Constructing the real number system by Dedekind cuts?

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Constructing the "real number system" by Dedekind cuts?

Hi all,

I've just been working through baby Rudin, and I've almost finished chapter 1. I am currently working through the appendix of chapter 1, which constructs the real number system from the rational number system by Dedekind cuts.

Here is my problem: the elements of the set of "real numbers, R" defined look nothing like a real number!

The elements of R are defined as cuts which are subsets of Q. My question is "How can a set of rational numbers be a real number?"

For instance, I know that 5 is a real number, so where is the "5" in that set R? The R defined in there has other sets as elements - not numbers.
 

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  • #2
mathwonk
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well this a standard trick in math, to forget we know about some gadget and then define that gadget in terms of some other gDGETS. of course we do niot get thge usual version, but we get some equivqlent thing that has all the same features as what we want. e.g. if you only know about integers yiou can define rational numbers as pairs of integers (p,q) where q ≠0, with an equivalence relation, i.e. (p,q) ≈ (r,s) iff ps = rq. etc......
 
  • #3
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well this a standard trick in math, to forget we know about some gadget and then define that gadget in terms of some other gDGETS. of course we do niot get thge usual version, but we get some equivqlent thing that has all the same features as what we want. e.g. if you only know about integers yiou can define rational numbers as pairs of integers (p,q) where q ≠0, with an equivalence relation, i.e. (p,q) ≈ (r,s) iff ps = rq. etc......
I understand your point about the "tricks". I saw something similar pertaining to definition of complex numbers (i.e. (a,b) instead of a + bi).

However, my problem is as follows: suppose I choose a real number (say 5), then which "cut" in R represents that number 5?
 
  • #4
mathman
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I understand your point about the "tricks". I saw something similar pertaining to definition of complex numbers (i.e. (a,b) instead of a + bi).

However, my problem is as follows: suppose I choose a real number (say 5), then which "cut" in R represents that number 5?
The real number 5 is given by all rational numbers < 5. The rational number 5 may be in the set as well.
 
  • #5
mathwonk
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a basic principle in euclidean geometry is that specifying a point, we can separate the line into two disjoint intervals, one open = the points to the left of that point, and one having an endpoint = the points to the right of and including that point. the converse is an axiom of "completeness", i.e. given any separation of the line into two disjoint intervals, the left one being open, then the right interval must have an endpoint.

we consider such a separation as equivalent to defining the unique endpoint. moreover the endpoint is also determined just by the equivalent separation of the rational points of the line.

thus given say only the rational points, we do the same thing. i.e. we define a real number as the separation of the rational points that is associated to that real number. so the real number 5 is defined as the pair of intervals (-infinity, 5) and [5, infinity), where the intervals only include rational points.

thus an irrational point is defined by a pair of rational intervals where neither interval has an endpoint.

and we really only need one of the two intervals, so...


i.e. every real number R is equivalent to the rational interval of all rationals less than R. or less than or equal if you prefer.
 
  • #6
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Oh, all right. Now I get it. Thank you very much.

It wasn't explained that explicitly in the book; it only proved that the real number system, so defined, is an ordered field with the least upper bound property, and that Q can be regarded as a subfield.
 
  • #7
mathwonk
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well, nothing is explained in that book. it has good information, but the explanations are almost non existent. i really dislike that book - in fact as books go it is probably the worst "good" book out there. i.e. from a teacher's perspective it is excellent, but from a learner's point of view it is terrible.
 
  • #8
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well, nothing is explained in that book. it has good information, but the explanations are almost non existent. i really dislike that book - in fact as books go it is probably the worst "good" book out there. i.e. from a teacher's perspective it is excellent, but from a learner's point of view it is terrible.
Yeah, I've heard such things before. Nevertheless, I still wanted to give it a shot since a few schools use this book for a first course in analysis. I'm planning to fill in the gaps in my knowledge by using Pugh or Apostol after finishing Rudin.
 
  • #9
mathwonk
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i see no reason at all to use that book. even if some schools use it, you are just making your life harder by using it when you are not forced to do so. much better to use pugh or something else helpful first, and then if you are ever forced to use this one you will be in the same position as the instructor, namely you will already know the material. what really is the point of writing a book that makes no effort at all to teach? much less reason to read such a book.
 
  • #10
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Yeah, I've heard such things before. Nevertheless, I still wanted to give it a shot since a few schools use this book for a first course in analysis. I'm planning to fill in the gaps in my knowledge by using Pugh or Apostol after finishing Rudin.
That's a bit backwards, in my opinion. It's a lot easier to read awesome books like Pugh first, and then read Rudin to fill in the gaps in your knowledge (if there are any).

Pugh is not an easy book, the exercises can be very challenging. Same for other good books. You will miss out on basically nothing. If you read Rudin, you'll miss out on a lot of intuitive and conceptual understanding (which is what math is about in the end).
 
  • #11
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i see no reason at all to use that book. even if some schools use it, you are just making your life harder by using it when you are not forced to do so. much better to use pugh or something else helpful first, and then if you are ever forced to use this one you will be in the same position as the instructor, namely you will already know the material. what really is the point of writing a book that makes no effort at all to teach? much less reason to read such a book.
Yeah I know, but my reason is that I'm more motivated to learn if the material isn't too easy. Anyway, I fully understand your point: if I run into another difficulty again while using this book, I'll order Pugh right away (which is highly probable).
 
  • #12
mathwonk
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suit yourself, but the problem is that there is no insight to be gained from rudin. that's why you are here asking us. he just presents the facts, with no understanding of why it is that way, so even if you do manage to plow through it you will probably not understand it well.

i.e. he is making it both harder and less helpful. even if you make it through his book, you still do not have the insight that should accompany the topic.
 

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