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Re: Doubts about Dedekind cuts
Posted:
Apr 18, 2014 1:11 PM


"pedro" <pedropfeiffer@gmail.com> wrote in message news:9c4263ae23cc49ca80110dc3b459a38f@googlegroups.com... > 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 a+b.
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 Mike.
> > Thanks.



