# What are Dedekind's Ketten?

• A
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.

## Answers and Replies

Samy_A
Homework Helper
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##.

Last edited:
Samy, does Dedekind also mean that K=ϕ(S)?

Samy_A
Homework Helper
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.

ProfuselyQuarky