How Big is Infinity

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How Big is Infinity?
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• Tim Hill
• 6.UAT Concept Talk
• 10/19/2006

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How Big is Infinity?

• VERY BIG!

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How Big is Infinity?

• VERY BIG!
• . . . even bigger

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Are these infinite?

• Stars in the sky
• about 3,000
• People on earth
• about 6,000,000,000
• Particles in the universe
• about 1080
• All of these have bounds, just add 1

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My favorite game

• I'm right
• I'm right 2

• I'm right 1
• . . .

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My favorite game

• I'm right
• I'm right 2
• I'm right 8!

• I'm right 1
• . . .

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A new infinity

• Infinity is still not really defined
• Sand, people, and stars didn't cut it, what set
  will?
• Let's play my favorite game starting at 1
• 1, 2, 3, 4, 5, . . .
• Call the set of all of them N
• N is infinite

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My favorite game

• I'm right
• I'm right 2
• I'm right N

• I'm right 1
• . . .

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My favorite game

• I'm right
• I'm right 2
• I'm right N

• I'm right 1
• . . .
• I'm right N 1
• Uh oh

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Back to the cavemen
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Back to the cavemen
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Back to the cavemen
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Cavemen and infinity

• In one way, N 1 N
• 0 1 2 3 4 . . .
• _ 1 2 3 4 . . .
• In another, N 1 N
• 0 1 2 3 4 . . .
• 1 2 3 4 5 . . .
• They are both unbounded

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My favorite game

• I'm right
• I'm right 2
• I'm right N

• I'm right 1
• . . .
• I'm right N 1

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My favorite game

• I'm right
• I'm right 2
• I'm right N

• I'm right 1
• . . .
• I'm right N 1
• I'm right N 2

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Cavemen and infinity

• What about other sets?
• Evens?
• 2 4 6 . . .
• 1 2 3 . . .
• Evens Odds N
• Evens Odds N
• 2 N N

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My favorite game

• I'm right
• I'm right 2
• I'm right N

• I'm right 1
• . . .
• I'm right N 1
• I'm right N 2

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My favorite game

• I'm right
• I'm right 2
• I'm right N

• I'm right 1
• . . .
• I'm right N 1
• I'm right N 2
• I'm right N 2

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My favorite game

• I'm right
• I'm right 2
• I'm right N

• I'm right 1
• . . .
• I'm right N 1
• I'm right N 2
• I'm right N 2

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MADNESS!

• CAN N BE STOPPED????

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A new challenger
1
0
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My favorite game

• I'm right
• I'm right 2
• I'm right N

• I'm right 1
• . . .
• I'm right C

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The challenger steps up

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

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The challenger steps up

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

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The challenger steps up

• 1 3 7 4 8 0 7 3 7 9 2 6 . . .
• 8 4 6 0 4 3 4 4 5 7 8 9 . . .
• 1 2 3 4 5 6 7 8 9 0 0 0 . . .
• 6 9 2 6 7 2 8 9 7 8 9 0 . . .
• . . .
• What about .2 5 4 7 . . . ?

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The challenger steps up

• 1 3 7 4 8 0 7 3 7 9 2 6 . . .
• 8 4 6 0 4 3 4 4 5 7 8 9 . . .
• 1 2 3 4 5 6 7 8 9 0 0 0 . . .
• 6 9 2 6 7 2 8 9 7 8 9 0 . . .
• . . .
• What about .2 5 4 7 . . . ?
• It can't be anywhere in the list

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The challenger steps up

• 1 3 7 4 8 0 7 3 7 9 2 6 . . .
• 8 4 6 0 4 3 4 4 5 7 8 9 . . .
• 1 2 3 4 5 6 7 8 9 0 0 0 . . .
• 6 9 2 6 7 2 8 9 7 8 9 0 . . .
• . . .
• What about .2 5 4 7 . . . ?
• It can't be anywhere in the list
• The list can't exist

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A new champion!

• C N

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My favorite game

• I'm right
• I'm right 2
• I'm right N

• I'm right 1
• . . .
• I'm right C

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My favorite game

• I'm right
• I'm right 2
• I'm right N
• I'm right C 1
• I'm right 2 C
• I'm right C 2
• I'm right 2 C

• I'm right 1
• . . .
• I'm right C

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My favorite game never ends

• There are an infinity of infinities
• How many infinities? N, or C, or . . .
• Are there hidden infinities?
• Maybe you can discover one of your own