Beyond the Limit: Exploring the Sizes of Infinity
Learn about countable vs. uncountable infinity, Cantor's Diagonal Argument, Hilbert's Hotel, and the role of infinity in calculus and physics.
Infinity
Noah & Keenan
What We Already Know
Infinity shows up in limits
It's not a number
Means "grows without bound"
Potential vs Actual Infinity
keeps going
complete infinite set
counting vs ℕ
Potential
Actual
ℕ
complete infinite set
Countable
Infinity
Can list elements
Natural numbers, integers, evens
All same "size"
Uncountable Infinity
Can't list all elements
Real numbers
Bigger than countable
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
2^𝔠
even larger...
𝔠
continuum / uncountable
ℵ₀
countable infinity
Why Infinity Matters
Defines real numbers
Used in physics
Used in computer science
A Question to Think About
Which infinity does the universe use?
- math
- infinity
- calculus
- set-theory
- georg-cantor
- hilberts-hotel
- science-education