Re: Can an infinite string have a beginning and end but no middle?

On Tue, 9 Oct 2007 bradc355113@xxxxxxxxx wrote:

If we number the elements like this:

(e1, e3, e5, ..., e6, e4, e2)

Does this construct make sense?

Let be D be the positive odd integers and E the positive even numbers.
Your set is D + E where + is order union. The order <<=, for D + E is

a <<= b when a in D, b in E
or a,b in D, a <= b (This is the usual <= order by magnitude.)
or a,b in E, b <= a

Exercise.
<<= for D + E is a discrete bounded linear order
which is neither well ordered nor complete.

Two ordered sets (S,<=) (T,<<=) are order identical or order isomorphic when

some bijection f:S -> T with for all x,y, (x <= y iff f(x) <<= f(y))

Exercise. (D + E, <<=) is order isomorphic to
S = { -1/n, 1/n | n in N } which as a
subset of R, is given the same order as R.

Exercise. Construct another unusual ordered set.

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