1. Georg Cantor

2. Famous For: Inventor of Set Theory
One-to-One Correspondences/Bijection
Theory of Transfinite Numbers
Cardinality of Infinite Sets

3. Background Information
Born in 1845 in St. Petersburg, Russia, to German Parents
Excellent violinist as a youth
Good student, graduated with honors
Completed dissertation at the University of Berlin
Professor at University of Halle for most of his life
Suffered from nervous breakdowns and depressions later on in life when rivals published papers that contradicted his work
Also ventured into philosophy and Elizabethan Literature
Died in 1918 in a sanatorium

4. Set Theory
Study of sets! Nothing more than it sounds, collections of objects
A set could be – prime numbers, even numbers, irrational numbers, etc
Set theory is its own branch of mathematics and has implications for the nature of numbers, infinity, and logic

5. Bijection One-to-one correspondence between sets
Each item in a set can be matched with an item in another set

6. Fun With Infinity
Are there more even numbers than integers?

7. Fun With Infinity
Intuition would tell us that there are twice as many integers as even numbers, because integers include all the even numbers plus the odd numbers.
BUT
Both sets are infinite – are they the same “size”?
Here’s where transfinite numbers come in – numbers that are infinite (larger than all finite sets), but not necessarily absolutely so.

8. Fun with infinity
{1, 2, 3, 4, 5, 6, 7, 8}
{2, 4, 6, 8, 10, 12, 14, 16}
The set of integers exhibits a one-to-one correspondence with the set of even numbers. Therefore, they are the same size!
They are both Aleph Naught ( )

9. Aleph Naught & Cardinality of Infinite Sets
The Aleph numbers are used to represent the degree of infinity of a set
Aleph Naught is the first infinite cardinal (the lowest infinity). It represents a set that is a countable infinity – a set that has bijection the set of natural numbers.
This includes prime numbers, rational numbers, perfect squares, etc

10. Rational Numbers Are rational numbers a countable infinity?
Here’s how to prove it:

11. Rational Numbers
How about this: Are there more numbers between 0 and 1 than there are natural numbers?
We can prove that there are using bijection and another principle of Cantor’s called the Diagonal Argument.
This produces an uncountable
Set of infinite numbers, so a set with
Greater cardinality than
Aleph Naught

12. That’s All!
Though we have passed through an uncountable infinite set of moments of time during this presentation, we have now finished. How is that possible?

13. Homework
Transcribe the complete set of countable infinite integers.