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From: WM <wolfgang.mueckenheim@tha.de>
Newsgroups: sci.math
Subject: Re: Incompleteness of Cantor's enumeration of the rational numbers
 (extra-ordinary)
Date: Wed, 15 Jan 2025 19:17:11 +0100
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On 15.01.2025 16:16, Jim Burns wrote:

> In each of our sets,
> each of its elements is in the set,
> each available place is occupied.

Therefore new numbers are not accepted.
> 
> A potentiallyᵂᴹ infiniteˢᵉᵗ set,
> the same as any other set,
> has all available places occupied
> and is completeᵂᴹ.

Potential infinity is growing. "In analysis we have to deal only with 
the infinitely small and the infinitely large as a limit-notion, as 
something becoming, emerging, produced, i.e., as we put it, with the 
potential infinite. But this is not the proper infinite. That we have 
for instance when we consider the entirety of the numbers 1, 2, 3, 4, 
.... itself as a completed unit, or the points of a line as an entirety 
of things which is completely available. That sort of infinity is named 
actual infinite." [D. Hilbert: "Über das Unendliche", Mathematische 
Annalen 95 (1925) p. 167]
> 
>> Half are new.
> 
> A step is never from finite to infinite.

∀n ∈ ℕ: 2n =/= n.

Regards, WM