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Unformatted text preview: Cardinality The introduction of cardinal numbers by Georg Cantor in the 1870s made it possible to compare the size of infinite sets, and say that some are larger than others. The crucial definition for this is that two sets are the same size have the same cardinality if there is a one-to-one mapping of one onto the other. For finite sets this means that both have the same number of elements, but it has surprising consequences for infinite sets. Here is a one-to-one mapping of the integers onto the even integers: 1 2, 2 4, 3 6, ... , n 2 n , ... So, even though the even integers are only half of all the integers, the set of even integers has the same cardinality as the set of all integers. People did not like this, and on Wikipedia you can read criticism from some very famous 19th century mathematicians. However, Cantors definition has proven extremely useful, and is fundamental today. The cardinality of N is denoted by , read aleph naught. That is the first infinite cardinal, and one calls sets with this cardinality countable or more precisely...
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