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Quiz #13 (Last Quiz!)
Consider the set of all perfect squares {1, 4, 9, 16, ...} and the set of all odd integers {..., ­3, ­1, 1, 3, ...}. If you pair up all the elements of these two sets, what odd number gets paired with the perfect square 144? And what perfect square gets paired with the odd number 17?
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2.5.2.notebook
November 28, 2016
2.5 Infinite Sets
Geometric Comparisons
Compare the set of points on the line segment 1 unit long with the set of points on the line segment 2 units long.
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Georg Cantor vs Leopold Kronecker
(late 1800's, about 140 years ago)
"scientific charlatan"
"renegade"
"corrupter of youth"
David Hilbert:
1900 International Congress of Mathematicians
Hilbert's 23 Problems
"No one shall expel us from the
Paradise that Cantor has created."
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So, isn't it just the case that all infinite sets are equivalent?
cantor's "dıagonal argument"
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Hilbert's Hotel
The hotel has an infinite number of rooms, but every room has been reserved.
a) A guest shows up without a reservation. A room can be found for them!
b) 10 guests show up without reservations. Can rooms be found for them?
c) A bus drives up with an infinte number of guests. Can rooms be found for them?
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