GEB class notes Jan 23

1
GEB class notes Jan 23
Infinite Sets Cantor, Hilbert, Russell,
Gödel The Infinite Hotel Infinite Sums and
Zenos Paradoxes
2
Cantor and Counting I

• Two Sets are kind of the same if they have the
  same number of elements.
• Infinite Sets are kind of the same if they can
  be put into one-to-one correspondence.
• The Set of Even Integers is kind of the same as
  the Set of All Integers.

3
Cantor and Counting II
Partitioning the Set of Sets
Sets having one element
Sets having two elements
. . .
. . .
Infinitely many elements

• In this context, kind of the same is
  isomorphic as sets ? FANCY!

4
Cantor and Counting III

• Cantor --- How do we distinguish the different
  kinds of infinite sets? (not a weird question!)
• Number of even integers Number of integers
• Number of powers of 2 Number of integers
  ie., 2, 4, 8,16,32,64, is kind of the same as
  1, 2, 3, 4, 5, 6,

5
The Infinite Hotel

• If youre in Room N, move to Room N1 always
  room for one more
• If youre in Room N, move to Room 2N room for
  infinite busload
• If youre in Room N, move to Room 2 N room for
  infinitely many infinite busloads

6
Cantor and Counting IV

• Number of integer fractions Number of
  integers (1873)
• Number of points on line segment is uncountable
  (1874)
• Number of points in square Number of
  points on line segment (1877) I see it, but I
  don't believe it!

7
Cantor Hilbert
1862 - 1943
1845 - 1918
8
Russell Whitehead
1872 - 1970
1861 - 1947
9
DOUBTS, FIXES, more DOUBTS

• Cantor we dont know how to count and we cant
  handle infinite sets.
• Russell what about the set of all sets which are
  not members of themselves?
• Russell and Whitehead Theory of Types and
  Principia Mathematica
• Hilbert desire to demonstrate that Principia
  Mathematica is both consistent and complete.
• Godel powerful and consistent implies incomplete.

1870s
1901
1910 - 1913
1922 - 1927
1931
10
Theory of Types

• The Set of Sets is an illegal concept.
• Get around the Set of Sets by referring to
  objects, sets of objects, collections of sets of
  objects, ensembles of collections of sets of
  objects,
• Hierarchy of language, metalanguage
  (statements about language), e.g.German
  sentences at the end verbs have metametalanguage
  (statements about statements about language),
  

11
Lets catch our breath with a blank slide
12
An Infinite Sum

• 1 ? ?

1
?
?
1
13

• 1 ? ? ?

?
1
?
?
1
14

• 1 ? ? ? ?

?
?
1
?
?
1
15

• 1 ? ? ? ?

?
?
1
?
?
1
16
Follow the Bouncing Ball

• Infinitely many bounces in finite time

http//www.pipey.com/freeflash/gravity.asp
17
Zenos Dichotomy Paradox

• In going from point A to point B, first you must
  go halfway.
• But before reach the halfway point, you must get
  halfway there.
• But before you get halfway to the halfway point,
  you must get halfway there
• Therefore motion is impossible!

18
Response to Zeno

• If A and B are 1 mile apart and you travel at 1
  mile per hour, then (even following Zenos
  description) your total elapsed time is
  ? ? ? ? 1 hour

A
B
?
?
?
?
...
ITS JUST NOT SO, ZENO!
19
The Tortoise and Achilles

• How did you handle this using ordinary algebra?

20
Achilles and the Tortoiserecursive formulas for
Zeno A n A n-1 ? (T n-1 - A n-1) t n
t n-1 ?(T n-1 - A n-1 ) (sA) T n T n-1
(t n - t n-1 )sT

• The nth stage can be computed from the (n -1)th
  stage.