Understanding Infinity: Mathematics and Paradoxes
Explore the concept of infinity through Cantor's diagonal argument, Hilbert's Hotel, Gabriel's Horn, and more. Learn why infinity matters in calculus.
Cantor's Diagonal Argument
Assume list of decimals
Change nth digit
New number not on list
Hilbert's Hotel
Infinite rooms
"Full" hotel
Still room for more
Infinity in Calculus
Limits → ∞
Infinite series
Improper integrals
Gabriel's Horn
Infinite surface area
Finite volume
"Fillable but unpaintable"
Sizes of Infinity
Countable < uncountable
Continuum
Even bigger infinities exist
countable infinity (ℕ, ℤ, ℚ)
continuum (ℝ)
larger infinities...
Why Infinity Matters
Defines real numbers
Used in physics
Used in computer science
Mathematics
Physics
Computer Science
A Question to Think About
Which infinity does the universe use?
∞
- mathematics
- infinity
- cantor
- calculus
- hilbert-hotel
- set-theory
- science