Re: Question about a minor recasting of Dedekind cuts.
- From: Douglas Eagleson <eaglesondouglas@xxxxxxxxx>
- Date: Wed, 25 Jul 2007 07:42:29 -0700
On Jul 20, 3:15 am, riderofgiraffes <mathforum.org...@xxxxxxxxxxxxxx>
wrote:
I've been explaining various constructions to a
maths club. I've constructed the integers as
equivalence classes of pairs of natural numbers,
where (a,b) == (c,d) if a+d=b+c, and I've
constructed the rationals similarly with ad=bc.
Now I'm constructing the reals in two ways.
Firstly I've used Cauchy sequences, equivalent
if their difference goes to zero. Easy enough.
With Cauchy sequences the limit doesn't have
to exist in the set from which the elements
are drawn - easy to prove.
Then I've used Dedekind cuts, but it's proving
really, really messy. I want to define the
negative, and subtraction, but having the limit
point (if it exists) in only one set means all
sorts of nasty conditions. Try it. Try to
define the quotient of two Dedekind cuts.
Why not simply say this:
A D-cut of the rationals is a pair of sets A,B
such that:
(using
u for union,
n for intersection,
e for element
)
a in A, b in B => a<=b
AuB = Q
(q e AnQ) & (a e A => a<=q) => q in B
(q e BnQ) & (b e B => q<=b) => q in A
Using shorter notation:
A <= B
AuB = Q
A<=q & q e A => q e B
q e B & q<=B => q e A
Now everything follows simply from the concept
that if A and B intersect then the intersection
is the rational they define, and if not, then
the irrational they define is "between" them.
Now all the arithmetic operations go through
without messy case analyses.
Thoughts?
Alot of what is likely written about the Reals appears coloquial?
A solution was to make all the appeared functional test. Meaning the
number was to cause each function class. A rational function therefor
differs from real number functions.
The short prose by the man Conway was to test the reader. DO you apply
the exact class as function?
A method of set size exists. A set of reals relative to rationals
MUST be critically understood.
ALL as infinite number was sized. And to make number relative without
anything other than sign direction relative to the current number is
the depth of formal theory.
A property of number called sign was used to size infinite sets. A set
negative relative to the example set was smaller!
SO the entire class of function appears assignable sign property. A
rational function is negative relative to the real functions for
example.
The implication is to follow the Conway texts to perform theory. All
real relative to rational defined which funct?
Relative sign was used profoundly.
Here is the definition of Socrate's class of function. Remind
yourself why the square root of two existed as a claim.
1. Aua
Abstract set A was in union with the element of A itself. Making the
theory of all Real in another fashion. Socrates simply counted
significant digits.
Now to understand the depth of Real Theory I will start a thread.
.
- References:
- Question about a minor recasting of Dedekind cuts.
- From: riderofgiraffes
- Question about a minor recasting of Dedekind cuts.
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