Lecture 4

Lecture 4 - Lecture 4 Infinite Cardinals Some Philosophy:...

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Lecture 4 Infinite Cardinals
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Some Philosophy: What is “2”? Definition 1: 2 = 1+1. This actually needs the definition of “1” and the definition of the “+” operation. Definition 2: Start with the concept of “two apples”, and remove all aspects of a single apple, e.g. redness, taste, etc. . You’ll be left with the number “2”. This definition is a bit problematic. Definition 3: 2 = The class of all sets of size 2 (this is indeed a very large class) Definition 4: 2 = {0,1}, where 1 = {0} and 0 = {}. Note: 2 is a particular set of size 2.
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Some History: What is “n”? Historically, people could not count beyond some (relatively small) finite number, e.g. 10. A (large) number “n” did not have a name, but people could access it by having a bag with n stones. If a shepherd wants to make sure the number of sheep was n, he matches the sheep with the stones. Thus, two sets have the same size if there is a bijection (one-to-one correspondence) between the elements of the sets. .
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Some Definitions The size of a set A is less than or equal to that of B, written A B iff there is an injective (one-to-one) function f:A B. The size of a set A equals the size of the set B,
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This note was uploaded on 11/11/2011 for the course MATH 112 taught by Professor Jarvis during the Winter '08 term at BYU.

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Lecture 4 - Lecture 4 Infinite Cardinals Some Philosophy:...

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