<text><span class="style10">orrespondence, Counting and Infinity (5 of 6)</span><span class="style7"></span><span class="style10">Infinite sets</span><span class="style7">The sequence of sets, 1, 1,2, 1,2,3, etc., has the property that every </span><span class="style26">finite set</span><span class="style7"> (i.e. a set with a finite number of members) corresponds to one of them. Thus every set corresponds to a unique set of natural numbers starting with 1 and in their natural order, and the size of the set is given by the largest number in that unique set.However, a set that does not have a largest member cannot be equivalent to any set that does, and this leads to the idea of an </span><span class="style26">infinite set</span><span class="style7"> as one that cannot be counted. In particular, the set of all the natural numbers cannot be counted, so it might seem possible to define </span><span class="style26">infinity</span><span class="style7"> in terms of equivalence to the set of all the natural numbers.All we need to do to show that two sets are equivalent is to show that their members can be put in one-to-one correspondence. But this immediately gives rise to a puzzle. It is easy to show that the mappings that associate each natural number with its double (and so, each even natural number with its half) are one to one. Thus, we have shown there are exactly as many </span><span class="style26">even</span><span class="style7"> natural numbers as there are natural numbers. However, since every natural number is either even or odd and there are as many even numbers as odd, there must be twice as many natural numbers as even natural numbers.This is only a paradox if we expect that a subset of a set must be smaller than the set itself; but since our idea of an infinite set means that it is possible to list members of the set forever, we can see why adding extra members at the end does not increase its size. Our intuition that a set must have more members than any of its subsets is false for infinite sets. In fact, the usual definition of an infinite set is that it can be put into one-to-one correspondence with some subset of itself.</span></text>
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<text>ΓÇó NUMBER SYSTEMS AND ALGEBRAΓÇó SETS AND PARADOXESΓÇó FUNCTIONS, GRAPHS AND CHANGE</text>
