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A What are Dedekind's Ketten?

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  1. May 18, 2016 #1
    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.
     
  2. jcsd
  3. May 18, 2016 #2

    Samy_A

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    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: May 18, 2016
  4. May 18, 2016 #3
    Samy, does Dedekind also mean that K=ϕ(S)?
     
  5. May 18, 2016 #4

    Samy_A

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    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.
     
  6. May 18, 2016 #5
    o.k. thanks for your answer.
     
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