What is the role of Dedekind cuts in constructing real numbers?

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In summary, Dedekind cuts are a way of constructing real numbers using sets of rational numbers that have certain properties. The real number is the boundary between these two sets, and while it may be in one of the sets, it is not in both. This approach is used to define real numbers in terms of structures involving rational numbers, and is a concept that is explored in the study of real analysis.
  • #1
Saph
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Hello,
I am teaching my self real analysis from Pugh's Real Mathematical Analysis, and I came acrosse the construction of reals through Dedekind cuts, but am confused about the following:

Dedekind cuts are defined as follows ( Pugh's page 11):
A cut in ##Q## is a pair of subsets ##A,B ## of ##\mathbb{Q}~## such that:
1) ##A\cup B = \mathbb {Q},~A\not=\phi,~B\not=\phi,~A\cap B=\phi.##
2) if ##a\in A## and ##b\in B## then ##a<b##.
3) ##A## contains no largest element.

Then few lines below he make the following definition:
A real number is a cut in ##\mathbb{Q}##.

What I don't understand is how a pair of sets ( which are infinite ) constitutes a real number.
I would like somebody to give me, if possible, a brief explanation about Dedekind cuts, too.
 
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All the points in A are less than all the points of B. The real number is the boundary between A and B, which is in A or B, but not both.
 
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  • #3
Saph said:
What I don't understand is how a pair of sets ( which are infinite ) constitutes a real number.

It is difficult to understand, but your interpretation is correct. The author intends to define "a real number" as a pair of sets of rational numbers that have certain properties. The "game" that is being played is to define real numbers in terms of structures involving the rational numbers.

Intuitively, you can think of this approach as establishing a real number like ##\sqrt{2}## as boundary between two sets of rational numbers ( those less than ##\sqrt{2}## and those greater), but you won't be able to follow the formal proofs unless you take the definition given for a real number literally - as a structure involving two sets of rational numbers.

The development of the real numbers via Dedekind cuts is a chapter in most books on real analysis, but it is a chapter that isn't used in later chapters. People negotiate the later chapters by thinking about real numbers the way they thought about them before they read about Dedekind cuts.
 
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After reading your replies, and looking through examples, things started to make sense, thank you very much :)
 
  • #5
Saph said:
Hello,
I am teaching my self real analysis from Pugh's Real Mathematical Analysis, and I came acrosse the construction of reals through Dedekind cuts, but am confused about the following:

Dedekind cuts are defined as follows ( Pugh's page 11):
A cut in ##Q## is a pair of subsets ##A,B ## of ##\mathbb{Q}~## such that:
1) ##A\cup B = \mathbb {Q},~A\not=\phi,~B\not=\phi,~A\cap B=\phi.##
2) if ##a\in A## and ##b\in B## then ##a<b##.
3) ##A## contains no largest element.

Then few lines below he make the following definition:
A real number is a cut in ##\mathbb{Q}##.

What I don't understand is how a pair of sets ( which are infinite ) constitutes a real number.
I would like somebody to give me, if possible, a brief explanation about Dedekind cuts, too.
Don't know if this helps but remember "infinity" is no specific number or formula..merely an idea or concept to help us deal with certain metaphysical/maths ideas..I.e. "the infinite" is just a very useful..idea.
 
  • #6
"All the points in A are less than all the points of B. The real number is the boundary between A and B, which is in A or B, but not both."

This is not correct for two separate reasons. A and B constitute a partition of the rational numbers ℚ into two sets, but condition 3) requires that A contain no largest element.

So A never contains the real number that the Dedekind cut represents, even if it is rational; in that case the represented number lies in B. But if the represented real number is irrational (for example, if A is all rationals less than √2 and B is all rationals greater than √2), then neither A nor B contains the represented number.
 
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  • #7
zinq said:
"All the points in A are less than all the points of B. The real number is the boundary between A and B, which is in A or B, but not both."

This is not correct for two separate reasons. A and B constitute a partition of the rational numbers ℚ into two sets, but condition 3) requires that A contain no largest element.

So A never contains the real number that the Dedekind cut represents, even if it is rational; in that case the represented number lies in B. But if the represented real number is irrational (for example, if A is all rationals less than √2 and B is all rationals greater than √2), then neither A nor B contains the represented number.
You are right. I should have said "may be in A or B, but not both". If the cut is a rational number, it will be in A or B. If it is irrational, neither.
 
  • #8
That it still wrong on two counts.

The real numbers are the cuts, which are pairs of sets. None of them appear in A or B for any cut.

The cut is a rational real number if it is the image of some ##q## in the set ##\mathbb{Q}## of rationals from which the reals are constructed under the mapping ##\phi(q)=\{A,B\}## and ##A=\{x\in\mathbb{Q}:x<q\}##, ##B=\{x\in\mathbb{Q}:x\geq q\}## (according to the definition).

Note that in this case ##q## (not the rational real image of ##q##) appears specifically in ##B##. We cannot accept both ##q\in A\wedge q\notin B## and ##q\in B\wedge q\notin A## as cuts, otherwise the system of reals constructed would have jumps at every rational real under its ordering ##C\leq C'## defined as ##A(C)\subseteq A(C')## (where ##A(X)## is the ##A## for cut ##X##).
 

1. What are Dedekind cuts and why are they important in mathematics?

Dedekind cuts are a mathematical concept introduced by German mathematician Richard Dedekind. They are used to define the real numbers in a rigorous and precise way. Dedekind cuts are important because they provide a foundation for understanding the real numbers and are used in many areas of mathematics, including analysis and algebra.

2. How do Dedekind cuts differ from other methods of defining the real numbers?

Dedekind cuts differ from other methods, such as Cauchy sequences, in that they do not rely on the concept of limit. Instead, Dedekind cuts define a real number as the division of the set of rational numbers into two non-empty subsets, known as the lower and upper sets. This approach avoids issues with infinite sequences and is considered to be more intuitive.

3. What is the purpose of using Dedekind cuts?

The main purpose of using Dedekind cuts is to provide a rigorous and axiomatic definition of the real numbers. This allows mathematicians to work with the real numbers in a precise and consistent manner, without relying on intuition or physical constructions. Dedekind cuts also have applications in other areas of mathematics, such as topology and measure theory.

4. Can Dedekind cuts be used to define other sets of numbers?

Yes, Dedekind cuts can be used to define other sets of numbers, such as the complex numbers and the p-adic numbers. However, the construction of these sets may differ slightly from the construction of the real numbers using Dedekind cuts. For example, in the case of the complex numbers, Dedekind cuts are used to define the field of quotients of the ring of Gaussian integers.

5. Are there any limitations or drawbacks to using Dedekind cuts?

One limitation of Dedekind cuts is that they can be difficult to visualize or conceptualize, especially for non-mathematicians. Additionally, some mathematicians may argue that other methods of defining the real numbers, such as Cauchy sequences or surreal numbers, may be more elegant or intuitive. However, Dedekind cuts remain a widely accepted and important tool in mathematics for defining the real numbers.

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