—developed through ancient, medieval, and modern debates.
Ports in time
Follow the scenes to see problems, notation, standards of proof, and applications changing across different times and places.
“Achilles can never overtake the tortoise” — can infinitely many steps be completed in a finite time? This was one of the earliest mathematical challenges involving infinity.
No reliable place is given, so time continues without an invented pin
Continue through the world of this yearGalileo observed that the natural numbers can be paired one-to-one with their squares. Infinity delivered an intuitive shock: a part can have the same size as the whole.
No reliable place is given, so time continues without an invented pin
Continue through the world of this yearCantor showed that the infinity of the real numbers is larger than that of the natural numbers. Infinity was no longer one undifferentiated idea, but a hierarchy of distinct sizes.
No reliable place is given, so time continues without an invented pin
Continue through the world of this yearGödel’s 1940 relative-consistency result for CH and Cohen’s 1963 forcing result for ¬CH together established that the standard ZFC axioms decide neither side.
No reliable place is given, so time continues without an invented pin
Continue through the world of this yearThe halting problem, countable versus uncountable sets, measure theory, and compactness in topology — modern mathematics operates throughout the hierarchy of infinities.
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.
The natural numbers provide the benchmark for the smallest infinite cardinality.
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The reals display a larger infinity that no list indexed by the naturals can exhaust.
Open the number voyage
Aleph notation breaks the intuition that infinity has only one size.
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Concept genealogy
Concepts arriving from before
Current port
Infinity
Concepts opened from here
Only direct editorial links are shown; this is not a complete learning order or historical influence line.
