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Am Thu, 12 Dec 2024 10:12:26 +0100 schrieb WM:
If a bijection with ℕ is possible, the sequence can be exhausted so that no natural numbers remains in an endsegment.The sequence is endless, has no end, is infinite.The end of the sequence is defined by ∀k ∈ ℕ : E(k+1) = E(k) \ {k}.
None of which are an infinite sets, so trying to take a "limit" of>
combining them is just improper.
Most endsegments are infinite. But if Cantor can apply all natural
numbers as indices for his sequences, then all must leave the sequence
of endsegments. Then the sequence (E(k)) must end up empty. And there
must be a continuous staircase from E(k) to the empty set.
It makes no sense not being able to „apply” numbers. Clearly Cantor does.He claims it. That means no numbers remain unpaired in endsegments.
The sequence IS continuous. It’s just that you misconceive of theCantor does. If the limit is not reachable, then complete bijections cannot be established.
limit as reachable.
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