Understanding Dedekind's Ketten: A Brief Explanation

[Stoney Pete]

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.

 

[Samy_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.

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 $\phi$.

 

[Stoney Pete]

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

 

[Samy_A]

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.

 

[Stoney Pete]

o.k. thanks for your answer.