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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.