Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)

On 13.12.2024 03:23, Richard Damon wrote:

On 12/12/24 9:25 AM, WM wrote:

if Cantor can apply all natural numbers as indices for his bijections, 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.
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But a segment that is infinite in length is, by definiton, missing at least on end.

That means that the premise "if Cantor can apply all natural numbers as indices for his bijections" is false.

So, which bijection from Cantor are you talking about? Of are you working on a straw man that Cantor never talked about?

There are many. The mapping from natumbers to the rationals, for instance, needs all natural numbers. That means all must leave the endsegments. Another example is Cantor's list "proving" uncountable sets. If not every natural number has left the endsegment and is applied as an index of a line of the list, the list is useless.
But if every natural number has left the endsegments, then the intersection of all endsegments is empty. Then the infinite sequence of endegments has a last term (and many finite predecessors, because of
∀k ∈ ℕ : ∩{E(1), E(2), ..., E(k+1)} = ∩{E(1), E(2), ..., E(k)} \ {k}).
Regards, WM