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26 - LECTURE 26 UNCOUNTABLE SETS(10.3 When you encounter...

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LECTURE 26: UNCOUNTABLE SETS ( § 10.3) When you encounter difficulties and contradictions, do not try to break them, but bend them with gentleness and time. Saint Francis de Sales (1567 - 1622) 1. Uncountable sets One way to think about uncountable sets is the following. Imagine a hotel with rooms numbered 1 , 2 , 3 , 4 , . . . . We can fit all sorts of sets into this hotel. We can even fit Q . An uncountable set is so big, that if we try to fit it into the hotel, there is just no way to do it. Or, in other words, if we fill each room, there will always be something from our set left over. (Hilbert’s hotel.) Theorem 1. If r 1 , r 2 , r 3 , . . . is a countable list of elements from (0 , 1) , then there is some element s (0 , 1) not in the list. Proof. We will write these real numbers using infinite decimal expansions. We will avoid duplicates by avoiding decimal expansions that end in repeating 9’s. So we have r 1 = 0 .r 11 r 12 r 13 r 14 . . . r 2 = 0 .r 21 r 22 r 23 r 24 . . . r 3 = 0 .r 31 r 32 r 33 r 34 . . .
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