Infinity and Cardinality

Infinity and Cardinality - MATH 135 Fall 2008 To Infinity...

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Unformatted text preview: MATH 135 Fall 2008 To Infinity and Beyond Introduction The concept of “infinity” is a strange one and difficult to get a handle on. We see it often in calculus in the context of limits. We have seen it in MATH 135 (“There are infinitely many prime numbers”). Strange things can happen with infinity, though. Hilbert’s Hotel You own a hotel with lots of rooms. In fact, there is one room for each positive integer and the rooms are numbered this way. Suppose that the rooms are all full and then a new guest comes. Can you give her a room? What if 10 new guests come? What if one new guest comes for every positive integer? In fact, in each of these cases you can accommodate them. If 1 new guest comes, move the guest from room n to room n + 1 and put the new guest in room 1. If 10 new guests come, move the guest from room n to room n + 10 and put the new guests in rooms 1 to 10. If a new guest comes for every positive integer, surprisingly we can still do this. Here is one way to do this. Move guest n to room 2 n- 1. This means that all of the odd-numbered rooms are occupied and all of the even-numbered rooms are empty. Put new guest m in room 2 m . This accommodates all of the guests! Does this mean that we can write ∞ + 1 = ∞ and ∞ + 10 = ∞ and 2 ∞ = ∞ ? NO!!! Infinity is not a number, so you can’t use it in equations in this way. What does it mean to talk about a set of infinite size? Suppose that A is a set. The cardinality of A is the number of elements in the set A . We denote the cardinality of A by | A | . If a set A contains finitely many elements, finding the cardinality is pretty straightforward – just count them! (For example, if A = { 1 , 2 , 3 , 4 , 5 } then | A | = 5.) The set of integers Z has an infinite number of elements. We give a special name to this cardi- nality: we say that | Z | = ℵ . (We read this “aleph nought” since ℵ is the Hebrew letter aleph.) How can we tell if two sets are the same size?How can we tell if two sets are the same size?...
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Infinity and Cardinality - MATH 135 Fall 2008 To Infinity...

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