Dedekind Cuts Question (Conceptual)

  • Thread starter mindarson
  • Start date
  • Tags
    Conceptual
In summary, the conversation discusses the use of Dedekind cuts to define the real numbers in terms of the rationals. The question is raised regarding how to construct the cut that corresponds to an irrational number, such as sqrt(2), without already having knowledge of the number. It is explained that the construction involves considering all limits in the rationals and taking all cuts where the rationals are smaller than the number when squared. The circular reasoning is addressed and it is clarified that the construction only involves rationals and their subsets, not irrational numbers.
  • #1
mindarson
64
0
Hi,

I have just begun a self-study program in analysis and have a question about Dedekind cuts.

My question is this: My understanding is that cuts are constructions by which to define the reals in terms of the rationals. I.e. the reals are a set of cuts, each of which is a set of rationals. And a cut is the set of rationals to the left of (or less than?) a certain point.

So my real question is: If we must use cuts to construct the reals, how do we know WHERE to cut in order to construct the cut that corresponds to an irrational, real number like sqrt(2)? How is this reasoning not circular? If we DO NOT HAVE the number yet, then how can we even talk about objects TO THE LEFT of the number (or oriented in any way with respect to it)?

Thanks for taking the time to read and consider.
 
Mathematics news on Phys.org
  • #2
What you actually do is consider all limits in the rationals. For instance to construct the square root of 2 just take all cuts such that all rationals in it are smaller than 2 when squared. Here no irrational number is used in any way just rationals and being able to square.

There can be no knowing where to cut, because there is in this construction nowhere to cut. You construct the reals as a set of subsets if the rationals namely all the subsets where is a rational is included also all rationals to the left of it are included.
 

What are Dedekind cuts and how are they used in mathematics?

Dedekind cuts are a mathematical concept introduced by German mathematician Richard Dedekind. They are used to define irrational numbers by dividing the set of rational numbers into two non-empty subsets. One subset contains all the rational numbers less than the irrational number and the other contains all the rational numbers greater than the irrational number. This allows us to define irrational numbers without using decimal expansions.

What is the significance of Dedekind cuts in the construction of the real numbers?

Dedekind cuts play a crucial role in the construction of the real numbers. They provide a rigorous and logical way to define irrational numbers, which cannot be expressed as a ratio of two integers. By using Dedekind cuts, we can define the real numbers as a complete ordered field, which means that every non-empty set of real numbers has a least upper bound and greatest lower bound.

How are Dedekind cuts related to the concept of continuity?

In calculus, continuity is a property of a function that implies that small changes in the input result in small changes in the output. Dedekind cuts are closely related to this concept because they allow us to define continuity in terms of sets. A function is continuous at a point if and only if the corresponding Dedekind cut is continuous, meaning that the cut separates the rational numbers into two sets that approach each other as the input values approach the point.

Can Dedekind cuts be used to define any irrational number?

Yes, Dedekind cuts can be used to define any irrational number, including algebraic and transcendental numbers. By using Dedekind cuts, we can construct the real numbers as a complete ordered field, which includes all irrational numbers. This means that we can define any irrational number as a Dedekind cut of the rational numbers.

Are there any limitations to the use of Dedekind cuts in mathematics?

While Dedekind cuts are a powerful tool in mathematics, there are some limitations to their use. One limitation is that they require a rigorous understanding of set theory and mathematical logic, which can be difficult for some individuals. Additionally, Dedekind cuts do not work well for defining certain types of numbers, such as complex numbers or infinitesimal numbers.

Similar threads

Replies
8
Views
1K
  • STEM Educators and Teaching
Replies
5
Views
2K
Replies
4
Views
523
Replies
6
Views
1K
  • Calculus and Beyond Homework Help
Replies
1
Views
674
  • General Math
Replies
5
Views
1K
Replies
12
Views
1K
  • Set Theory, Logic, Probability, Statistics
Replies
7
Views
2K
  • Classical Physics
3
Replies
85
Views
4K
  • Topology and Analysis
Replies
3
Views
1K
Back
Top