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Transcript 
How many infinities are there?
A
none
B
one
C
more than one
D
infinity
Georg Cantor
500 BC
0
Aristotle: 2
Surya Prajnapti: 7
500
1000
1500
Descartes: 1
2000
How many infinities are there?
A
none
B
one
C
more than one
D
infinity
Georg Cantor
500 BC
0
Aristotle: 2
Surya Prajnapti: 7
500
1000
1500
Descartes: 1
2000
How many infinities are there?
A
none
B
one
C
more than one
D
infinity
Georg Cantor
500 BC
0
Aristotle: 2
Surya Prajnapti: 7
500
1000
1500
Descartes: 1
2000
Infinity
≈ Number
Number ≈ Property of a set
Assumption 1. Infinite sets exist
How do you count an infinite set?
When are two sets equal in size?
Assumption 2. Two sets are equal in size if
all elements can be paired together
Infinity
≈ Number
Number ≈ Property of a set
Assumption 1. Infinite sets exist
How do you count an infinite set?
When are two sets equal in size?
Assumption 2. Two sets are equal in size if
all elements can be paired together
Infinity
≈ Number
Number ≈ Property of a set
Assumption 1. Infinite sets exist
How do you count an infinite set?
When are two sets equal in size?
Assumption 2. Two sets are equal in size if
all elements can be paired together
Infinity
≈ Number
Number ≈ Property of a set
Assumption 1. Infinite sets exist
How do you count an infinite set?
When are two sets equal in size?
Assumption 2. Two sets are equal in size if
all elements can be paired together
Infinity
≈ Number
Number ≈ Property of a set
Assumption 1. Infinite sets exist
How do you count an infinite set?
When are two sets equal in size?
Assumption 2. Two sets are equal in size if
all elements can be paired together
Infinity
≈ Number
Number ≈ Property of a set
Assumption 1. Infinite sets exist
How do you count an infinite set?
When are two sets equal in size?
Assumption 2. Two sets are equal in size if
all elements can be paired together
Infinity
≈ Number
Number ≈ Property of a set
Assumption 1. Infinite sets exist
How do you count an infinite set?
When are two sets equal in size?
Assumption 2. Two sets are equal in size if
all elements can be paired together
HOTEL
HOTEL
HOTEL
Infinity
≈ Number
Number ≈ Property of a set
Assumption 1. Infinite sets exist
How do you count an infinite set?
When are two sets equal in size?
Assumption 2. Two sets are equal in size if
all elements can be paired together
Assumption 1. Infinite sets exist
Assumption 2. Two sets are equal in size if
all elements can be paired together
An infinite number:
size of set of natural numbers (1, 2, ... )
Assumption 1. Infinite sets exist
Assumption 2. Two sets are equal in size if
all elements can be paired together
An infinite number:
size of set of natural numbers (1, 2, ... )
Assumption 1. Infinite sets exist
Assumption 2. Two sets are equal in size if
all elements can be paired together
An infinite number:
size of set of natural numbers (1, 2, ... )
Assumption 1. Infinite sets exist
Assumption 2. Two sets are equal in size if
all elements can be paired together
An infinite number:
size of set of natural numbers (1, 2, ... )
“Aleph-zero”
HOTEL
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guests
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guests
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HOTEL
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guests
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HOTEL
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guests
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HOTEL
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guests
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1
More weird facts
but how big exactly?
undecidable!
true for any
(Cantor)
there are always bigger infinities!
The set of all infinities does not exist (Russell)
there are more than infinite infinities!
More weird facts
but how big exactly?
undecidable!
true for any
(Cantor)
there are always bigger infinities!
The set of all infinities does not exist (Russell)
there are more than infinite infinities!
More weird facts
but how big exactly?
undecidable!
true for any
(Cantor)
there are always bigger infinities!
The set of all infinities does not exist (Russell)
there are more than infinite infinities!
More weird facts
but how big exactly?
undecidable!
true for any
(Cantor)
there are always bigger infinities!
The set of all infinities does not exist (Russell)
there are more than infinite infinities!
More weird facts
but how big exactly?
undecidable!
true for any
(Cantor)
there are always bigger infinities!
The set of all infinities does not exist (Russell)
there are more than infinite infinities!
More weird facts
but how big exactly?
undecidable!
true for any
(Cantor)
there are always bigger infinities!
The set of all infinities does not exist (Russell)
there are more than infinite infinities!
Georg Cantor
(1845-1918)
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