1. Donald Byrd rev. 28 November 2012
Cantor’s Diagonal Proof and Uncountable Numbers: To Infinity and Beyond!
Donald Byrd
rev. 28 November 2012

2. Hilbert’s Hotel and Infinite Sets
Explanation of infinite sets by David Hilbert
Hotel with finite rooms, all occupied
Can’t accommodate a new guest
Hotel with infinite rooms, all occupied
Can accommodate a new guest
Move each guest from room n to n+1
Can accommodate infinite no. of new guests!
How?
28 Nov. 2012

3. One-to-One Correspondence
How can you compare size of collections of things if there are too many to count?
…for example, infinite collections
Georg Cantor (1891):
Put items in each set (collection) in a list
Try one-to-one correspondence
There are infinitely many integers, but…
Cantor proved there are more real nos.!
28 Nov. 2012

4. One-to-One Correspondence between Infinite Sets (1)
Even
N Numbers Integers
0 0 0
1 2 -1
2 4 1
3 6 -2
4 8 2
etc. etc. etc.
28 Nov. 2012

5. One-to-One Correspondence between Infinite Sets (2)
Even Positive
N Numbers Integers Squares Rationals /2 /1 /3 /1 /4 /3 /2 /1
etc. etc. etc. etc. etc.
28 Nov. 2012

6. “Complete List of Real Numbers”?
etc., etc.
28 Nov. 2012

7. “Complete List of Real Numbers”?
Just real numbers between 0 and 1 is enough.
List might start like this:
28 Nov. 2012

8. No “Complete List of Real Numbers”!
etc.
Make a new number:
etc.
• New Number = … isn’t in the list!
• How do we know?
28 Nov. 2012

9. Different Sizes of Infinity
Proof by contradiction
Cantor’s conclusion: there are more reals between 0 and 1 than there are integers!
Integers are countable
…also even nos., rational nos., etc.
Reals (and larger infinities) are uncountable
No. of integers = aleph-0; of reals, aleph-1
Amazed mathematicians
Led to set theory, new branch of math
28 Nov. 2012

10. Conclusion: Let’s Sing! (1)
Some versions of A Hundred Bottles of Beer for really long car trips 
Cf.
Basic transfinite version 1  
Infinite bottles of beer on the wall, infinite bottles of beer;
If one of those bottles should happen to fall,
infinite bottles of beer on the wall.
(etc.)
28 Nov. 2012

11. Conclusion: Let’s Sing! (2)
Basic transfinite version 2 (generalization of ver. 1)  
Infinite bottles of beer on the wall, infinite bottles of beer;
If finite bottles should happen to fall,
infinite bottles of beer on the wall.
(etc.)
28 Nov. 2012

12. Conclusion: Let’s Sing! (3)
Larger-infinity version  
Uncountable bottles of beer on the wall, uncountable bottles of beer;
If countable bottles should happen to fall,
uncountable bottles of beer on the wall.
(etc.)
28 Nov. 2012

13. Conclusion: Let’s Sing! (4)
General transfinite version  
Aleph-n bottles of beer on the wall, aleph-n bottles of beer;
If, where m < n, aleph-m bottles should happen to fall,
aleph-n bottles of beer on the wall.
(etc.)
Transfinite and indeterminate version (by Richard Byrd)
Infinite bottles of beer on the wall, infinite bottles of beer;
If infinite bottles should happen to fall,
indeterminate bottles of beer on the wall.
(The End)
28 Nov. 2012