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GEORG CANTOR

FAMOUS FOR:

• Inventor of Set Theory•One-to-One Correspondences/Bijection• Theory of Transfinite Numbers• Cardinality of Infinite Sets

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

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

BIJECTION

• One-to-one correspondence between sets• Each item in a set can be matched with an item

in another set

FUN WITH INFINITY

• Are there more even numbers than integers?

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.

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 ( )

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

RATIONAL NUMBERS

• Are rational numbers a countable infinity?• Here’s how to prove it:

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 withGreater cardinality thanAleph Naught

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?

HOMEWORK

• Transcribe the complete set of countable infinite integers.