Cantors Diagonal Proof Cantor introduced the idea of sets. A set was, for Cantor, a purely intuitive concept, not formally defined. A set is a kind of collection of objects that we can imagine intuitively in our minds. We can speak, for instance, of the set of all red things. Or a set can contain other sets, such as the set of all sets of coloured things. . However, Cantor asked us to consider that there could be larger numbers than infinity. These numbers would, of course, also be infinite, but would be larger infinities than the standard infinity, , that we are used to. Cantor called these numbers "transfinite" numbers. After , we get ( + 2), and so on. So what is the cardinality, or size, of ( . But, in fact, I can place all the elements of the first set into a one-to-one correspondence with the second, like so:
Cantor defined the size of his sets so that any two sets that could be matched up one-to-one like this were considered to be the same size. The same holds true for any finite number of new elements I insert into the set. Note that it does not really matter that I placed the new element at the beginning, since the one-to-one match-up would still hold no matter where I inserted the new element. So it really makes no sense to say that one of these sets is larger than the other. In other words, + 1, and we do not seem to have created a larger number with | |
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