Re: Courage?

For finite sets we can decide whether they have an equal number by comparison. If we match the fingers of the left hand with those in the right hand we say both have the same "number" of members.
This same approach can be done with infinite sets. If we can match the elements of one set with another we say both sets have the same "cardinal". By "match" we mean there is a 1:1 mapping of one set ONTO the other. For example the positive even integers have the same cardinal as the positive integers because the map n->2n is a 1:1 map from the positive integers onto the positive even integers.
Any set that has the same cardinal as the positive integers is called "countable" so clearly the positive even integers are countable. It is a little harder to see that the rationals are countable but they are. The reals are NOT countable however. Even the positive reals between 0 and 1 are not countable. The proof of this is to imagine they are countable by putting them, using their decimal expressions, in a countable list in which the first is the one associated with 1 the second with 2 etc. By then writing the number whose first digit differs from the first digit of the first number and whose second digit differs from the second digit of the second number number etc. we will have found a number not included in the list.
This contradiction proves the reals are not countable.
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