Georg Cantor's theories on infinity and set theory revolutionized mathematics despite facing resistance from contemporaries. He defined different types of infinities and introduced concepts like cardinality, demonstrating that not all infinities are equal, particularly through his proof of the uncountability of real numbers. His work also led to the formulation of Cantor's theorem and the continuum hypothesis, establishing the complexity and hierarchy of infinite sets.

1. To Infinity and Beyond!
Cantor’s Infinity theorems
Oren Ish-Am

2. Georg Cantor (1845-1918)
 His theories were so counter-intuitive that met with much resistance from
contemporary mathematicians (Poincaré, Kronecker)
 They were referred to as “utter nonsense” “laughable” and “a challenge to the
uniqueness of God”.
 Invented Set Theory – fundamental in mathematics.
 Used in many areas of mathematics including topology,
algebra and the foundations of mathematics.
 Before Cantor – only finite sets in math - infinity was a
philosophical issue
 Cantor, a devout Lutheran, believed the theory had been
communicated to him by God

3. Set Theory – basic terminology
Set – collection of unique elements – order is not important
A = 3, , B = 8,
Member: 𝒂 ∈ 𝑨 means 𝑎 is a member of the set 𝐴
∈ A 8 ∈ B
Subset: 𝑺 ⊂ 𝑨 means 𝑆 contains some of the elements of the set 𝐴
S = 3, S ⊂ A
Union: 𝑪 = 𝑨 ∪ 𝑩 means 𝐶 contains all the elements 𝐴 and all elements of 𝐵
C = 3, , 8,,

4. Set Theory – basic terminology
Set Size – |𝑨| is the number of elements in 𝐴
Empty Set: {} or 𝝓 is the empty set – set with no items. 𝜙 = 0
|A| = 3 |B| = 2
Natural Numbers: ℕ = {1,2,3…}
Rational Numbers: ℚ =
1
3
,
101
23
,
6
6
,
10
5
, …
Real Numbers : ℝ = ℚ ∪ all numbers that are not rationals like π,e, 2 …

5. Functions – basic terminology
‫על‬ ‫חח‬"‫ע‬ ‫חח‬"‫ע‬‫ועל‬

6. Counting (it’s as easy as 1,2,3)
How can we tell the size of a set of items?
Just count them:
21 3 4
A B
 For finite sets, if A is a proper subset of B (A ⊂ B)
 Then the size of A is smaller then the size of B ( A < B )

7. Counting (it’s not that easy…)
Surely there are less squares than natural numbers (1,2,3,…) – right?
What can be say about an infinite subset of an infinite set?
Let’s try another counting method - comparison
1 2 3 4 5 6 7 8 9 …
…
 Can we count infinite sets?
 For example - what is the “size” of the set of all squares?

8. Assume we have a classroom full of chairs and students
We know all chairs are full and - there are students standing.
Without having to count, we know there are more students then chairs!
Counting (it’s not that easy…)
 Similarly, if there is exactly one student on each chair – the
set of students is the size of the set of chairs.
 This type of mapping is called bijection ( ‫חד‬‫חד‬‫ועל‬ ‫ערכי‬ )

9. Cardinality )‫עוצמה‬(
Definition: Two sets have the same cardinality iff there exists a
bijection between them
We can now measure the size of infinite sets! Let’s try an example:
The set of natural numbers and the set of even numbers
2 4 6 8 10 12 14 16 …f(x)
1 2 3 4 5 6 7 8 …x
 A simple function exists: 𝑓 𝑥 = 2𝑥, a bijection.
 Therefore the two sets have the same cardinality.

10. Cardinality )‫עוצמה‬(
How about natural number and integers?
1 -1 2 -2 3 -3 4 …f(x) 0
1 2 3 4 5 6 7 …x 0
𝑓 𝑥 =
𝑥/2 𝑥 ∈ odd
− 𝑥 2 𝑥 ∈ even
 Cantor was sick of Greek and Latin symbols
 Decided to note this cardinality as ℵ 𝟎
 This is the cardinality of the natural numbers, called countably infinite

11. Cardinality of ℚ
The rationales are dense;
there is a fraction between every two fractions.
There are infinite fractions between 0 and 1 alone!
In fact, there are infinite fractions between any two fractions!
Could we find a way to match a natural number to each fraction?
 Ok, we get it, so ℵ0 + 𝑐𝑜𝑛𝑠𝑡 = ℵ0 and ℵ0 ⋅ 𝑐𝑜𝑛𝑠𝑡 = ℵ0
 What about the rationals - ℚ?

12. Cardinality of ℚ
Naturals 1 2 3 4 5 6 7 8 9 10
numerator 1 1 2 3 2 1 1 2 3 4
denominator 1 2 1 1 2 3 4 3 2 1
 We map a natural number to a numerator and denominator (fraction)
 Eventually, we will reach all possible fractions – Bijection! Hurray!

13. Countably Infinite - ℵ 𝟎
Challenge: find the bijection from natural numbers to fractions.
This is the cardinality of
All finite strings
All equations
All functions you can write down a description of
All computer programs.
 Seems like one can find a clever bijection from the natural numbers
to any infinite set….right? Are we done here?
 Well… not so fast…

14. Cardinality of Reals - ℝ
List might start like this:
1. .1415926535...
2. .5820974944...
3. .3333333333...
4. .7182818284...
5. .4142135623...
6. .5000000000...
7. .8214808651...
 Cantor had a hunch that the Reals were not countable.
 Proof by contradiction – let’s assume they are and make a
list:
This method is called
Proof By Diagonalization

15. Cardinality of Reals - ℝ
New Number = 0.0721097… isn’t in the list! Contradiction!
The reals are not countable – more than one type of infinity!!
“Complete List”
1. .1415926535...
2. .5820974944...
3. .3333333333...
4. .7182818284...
5. .4142135623...
6. .5000000000...
7. .8214808651...
etc.
Make a new number:
1. .0415926535...
2. .5720974944...
3. .3323333333...
4. .7181818284...
5. .4142035623...
6. .5000090000...
7. .8214807651...
etc.

16. Cardinality of Reals
Easy! Just de-interleave the digits. For example
0.1809137360 … → 0.1017664 … , 0.893301 …
Therefore - ℵ is the cardinality of any continuum in ℝ 𝑛
Cantor: “I see this but I do not believe!”
0 1
 Cantor called this cardinality ℵ (or ℵ1 as we will soon see).
 ℵ is the cardinality of the continuum –all the points on the line:
0,1 , (−∞, ∞) both have cardinality ℵ. Can you find the bijection?
 What about the Unit Square? Can we map 0,1 → 0,1 × [0,1]?

17. Cardinality of Reals
This leads to some interesting “paradoxes” like the famous Banach-Tarsky:
Given a 3D solid ball, you can break it into a finite number of disjoint subsets,
which can be put back together in a different way to yield two identical copies
of the original ball.
 The reconstruction can work with as few as five “pieces”.
 Not pieces in the normal sense – more like sparse infinite sets of points

18. Cardinality of Reals
 ‫ספור‬ ‫אין‬ ‫או‬ ‫סוף‬ ‫אין‬?
Implications of ℵ0 < ℵ:
Most real numbers do not have a name.
They will not be the result of any formula or computer program.
Can we say they actually “exist” if they can never be witnessed??
 This has actual implications in mathematics!
We need to add the Axiom of Choice

19. More Power!
So are we done now? Countable infinity and continuum?
Power Set: For a set 𝑆 the power set 𝑃(𝑆) is the set of all subsets of 𝑆
including the empty set (𝜙 or {}) and 𝑆 itself.
 Example:
𝑆 = 1,2,3
𝑃 𝑆 = 𝜙, 1 , 2 , 3 , 1,2 , 1,3 , 2,3 , 1,2,3
 Challenge: prove that for any finite set 𝑆, 𝑃 𝑆 = 2 𝑆 (hint – think binary)
 What about infinite sets? Here Cantor came up with a simple and ingenious
proof that shows for any infinite set 𝐴: 𝑃 𝐴 > |𝐴|

20. Cantor’s Theorem
Theorem: Let 𝑓 be a map from set 𝐴 to 𝑃(𝐴) then 𝑓: 𝐴 → 𝑃(𝐴) is not
surjective (‫)על‬
‫על‬ ‫חח‬"‫ע‬ ‫חח‬"‫ע‬‫ועל‬

21. Cantor’s Theorem (proof from “the book”)
Proof:
a. Consider the set 𝐵 = {𝑥 ∈ 𝐴 ∣ 𝑥 ∉ 𝑓 𝑥 }.
Set of all items from 𝐴 which are not an element of the subset they
are mapped to by 𝑓
a. Note that 𝐵 is a subset of 𝐴 the therefore B ∈ 𝑃(𝐴)
b. Therefore there exists an element 𝑦 ∈ 𝐴 such that 𝑓 𝑦 = 𝐵
c. Two options:
 𝑦 ∈ 𝐵 in this case by definition of 𝐵, 𝑦 ∉ 𝑓 𝑦 = 𝐵:
contradiction!
 𝑦 ∉ 𝐵 in this case 𝑦 ∉ 𝑓(𝑦) which means 𝑦 ∈ 𝐵: contradiction!
So such a function 𝑓 cannot exist and there can not be a bijection:
𝑃 𝐴 > 𝐴
1
2
3
4
5
.
.
.
{1,2}
{3,17,5}
{8}
{17,4}
{1,2,5}
.
.
.
y
.
B={2,3,…}
.
A P(A)

22. Cardinality of Reals – connection to subsets of ℕ
Challenge: Prove that 2 ℕ = |ℝ| (hint – think binary. Again)
Let’s look at [0,1]: The binary representation of all numbers in [0,1] are all
the subsets of ℕ:
0.11000101… is the set {1,2,6,8,…}

23. Other cardinalities
Why not go further? Define ℶ = 2ℵ
the cardinality of the power set of ℝ. This
is the cardinality of all functions 𝑓: ℝ → ℝ
 We can continue forever increasing the cardinality:
ℵ0, ℵ1 = 2ℵ0, ℵ2 = 2ℵ1, … , ℵ 𝑛+1 = 2ℵ 𝑛, …
 There is no largest cardinality!
 Wait – what about the increments? Is there a cardinality between the
integers and the continuum?
 This was Cantor’s famous Continuum Hypothesis

24. Continuum Hypothesis
CH is independent of the axioms of set theory. That is, either CH or its
negation can be added as an axiom to ZFC set theory, with the resulting
theory being consistent.
 Continuum Hypothesis (CH): There is no set whose cardinality is strictly
between that of the integers and the real numbers.
 Stated by Cantor in 1878 is became one of the most famous unprovable
hypotheses and drove Cantor crazy.
 Only in 1963 it was proved by Paul Cohen that….

25. Cardinality of Cardinalities
Just one last question to ask – exactly how many different cardinalities are
there?
 By iterating the power set we received the sets
ℕ , 𝑃 ℕ , 𝑃 𝑃 ℕ , P P 𝑃 ℕ , …
 With the cardinalities
ℵ0, ℵ1, ℵ2, ℵ3, …
 There are at least ℵ0 different cardinalities.
Are we good Vincent?

26. 𝑩 = 1,17, 4,9 , 102,4,22 , 2 , 3,6 , 96,3,21 , …
Cardinality of Cardinalities
Ok, hold on to your hats…
Let’s look at the set 𝐴 that contains all the power sets we just saw:
𝐴 = ℕ , 𝑃 ℕ , 𝑃 𝑃 ℕ , P P 𝑃 ℕ , …
 And now let’s look at the set 𝑩 = ⋃𝑨 which is the union of all the elements
of all the sets in 𝐴. Here is what 𝐵 may look like
Elements
from ℕ
Elements from
𝐏 ℕ
Element from
𝐏 𝐏 ℕ
 𝐵’s cardinality is as large as the largest set in 𝐴

27. Cardinality of Cardinalities
The set 𝐵 𝑏𝑖𝑔𝑔𝑒𝑟 = 𝑃 𝐵 = 𝑃(⋃𝐴) - that cardinality is not in the ℵ 𝑛 list
In fact – for any set 𝑆 of sets, the set 𝑆 𝑏𝑖𝑔𝑔𝑒𝑟 = 𝑃 ⋃𝑆 will have a cardinality
not in 𝑆!
No set, no matter how large cannot hold all cardinalities.
So the collection of possible cardinalities is… is not even a set because it does
not have a cardinality.
Mathematicians call this a proper class, something like the set of all sets
(which is not permitted in set theory to avoid things like Russel’s paradox).

28. Thank You!