Dedekind Cuts

Date: 10/23/96 at 19:30:15
From: mat stern
Subject: Dedekind cut

I have already figured out Dedekind's theory of the rings and
number notation.  I still cannot figure out what his theory of
the Dedekind cut is.

Could you please send me a couple of sentences of what his cut is all
about--so that a 7th grader can understand?  I have contacted several 
mathamatic instructors and they know but cannot tell me in language 
that I can understand.

Date: 10/24/96 at 10:22:13
From: Doctor Jerry
Subject: Re: Dedekind cut

Dedekind invented the idea of a Dedekind cut to show how the real 
numbers can be constructed from the rational numbers.  The idea is 
that if you have the set of rational numbers (all numbers of the form 
p/q, where p and q are integers and q is not zero; the integers can be 
positive or negative), it ought to be possible to somehow create 
numbers like e and pi from this raw material.

For example, Dedekind said that it would be possible to think of the 
number sqrt(2), which is not a rational number, as two sets of 
rational numbers.  Let L (for left) be the set of all rational numbers 
whose squares are less than 2 and let R (for right) be the set of all 
rational numbers whose squares are bigger than 2.  Both L and R can be 
entirely defined and described within the rational number system.  We 
construct the number sqrt(2) by agreeing that sqrt(2) = {L,R}.  This 
new object, a pair of sets, is what we will say is sqrt(2).  

The entire set of reals can be constructed by taking all possible 
pairs of subsets {L,R} of the rational numbers, where L and R must 
satisfy certain conditions (for example, every member of L must be 
less than every member of R).

After creating the set of Dedekind cuts, one then defines how to add 
and multiply Dedekind cuts.  

There is a very nice book on this and related constructions.  It's 
called Foundations of Analysis, by Edmund Landau. I have a copy 
published in 1951 by Chelsea Publishing Company.

I see that I've written more than a couple of sentences.  I hope that 
I've helped you understand a little about Dedekind cuts.

-Doctor Jerry,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/   

Date: 11/25/96 at 15:23:57
From: Doctor Ceeks
Subject: Re: Dedekind cut

Hi,

There came a time in the history of mathematics where people became
very concerned that certain results were not properly proven.

Mathematics can take pride in the absolute correctness of its results
these days, but in the early days, many people made very loose 
arguments and "proved" some things which really make no sense.  Even 
Leonard Euler provided examples of such nonsense.

When people realized these loose arguments were becoming a real 
threat to the absolute correctness of mathematical theorems, some of 
them decided to try to develop mathematics very, very carefully.

This meant rebuilding the counting numbers, and then the integers, and
then the rational numbers very, very carefully to ensure that 
mathematics had a solid and secure foundation.

After constructing the rational numbers, the next step was to 
construct the real numbers, of which pi, e, and the square root of 2, 
are non-rational examples.  Have you ever wondered what exactly the 
real numbers are?  You may think it is obvious, but there were many 
things which early mathematicians thought were obvious and based their 
arguments upon, only later, to find out that what they thought was 
obvious, was not even true.  Examples of such things usually do not 
appear in high school mathematics because there, mathematical 
structures are simple enough that our intuition tends to guide us sure 
and true.

In any case, the process of making mathematics "rigorous" included
constructing in a systematic way, the real number system.

Dedekind devised a way of constructing the real numbers from the 
rationals by utilizing the "Dedekind Cut".  What Dedekind felt 
intuitively was that the set of rational numbers less than a given 
real number should determine that real number uniquely.

One fact which Dedekind wanted to be true about real numbers was that
between any two real numbers, there should be a rational number.  In
modern terminology, this means that Dedekind wanted the rational 
numbers to be "dense" amongst the real numbers.  This fact is 
consistent with Dedekind's idea in the previous paragraph, because 
assuming this fact, one can see that if X and Y are different real 
numbers, then the set of rationals less than X must differ from the 
set of rationals less than Y, since there must be a rational number in 
between X and Y.

Thus, Dedekind was led to make the following definition:

Let R be the set of subsets S of the rational numbers with the 
following property:

1. S has no largest element.

2. If the rational number q is in S, then every rational number less 
   than q must also be in S.

Dedekind hoped that he could make R into a model of the real number 
system.

To do this, he had to define a way of adding and multiplying elements 
of R.

He found that he could do this, and that the resulting addition and
multiplication satisfy all the laws we want the real numbers to 
satisfy.

He then declared that R, along with the addition and multiplication he
devised, be called the "Real Number System".

Now, mathematicians can rest assured that the real number system 
exists. But the point is, if a mathematician is trying to prove 
something and needs to use a fact about real numbers which s/he isn't 
sure about, then all s/he has to do is see if s/he can deduce that the 
fact holds for the system R defined above.  S/he doesn't have to make 
it a guessing game.

Please write back if you have questions about the above.

-Doctor Ceeks,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/