Jim Choate <ravage@EINSTEIN.ssz.com> writes:
One can construct the reals from the rationals quite easily using any of several well-known methods, such as equivalence classes of Cauchy sequences, or Dedikind cuts.
So you are saying that the Reals are a subset (ie can be constructed from) of the Rationals?
Yes, the Reals can be constructed from the Rationals. No, the Reals are not a subset of the Rationals. Fortunately, construction in mathematics is not simply limited to the taking of subsets. The Rationals can be constructed from the Integers, for instance, by multiplication and the taking of appropriate equivalence classes. In fact, everything can be built out of two sets and the axioms of Set Theory.
I can create a number which is not representable by the ratio of two integers from two numbers which are representable by ratios of two integers?
Er, no. But you can create a number which is not representable as a fraction as a limit point of very many fractions.
That's a nifty trick indeed, I am really impressed.
Thank-you. :)
Cauchy produced a test for testing convergence. I fail to see the relevance here, but please expound...
Cauchy sequences are useful for adding limit points to a set of things because their convergence criteria is very simple, and it is conceptually easy to take all Cauchy sequences whose elements come from a given set. If we then consider equivalence classes of those Cauchy sequences which converge to the same limit, and consider an element of the original set to correspond to the class containing the sequence all of whose members are that element, we can consider the classes to form a "completion" of the original set by addition of all its limit points. Similarly, the Reals are the completion of the Rationals.
Dedekind Cut:
"Thus a nested sequence of rational intervals give rise to a seperation of all rational numbers into three classes."
Just exactly where does this allow us to create Reals?
Finite ordered sets have maximum elements. Bounded infinite ordered sets have Least Upper Bounds, which may be a limit point as opposed to being an actual member of the set. Dedekind Cuts are a simple abstraction, often used to construct the Reals from the Rationals in undergraduate calculus courses. Conceptually, one makes a single "cut" in the set of Rationals, dividing it into two parts, all of the members of one part being greater than all of the members of the other. The number of ways of doing this correspond to the Reals. -- Mike Duvos $ PGP 2.6 Public Key available $ mpd@netcom.com $ via Finger. $