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

On 23.12.2024 15:31, Richard Damon wrote:

On 12/23/24 3:52 AM, WM wrote:

On 23.12.2024 01:05, FromTheRafters wrote:

WM brought next idea :
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The function E(n) decreases from infinity to zero because in set theory ℕ \ {1, 2, 3, ...} =  { } is an accepted formula.

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In what way is it decreasing?

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One by one. Every endsegment transforms a natural number from content inside to index outside. That is guaranteed by mathematics:
∀k ∈ ℕ : ∩{E(0), E(1), E(2), ..., E(k+1)} = ∩{E(0), E(1), E(2), ..., E(k)} \ {k+1}. Note the universal quantifier.

Except that you don't understand that infinitiy minus 1 is still infinity, and thus did not "decrease".

The sets E(n) decrease. If the sequence (E(n)) could not get empty one by one then Cantor could not set up an infinite sequence using all indices n of that sequence.
Regards, WM