sonsuzluk

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The Mathematics of Infinity

• Georg Cantors Theory of Sets

To see the world in a grain of sand. And heaven
in a wildflower Hold infinity in the palm of
your hand, And eternity in an hour. --Willia
m Blake
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From the paradise created for us by Cantor no
one will drive us out. David Hilbert
Georg Cantor
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Zenos paradox
2 1
1/2 1/4 1/8 1/16
Suppose a hare can run twice as fast as a
tortoise. The hare starts 2 miles from the
finish and the tortoise starts at one mile from
the finish.
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Zenos paradox
2 1
1/2 1/4 1/8 1/16
By the time the hare gets to where the tortoise
starts, the tortoise has gone 1/2 mile.
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Zenos paradox
2 1
1/2 1/4 1/8 1/16
By the time the hare gets from the 1 mile marker
to the 1/2 mile marker, the tortoise has gone 1/4
mile.
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Zenos paradox
2 1
1/2 1/4 1/8 1/16
This could go one forever, by the time the hare
gets to a marker, the tortoise has moved on to
the next. The hare never seems to catch up with
the tortoise.
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Zenos paradox
How many parts are there in a finite
object? 1/2 1/4 1/8 ... 1
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Hilberts Hotel
and still everyone gets a room!
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Hilberts Hotel
1 2 3
4 5
No Vacancies
1 2 3
4 5
1 moves to 3, 2 moves to 5, 3 moves to 7,
Even an infinite number of guests can be served
at Hilberts Hotel.
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What does Infinity mean?

• The known is finite, the unknown infinite
  intellectually we stand on an island in the midst
  of an illimitable ocean of inexplicability. Our
  business in every generation is to reclaim a
  little more land.
• --Thomas H Huxley

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An Introduction to Sets

• A 2, 3, 5, 7, 11, 13, 17, 19
• N 1, 2, 3, 4,
• Z -3, -2, -1, 0, 1, 2, 3,
• Q a/b a, b ? Z, b ? 0
• A ? N, N ? Z, and Z ? Q

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Equivalent Sets

• Two sets are equivalent if there exists a 1-1
  correspondence between their elements
• A 2,4,6 and C 1,2,3 are equivalent

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Infinite Sets

• A set A is finite if it is empty or if there is
  a natural number n such that A is equivalent
  to 1,2,3, ,n
• A set is infinite if it isnt finite.

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Question Are all infinite sets equivalent?

• In other words, can two infinite sets always be
  put in 1-1 correspondence?
• Consider the Natural numbers and the even
  positive integers

A 1-1 correspondence is shown so these are
equivalent sets.
1, 2, 3, 4, , n, ...
2, 4, 6, 8, , 2n, ...
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N is equivalent to Z
1, 2, 3, 4, 5, 6, 7, 8, 9,
Here is a 1-1 correspondence between the Natural
numbers and the Integers.
0, 1, -1, 2, -2, 3, -3, 4, -4,
So the Natural numbers and the Integers have the
same number of numbers in them.
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The Rational Numbers
What about Q, the set of all rational numbers (ie
fractions), which is clearly much bigger than N?
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The Rationals
The rational numbers and the natural numbers are
sets with the same number of elements.
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More about Rational Numbers

• By definition, any number which may be expressed
  as a fraction is a Rational number.
• It may also be shown that any decimal which
  terminates or repeats may be made into a fraction
  and is thus a Rational number.

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N is not equivalent to R.
In fact by showing that the interval 0, 1 is
larger than N, then R must be larger than N.

• N 1, 2, 3, 4,

R
-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1
2 3 4 5 6
0
1
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N is not equivalent to R.

• Proof by contradiction
• Assume N is equivalent to R. Then there must
  be a 1-1 correspondence between them.

N 1, 2, 3, 4,
R
-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1
2 3 4 5 6
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N is not equivalent to R.

• Proof by contradiction
• Assume N is equivalent to R. Then there must
  be a 1-1 correspondence between them.
• This correspondence will pair each Natural number
  with a Real number.

N 1, 2, 3, 4,
R
-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1
2 3 4 5 6
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N is not equivalent to R.

• Proof by contradiction
• Assume N is equivalent to R. Then there must
  be a 1-1 correspondence between them.
• This correspondence will pair each Natural number
  with a Real number.
• We will show that this assumption leads to a
  contradiction by constructing a Real number which
  has not been included in the correspondence.

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A possible correspondence

• N 0,1
• 1 0 . 3 0 1 2 5 9 4
• 2 0 . 1 6 6 5 2 1 8
• 3 0 . 4 1 1 2 1 0 7
• 4 0 . 2 0 5 0 9 6 3
• . .
• . .
• . .

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Constructing a new number(not in the
correspondence)
N 0,1 1 0 .(3)0 1 2 5 9 4 2 0 . 1(6)6 5
2 1 8 3 0 . 4 1(1)2 1 0 7 4 0 . 2 0 5(0)9
6 3 . . . . . .
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Constructing a new number(not in the
correspondence)
N 0,1 1 0 .(3)0 1 2 5 9 4 2 0 . 1(6)6 5
2 1 8 3 0 . 4 1(1)2 1 0 7 4 0 . 2 0 5(0)9
6 3 . . . . . .
If a decimal place is not a 1, put a 1 in that
decimal place. If the decimal place is a 1, put a
2 there.
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Constructing a new number(not in the
correspondence)
N 0,1 1 0 .(3)0 1 2 5 9 4 2 0 . 1(6)6 5
2 1 8 3 0 . 4 1(1)2 1 0 7 4 0 . 2 0 5(0)9
6 3 . . . . . .
w 0 . 1 1 2 1
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Constructing a new number(not in the
correspondence)
N 0,1 1 0 .(3)0 1 2 5 9 4 2 0 . 1(6)6 5
2 1 8 3 0 . 4 1(1)2 1 0 7 4 0 . 2 0 5(0)9
6 3 . . . . . .
w 0 . 1 1 2 1 Every decimal place of w is
different than every number in this (supposedly)
complete list.
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A Contradiction!!

• So w is different from every Real number in the
  original correspondence.
• For every possible correspondence, there is a way
  to find a number it missed.
• Thus, N is not equivalent to R.

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A Contradiction!!

• So w is different from every Real number in the
  original correspondence.
• For every possible correspondence, there is a way
  to find a number it missed.
• Thus, N is not equivalent to R.
• This means that there are at least two sizes or
  classes of infinite sets.

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Sizes of Infinity ?0 lt ?1 lt ?2 lt

• Just as the cardinal number 3 is used to describe
  the size of any set equivalent to a, b, c,
• ?0 is used to describe the size of any infinite
  set equivalent to N.
• And c is used to describe the size of any
  infinite set equivalent to R.

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So now, we have at least two sizes of infinity

• Are there others?
• It can be shown that there are an infinite number
  of infinities of larger and larger sizes, but
  that is enough for now.

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I could be bounded in a nutshell, and count
myself a king of infinite space.
--Shakespeare
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TOMORROW THE SHAPE OF THE UNIVERSE
Please bring one item that is 1 inch to 3 inches
tall with a string attached that is about 4
inches long. Bright colors are preferred.
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TOMORROW THE SHAPE OF THE UNIVERSE
Bring a small flashlight if you have one. The
smaller, the better.
Also, we will be working with mirrors, so if you
want to, bring gloves.
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Thank You