Yes, the natural numbers are well-ordered. But they're only a piece of it. Crucial to the idea of well-ordered sets is that for every two well ordered sets V and W, at least one of them has the same shape as a starting piece of the other one.
U = {0, 1, 2, 3} and V = {"first", "second", "third"}
are an example, if you call "first" 0, and "second" 1, and "third" 2 then V is just {0, 1, 2} with different labels, which is isomorphic to the first part of U under the ordering 0 < 1 < 2 < 3.
The first so-and-somany items of any (large enough) well ordered set will always have the same ordering as N, the set of natural numbers. But you can go further. For example
{0, 1, 2, 3, ... "something else"}
with ordering 0 < 1 < 2... as usual, and every natural number < the "something else" item is also a well-ordered set. Usually we call an element in the position of "something else" w (that's really an omega). You can have yet something else, usually called w+1, and so on. Then you can have
0, 1, 2, 3... w, w+1, w+2, ... 2*w, 2*w+1... 3*w
and soon you find yourself needing w^2 = w*w and w^3 and w^4 and... w^w by which time people usually go for a different letter ;-)
*Note* the "w" and "2*w" and "w^w" etc. are just names, you could just as well say "aardvark", "purple", "they" etc. What matters is the pattern of ordering. Those ... for example (look up "limit ordinal").
--regards, marijke
--regards, marijke
http://web.mat.bham.ac.uk/marijke/ |
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