M3T Cardinalities

Prepared by: Kee Wen Ng

What Do You Know About Infinity?

Does ∞ + 1 = ∞?

Does ∞ + ∞ = ∞?

Is ∞ * ∞ > ∞?

Are there different kinds of infinities?

If so, how many different kinds of infinity exist?

What is ∞?

Hilbert’s Grand Hotel

HGH has infinitely many rooms, numbered {0, 1, 2, 3, 4, ... }

HGH’s aim: accommodate as many customers as possible

All the rooms are occupied

All the customers are cooperative

What happens if they need to accommodate a new customer?

Methods of Counting

Given 2 boxes of marbles, determine which one has more marbles.

Bijective Functions

1

2

3

A bijection (one-to-one correspondence) is a function between 2 sets, where all element from one set is paired with a unique element from the other set, and no elements in both sets are unpaired.

Back to Help the Lobby Boy

v

Think About the Bijection

{ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, … } ∪ { 0 }

...

~~~~~~~~~~~f(x) = x + 1~~~~~~~~~~~~~~

…

{ 1, 2, 3, 4, 5, 6, 7, 8, 9 ,10, 11, 12, 13, 14, … } 1k

More Customers!!!

An infinitely big bus with infinitely many passengers arrive:

f(x)=2x

Infinitely many busses all with infinitely many passengers arrive:

Infinitely many ferries with infinitely many buses with infinitely passengers arrive???

ℕ vs ℤ vs ℚ vs ℝ vs ℝ/ℚ

Natural Numbers, ℕ = { 1, 2, 3, 4, 5, 6, 7, … }

Integers, ℤ = { … , -5, -4, -3, -2, -1, 0 , 1, 2, 3, 4, 5, … }

Rational Numbers, ℚ = { a/ b| a ∈ ℤ, b ∈ ℕ }

Real Numbers, ℝ = ( -∞, +∞), e.g.: 3, -4.1, π, -e/3.22, 0…

Irrational Numbers ℝ/ℚe.g.: π, (π)^(1/2), sqrt(2), e, 0.101001000100001000001…

Natural vs Integers

Comparison with the Hilbert’s Hotel Problem

ℤ = { … , -5, -4, -3, -2, -1, 0 , 1, 2, 3, 4, 5, … } = {… , -5, -4, -3, -2, -1} ∪ { 0 } ∪ {1, 2, 3, 4, 5, … } = -ℕ ∪ { 0 } ∪ ℕ

=>|ℤ|= |ℕ|+1+|ℕ| = 2 1.+א

א = . =|ℕ|

We call these sets countably infinite

What about Rational numbers vs Natural numbers or Integers?

{ … , -5, -4, -3, -2, -1, 0 , 1, 2, 3, 4, 5, …}

{ 0 , 1, 2, 3, 4, 5, …}

{1, 2, 3, 4, 5, 6,…}

Rational vs Natural/ Integers

Is there a way to send { … , -1/2, -56/37, 0, 1/8, 2/3, 4, …}to { 1, 2, 3, 4, 5, 6, 7, …}? (George Cantor)

Take a look at the positive rationals:

Rational vs Natural/ Integers

Re-arrange into: { 1, 2/1, 1/2, 1/3, 3/1, 4/1, 3/2, 2/3, 1/4, 1/5, 5/1, … }

Map to { 1, 2, 3, 4, 5, 6, 7, 8, …}

=> |positive ℚ| = |ℕ| = א. => |ℚ| = |positive ℚ| + |negative ℚ| + |{0}|

1+ א. . + א =

א = . => ℚ is countably infinite

Is there a bigger infinity?

Real numbers vs Natural

Can real numbers be paired one-to-one with Natural numbers?

i.e. |ℝ| = א.? Let’s just check the small subset of ℝ, ( 0, 1)

Assume the real numbers in ( 0, 1) is countable, i.e. a one-to-one relationship between ℕ and the real numbers in ( 0, 1):

12345678…

ℕ0.13598431619840237415710384143513418609538642316…0.82414651614276424342429358252657523735757865972…0.54935683783245254736586984474452345234676586787…0.30000000000000000000000000000000000000000000000…0.31415926535897932384626433832795028841971693993…0.10100100010000100000100000010000000100000000100…0.93452034000000000000000000000000000000000000000…0.42654646376478598794252345546354769758985732423… …

( 0, 1 )

…

Real numbers and Aleph-1

123456789…

0. 1 3 5 9 8 4 3 1 6 1 9 8 4 0 …0. 8 2 4 1 4 6 5 1 6 1 4 2 7 6 …0. 5 4 9 3 5 6 8 3 7 8 3 2 4 5 …0. 3 0 0 0 0 0 0 0 0 0 0 0 0 0 …0. 3 1 4 1 5 9 2 6 5 3 5 8 9 7 …0. 1 0 1 0 0 1 0 0 0 1 0 0 0 0 …0. 9 3 2 0 0 0 0 0 0 0 0 0 0 0 …0. 3 3 3 3 3 3 3 3 3 3 3 3 3 3 …0. 1 4 2 8 5 7 1 4 2 8 5 7 1 4 … …

( 0, 1 )

…0. 2 3 0 1 6 2 1 4 3 … ( 0, 1 )New Number:

+

1

+

1

+

1

mod10 Doesn’t

EqualCONTRADICTION!!!!!

Irrational Numbers

( 0, 1) subset of ℝ has more elements than |ℕ| = א. =>|ℝ| ≥ |( 0, 1)| > |ℕ| = א.

|ℝ| = א 2= 1א . (We say that ℝ is an uncountable set)

What can we deduce about Irrational Numbers?

|ℝ/ℚ| + |ℚ| = |ℝ|

There are more Irrational Numbers than Rational Numbers!(In fact, you can prove this by using the same trick to prove that the real numbers are uncountable!)

CountableUncountable

Uncountable

Bigger cardinalities!!!

Are there bigger infinities than Aleph-1?

YES! In fact there are INFINITELY many bigger infinities!

Notes:

Is the smallest infinity .א

א 2 = 1א .

In fact, we can create bigger infinities by taking

k-1א k = 2א