A Dedekind cut is a division of the rational numbers into two nonempty sets, A and B,
such that everything in A is less than everything in B, the union is all of Q,
and B does not contain its greatest lower bound.
The collection of Dedekind Cuts is then a construction of the real numbers.
Defining "addition" and "negation" is relatively straightforward, although
care must be exercised regarding subtraction, because if A does contain
its least upper bound, that point has to be transferred to B.
"Multiplication" and "multiplicative inverse" need more care, and are quite
messy in some cases.
An alternative formulation is this:
Version I
 A and C are two sets of rational numbers
 Everything in A is strictly less than everything in C
 $A\cup{C}$ is all of Q with the exception perhaps of a single point b.
 If b exists, it is larger than everything in A and smaller than everything in C.

Version II
 $Q=A\cup{B}\cup{C},\quad\text{with}\quadB\le{1}$
 $A<B,\quad~B<C,\quad~A<C$
 where X<Y means everything in X is less than everything in Y
 BEWARE: This is nontransitive! Consider the case when B is empty.

Although this seems a little messy, the intent is clear. We cut the rationals into two,
and if the cut is exactly at a rational then we leave it out.
With this formulation some of the technicalities are avoided, but others do arise.
As with many theorems and definitions in mathematics it is useful to have several,
equivalent forms, and then use the form that is most convenient.
Euclid and Hamilton both came close to the Dedekind Cut, but never used it as a definition
of the Real Numbers.
 Prove that the two versions given above are equivalent.
 For each version, define how to perform addition, negation, multiplication and reciprocal.
 Show that the collection of Dedekind cuts is isomorphic to the real numbers.
http://mathworld.wolfram.com/DedekindCut.html
http://en.wikipedia.org/wiki/Dedekind_cut
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