Understanding Dedekind's Ketten: A Brief Explanation

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In summary, Dedekind's notion of a Kette is a subset of a set that remains unchanged when mapped onto itself by a given function. It is not necessary for the subset to be equal to the function's output. For example, with the function f(n)=2n, the subset of even numbers is a kette, while the subset of odd numbers is not.
  • #1
Stoney Pete
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Hi everybody,

I am struggling to precisely understand Dedekind's notion of a Kette. Perhaps you can help me.

I know a Kette has to do with how certain functions from N to N map N onto proper subsets of itself. Thus e.g. f(n)=2n maps N onto the set of the even numbers. Now my intuition is that a Kette for Dedekind is the infinite set of such subsets that result from recursive application of the function. So if we have f(n)=2n and recursively apply it to its own output, we get the following sets:

{2, 4, 6, 8,...}
{4, 8, 12, 16...}
{8, 16, 24, 32,...}
{16, 32, 48, 64,...}
Etc.

The Kette belonging to f(n)=2n would then be the set of all those subsets of N. Is this correct? Thanks for your answers.
 
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  • #2
Stoney Pete said:
Hi everybody,

I am struggling to precisely understand Dedekind's notion of a Kette. Perhaps you can help me.

I know a Kette has to do with how certain functions from N to N map N onto proper subsets of itself. Thus e.g. f(n)=2n maps N onto the set of the even numbers. Now my intuition is that a Kette for Dedekind is the infinite set of such subsets that result from recursive application of the function. So if we have f(n)=2n and recursively apply it to its own output, we get the following sets:

{2, 4, 6, 8,...}
{4, 8, 12, 16...}
{8, 16, 24, 32,...}
{16, 32, 48, 64,...}
Etc.

The Kette belonging to f(n)=2n would then be the set of all those subsets of N. Is this correct? Thanks for your answers.
In Was sind und was sollen die Zahlen? Dedekind defines a Kette as follows (page 12):
If ##\phi: S \to S## is a function, then a subset ##K\subseteq S## is a kette if ##\phi(K) \subseteq K##. Dedekind uses the notation ##K'## for ##\phi(K)##.
Today we probably would call that a subset that is invariant under the function ##\phi##.
 
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  • #3
Samy, does Dedekind also mean that K=ϕ(S)?
 
  • #4
Stoney Pete said:
Samy, does Dedekind also mean that K=ϕ(S)?
No, K can be any subset of S. It will be a kette if it is mapped to itself by ϕ. He doesn't even assume that for a kette K, K=ϕ(K). All that is needed is that ϕ(K) ⊆ K.

To take your example, ##\phi: \mathbb N \to \mathbb N: n \mapsto 2n##.
The subset of even numbers is a kette, the subset of all multiples of 7 is a kette, ...
The subset of odd numbers is not a kette.
 
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Likes ProfuselyQuarky
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o.k. thanks for your answer.
 

1. What is Dedekind's concept of Ketten?

Dedekind's Ketten, also known as Dedekind's Chains, is a mathematical concept introduced by German mathematician Richard Dedekind. It refers to a totally ordered set, where every element has a predecessor and a successor. This concept is important in the study of number systems, particularly in the development of the real numbers.

2. How is Dedekind's Ketten different from Cantor's notion of sets?

Dedekind's Ketten is different from Cantor's notion of sets in that it focuses on the ordering of elements within a set, rather than the elements themselves. In Dedekind's Ketten, the elements are arranged in a specific order, whereas in Cantor's sets, the elements are simply grouped together without any particular order.

3. What is the significance of Dedekind's Ketten in mathematics?

Dedekind's Ketten is significant in mathematics because it allows for a more rigorous and systematic approach to the study of number systems. It also serves as the foundation for the development of real analysis and the concept of continuity.

4. How is Dedekind's Ketten related to the concept of completeness?

Dedekind's Ketten is closely related to the concept of completeness in mathematics. A set is considered complete if it contains all of its limit points. In Dedekind's Ketten, completeness refers to the property of having no gaps or holes in the ordering of elements, and every element having a predecessor and a successor.

5. Can Dedekind's Ketten be applied to other fields besides mathematics?

While Dedekind's Ketten was originally developed in the context of mathematics, it has also been applied to other fields, such as computer science and linguistics. In computer science, it has been used in the development of data structures and algorithms. In linguistics, it has been used in the study of language and its structure.

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