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

On 28.11.2024 09:34, Jim Burns wrote:

Consider the sequence of claims.
⎛⎛ [∀∃] for each end.segment
⎜⎜ there is an infinite set such that
⎜⎝ the infinite set subsets the end.segment

and its predecessors!
If each endsegment is infinite, then this is valid for each endsegment with no exception. because all are predecessors of an infinite endsegment. That means it is valid for all endsegments.
The trick here is that the infinite set has no specified natural number (because all fall out at some endsegment) but it is infinite without any other specification.

⎜
⎜⎛ [∃∀] there is an infinite set such that
⎜⎜ for each end.segment
⎝⎝ the infinite set subsets the end.segment
 We cannot SEE,
just by looking at the claims,
that, after [∀∃], [∃∀] is not.first.false.

I have proved above that [∃∀] is true for all infinite endsegments.
A simpler arguments is this: All endsegments are in a decreasing sequence. Before the decrease has reached finite endsegments, all are infinite and share an infinite contents from E(1) = ℕ on. They have not yet had the chance to reduce their infinite subset below infinity.
Regards, WM