Minor question about solution to Dedekind-Cut proof

[s3a]

Homework Statement

Problem:
Dedekind defined a cut, section or partition in the rational number system as a separation of all rational numbers into two classes or sets called L (the left-hand class) and R (the right-hand class) having the following properties.:

I. The classes are non-empty (i.e. at least one number belonds to each class).
II. Every rational number is in one class or the other.
III. Every number in L is less than every number in R.

Prove each of the following statements.:

(a) There cannot be a largest number in L and a smallest number in R.

(b) It is possible for L to have a largest number and for R to have no smallest number. What type of number does the cut define in this case?

(c) It is possible for L to have no largest number and for R to have a smallest number. What type of number does the cut define in this case?

(d) It is possible for L to have no largest number and for R to have no smallest number. What type of number does the cut define in this case?

(Textbook's) Solution:
(a) Let a be the largest rational number in L and b the smallest rational number in R. Then either a = b or a < b. We cannot have a = b, since, by definition of the cut, every number in L is less than every number in R. We cannot have a < b, since by [another problem in the textbook], ½ (a + b) is a rational number which would be greater than a (and so would have to be in R) but less than b (and so would have to be in L), and, by definition, a rational number cannot belong to both L and R.

(b) As an indication of the possibility, let L contain the number 2/3 and all rational numbers less than 2/3, while R contains all rational numbers greater than 2/3. In this case, the cut defines the rational number 2/3. A similar argument replacing 2/3 by any other rational number shows that in such a case the cut defines a rational number.

(c) As an indication of the possibility, let L contain all rational numbers less than 2/3, while R contains all rational numbers greater than 2/3. This cut also defines the rational number 2/3. A similar argument shows that this cut always defines a rational number.

(d) As an indication of the possibility, let L consist of all negative rational numbers and all positive rational numbers whose squares are less than 2, while R consists of all positive numbers whose squares are greater than 2. We can show that if a is any number of the L class, there is always a larger number of the L class, while if b is any number of the R class, there is always a smaller number of the R class. A cut of this type defines an irrational number.

From parts (b), (c) and (d), it follows that every cut in the rational number system, called a Dedekind cut, defines either a rational or an irrational number.

By use of Dedekind cuts, we can define operations (addition, multiplication, etc.) with irrational numbers.

Just in case I made a mistake when typing the problem and its solution, I am attaching the TheProblemAndSolution.jpg file.

Homework Equations

Definition of Dedekind cut.

The Attempt at a Solution

Based on what the question asked, I just wanted to know if, for part (c), it should have said that R contains all rational numbers greater than or equal to 2/3 instead of just greater than 3/2?

Or is there something I'm not grasping?

 

[pbuk]

Based on what the question asked, I just wanted to know if, for part (c), it should have said that R contains all rational numbers greater than or equal to 2/3 instead of just greater than 3/2?

Yes. The argument in (c) should be an exact mirror of (b).

 

[verty]

I read Dedekind's paper a long time ago, I can't find it now but IIRC he said, if there is a smallest rational ##r## not in ##L##, this places the cut at ##r##, but whether ##r## is placed in ##R## or ##L## is irrelevant, it is the same cut. He was using a geometric intuition where the cut is at ##r## and ##r## is just a point without width, the cut would be the same regardless.

 

[HallsofIvy]

The requirement that "II. Every rational number is in one class or the other" assures us that is "2/3" is not in L then it must be in R and that if "2/3" is not in R it must be in L.

If I remember correctly Dedekind's actual definition required that "R has no smallest member".

 

[s3a]

Thanks for the replies.

I have no difficulty with what was said, but I now have another issue.

For parts (a), (b) and (c), I'm not confused, because the number that's splitting the one set to two sets is rational, but what if that is not the case, as in part (d)?

Am I supposed to assume that sqrt(2) is irrational (which I know from another proof – so would I just say “by another proof” or something of the sort?) to form sets L and R, or am I supposed to prove that sqrt(2) is irrational by “defining it”?

If I'm supposed to prove that sqrt(2) is irrational, then that means I can't assume it's irrational, so how would I justify that it's excluded from both L and R?

I can't just say “it's irrational because it's not included in either L or R”, because it would have been excluded from L and R because I assumed that it's irrational.

 

[pbuk]

The requirement that "II. Every rational number is in one class or the other" assures us that is "2/3" is not in L then it must be in R and that if "2/3" is not in R it must be in L.

If I remember correctly Dedekind's actual definition required that "R has no smallest member".

No, he dealt with both alternatives simultaneously.

... it was pointed out that every rational number a effects a separation of the system R into two classes such that every number a₁ of the first class A₁ is less than every number a₂ of the second class A₂; the number a is either the greatest number of the class A₁ or the least number of the class A₂ ... every rational number produces one cut or, strictly speaking, two cuts, which, however, we shall not look upon as essentially different; this cut possesses, besides, the property that either among the members of the first class there exists a greatest or among the numbers of the second class a least number.

Most people find this unnecessarily confusing and so subsequent distillations of his work just pick one alternative and stick to it.

 

[s3a]

[. . .]

No, he dealt with both alternatives simultaneously.

[. . .]

Most people find this unnecessarily confusing and so subsequent distillations of his work just pick one alternative and stick to it.

That's interesting to know (and I'm not being sarcastic or anything of the sort).


However, I'm still stuck on what I said in my last post (about part d), and I would appreciate any help with that.

 

[pbuk]

However, I'm still stuck on what I said in my last post (about part d), and I would appreciate any help with that.

This question is not asking you to prove the existence of irrational numbers, it is simply asking you to prove that "It is possible for L to have no largest number and for R to have no smallest number", and by choosing to produce the cut from an irrational number you have shown that this is possible.

 

[s3a]

This question is not asking you to prove the existence of irrational numbers, it is simply asking you to prove that "It is possible for L to have no largest number and for R to have no smallest number", and by choosing to produce the cut from an irrational number you have shown that this is possible.

Okay, so then I do assume that sqrt(2) is irrational (from the very beginning of the proof), right?

If so, then what is meant by saying that the cut “defines” the irrational number sqrt(2)?

 

[verty]

Okay, so then I do assume that sqrt(2) is irrational (from the very beginning of the proof), right?

If so, then what is meant by saying that the cut “defines” the irrational number sqrt(2)?

You call the cuts real numbers and then investigate the "real number" sqrt(2). You ask the question, is sqrt(2) rational, and you answer, nope, the cut sqrt(2) is not rational. I mean, the cuts become the real numbers, there is no difference between them.

 

[pbuk]

Okay, so then I do assume that sqrt(2) is irrational (from the very beginning of the proof), right?

I would rather say that you should state that it has been proven that sqrt(2) is irrational.

If so, then what is meant by saying that the cut “defines” the irrational number sqrt(2)?

If the word "defines" bothers you, you can say it "corresponds to" ...

 

[s3a]

Sorry, I double-posted.

 

[s3a]

MrAnchovy, what you said seems clear to me. :)

verty, so, basically you're implying that the set of all real numbers can be defined as the collection of all Dedekind cuts, right?

 

[verty]

MrAnchovy, what you said seems clear to me. :)

verty, so, basically you're implying that the set of all real numbers can be defined as the collection of all Dedekind cuts, right?

This is a good question. I want to give you an accurate answer, so this may be a long post.

#1) What does define mean? To me it means, pick out or make noticeable. For example, "that jacket really defines him" means you would notice him in a crowd because of the jacket's effect. Also, we often hear about defining moments, the moments in which it all became clear or in which the entity became noticeable. So if we think in this way, cuts can be said to define the real numbers in the sense that cuts are things one can think about or intuit, they are objects. In thinking about cuts and what properties they have, we understand the real numbers better. So they have this defining role, we can say that they define the real numbers. So this gives "the cuts define the real numbers", but I want something in the other direction.

#2) A definition is a description or way to interpret a new name or symbol. What do we do if we define the real numbers? There is a close link to #1 in the sense that we might use objects like cuts to aid understanding of the real numbers. That would be defining them by #1. That gives, "we define the real numbers by cuts". But this probably not canonical. Another meaning is, we describe the real numbers. Whatever the real numbers are, we describe them, we explain how they behave. So a definition in this sense is not an object but a list of properties or rules that real numbers possesses or obey. In this sense, a definition is a specification; we can construct objects to match the specification. Anything that matches the specification is called a model, so cuts are a model of the real numbers.

But then, what are the real numbers and how do cuts model them? In specifying how real numbers behave, we must have something in mind that is a real number. I mean, something must behave that way for there to be real numbers. But mathematicians take this approach, they define a term and then prove existence, that there is a model. If there is a model of the definition, the definition is consistent, the term has been well-defined and that is that. Somehow, it doesn't matter what real numbers are as long as some things are real numbers.

#3) By that interpretation, 1+1=2 means: in any model of the real numbers, 1+1=2. But if there are models of the real numbers, there must be canonical real numbers that the models are models of, I think. And some books define a process called "completion of a field" which is closely related to cuts. Given a field, the completion of the field is the field of cuts. At least, this is how I've seen it defined. They will then say, the real number field is the completion of the rational number field. Now there are canonical real numbers, cuts of the rationals. So this blurs things: on the one hand, we define the real numbers as a complete analogue of the rationals, but there are canonical objects that we have in mind: cuts of rationals. But one can also look at this in a purely specificatory sense: if we are given a model of the rationals, completing it will give a model of the reals.

#4) I've just thought of another option, this language: we construct the real numbers as cuts. Now the real numbers are templates or ideal forms, and we construct or instantiate them to be become actual. Then we can't say, "we define the real numbers as cuts"; we can say "we define the real numbers by cuts", this now very clear. But we can say, "we define the real numbers by this specification and prove there is a model of the real numbers", this is now understandable.

In conclusion. "We define the real numbers as cuts" is probably not correct. There is a theory of meaning called denotational semantics where meanings are understood to be objects, so this would make sense according to that theory. But it is a very old idea and people don't speak that way. "We define the real numbers by cuts" works for two reasons, because cuts demonstrate them and using cuts as a template works. Or we can say, "we define them as the complete analogue of the rationals", understanding that this means, behaving like cuts of rationals. Or, we can say "we define the real numbers by these axioms, and prove by constructing them via cuts that they are well defined". Many books do this. Perhaps it is the best thing to say.