Understanding Infinity: Paradoxes, Calculus, and Cantor

Explore the beauty of mathematical infinity, from Hilbert's Hotel and Gabriel's Horn to Cantor's diagonal argument and sizes of infinity.

#mathematics#infinity#calculus#set-theory#cantor#science#education

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01

6

Cantor's Diagonal Argument

Assume a complete list of decimals
Change the n-th digit of the n-th number
The new number differs from every list entry

               
            
                       n1 =
                       0.
                    
3
1
4
1
5
9
...

                       n2 =
                       0.
                    
7
1
8
2
8
1
...

                       n3 =
                       0.
                    
2
7
1
8
2
8
...

                       n4 =
                       0.
                    
1
4
1
5
9
2
...

                       n5 =
                       0.
                    
6
1
8
0
3
4
...

                       n6 =
                       0.
                    
8
4
1
4
7
1
...

                       nnew =
                       0.
                    
1
2
2
1
1
2
...

                       Rule: 1 → 2, x → 1
                    

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02

7

Hilbert's Hotel

Infinite rooms
"Full" hotel
Still room for more

New Guest

New → 1

Room 1

1 → 2

Room 2

2 → 3

Room 3

3 → 4

Room 4

4 → 5

Room 5

5 → 6

Room 6

6 → 7

Room 7

n → n+1

Room 8

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03

8

Infinity in Calculus

Limits → ∞
Infinite series
Improper integrals

lim → ∞

Convergent

Divergent

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04

9

Gabriel's Horn

Infinite surface area
Finite volume
"Fillable but unpaintable"

y =

1 x

x

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05

10

Sizes of Infinity

Countable < uncountable
Continuum
Even bigger infinities exist

ℵ₁, ℵ₂, ...

Larger Infinities...

Even larger cardinals

c = 2^ℵ₀

Continuum

(real numbers)

ℵ₀

Countable Infinity

(integers, rationals)

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06

11

Why Infinity Matters

Defines real numbers
Used in physics
Used in computer science

∞

⃛

∑

∫

Mathematics

E=mc²

Physics

Comp Sci

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07

A Question to Think About

Which infinity does the universe use?

12

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