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Topic: Doubts about Dedekind cuts
Replies: 5   Last Post: Apr 19, 2014 5:33 AM

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Mike Terry

Posts: 767
Registered: 12/6/04
Re: Doubts about Dedekind cuts
Posted: Apr 18, 2014 1:11 PM
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"pedro" <> wrote in message
> Can anyone help me with two doubts about Dedekind cuts? See if you can
understand my doubts.
> Here is the definition of a Dedekind cut: a Dedekind cut is defined to be

a set of rationals alpha such that it's different from the empty set and the
set of all rationals; and for all p and q in the rationals and p belonging
to alpha, if q<p then q belongs to alpha; and alpha doesn't have a maximum.
> Now there is a picture of a Dedekind cut; it can be proven that a set is a

Dedekind cut if and only if it's the set of all rationals smaller than a
real number (assuming that I already have a conception of real numbers).
This is one of my doubts. Is there no way out? In order to have this picture
of a Dedekind cut I must already understand what a real number is?

The picture is not part of the formal development that you are following.
It is intended only to provide a motivation for the definition and
subsequent definitions of operations. The thinking goes that once the
motivation is clear, the definitions should become mostly obvious, but from
a *formal* perspective you could ignore the motivation and just confirm that
all the subsequent facts that are proved about Dedekind cuts follow from the
definitions. Once sufficient of these facts (i.e. properties of Dedekind
cuts) are proved, you could then come to accept that they characterise what
you understand as the "real numbers".

> My second doubt is about the definitions of the operations on Dedekind

cuts. For exemple, addition of two Dedekind cuts alpha and beta is defined
to be the set {a+b : a in alpha and b in beta}. But how can I know this
really is addition? It initially looks as a kind of random definition. It
seems that in order to understand that this is addition I must already have
a conception of addition and prove that the set {a+b : a in alpha and b in
beta} is equal to the rationals smaller than the sum of alpha and beta. Here
is another kind of circular reasoning. To understand addition I must already
understand addition.

That's misleading because you're confusing too meanings of the word
"addition". What is true is that to understand addition of *Dededing cuts*
(aka "real numbers") you need to understand addition of *rational numbers*.
That's not circular reasoning. The author is taking the rationals as
previously given, and understood by the reader.

Given the motivation the author has given, the definition should not seem
random to you, but rather it should be seen as the obvious definition to
use. You ask how you can know the definition "really is" addition. This is
not a good starting question, because the author is giving a definition of
*new* operations PLUS and TIMES (or whatever the auther calls them). At
this point it strictly does not matter what PLUS "really is" - you can just
accept the definition (without committing to what the operation "really is")
and follow the proofs of its properties that follow. As you go along, you
will clearly want the proved facts to include things like following:

a) when we have two Dedekind cuts s, t corresponding to rationals a, b
respectively, then the cut PLUS(a,b) corresponds to the cut for the rational

b) similar result for TIMES, the authors new "multiplication" operation on
cuts (which is more fiddly to define, since there are some complications
dealing with negative numbers)

When you have proved these results, at least you will be able to say the new
operations agree with your understanding of addition and multiplication when
applied to cuts corresponding to the familiar rational numbers.

Eventually you will have confirmed that there is a set of Dedekind cuts,
equipped with operations PLUS and TIMES that obey all the rules normally
associated (and characterising) the real numbers with operations of addition
and multiplication. You will also have shown that within this set there is
a subset corresponding to the rational numbers, and that behaves in the
expected way regarding addition/multiplication of rational numbers.

However, it's hard work doing this with no motivation and all the
definitions seemingly "plucked out of the air", which is why the author
offers the underlying motivation for the definitions - then all the checking
of required properties becomes almost a rote exercise... (So the
"motivation" is not strictly necessary, but is given to make your work
easier, and keep you "on track" with the overall program! :)

Hope this helps

> Thanks.

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