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From: WM <wolfgang.mueckenheim@tha.de>
Newsgroups: sci.math
Subject: Re: The set of necessary FISONs
Date: Mon, 27 Jan 2025 13:55:24 +0100
Organization: A noiseless patient Spider
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On 26.01.2025 23:54, Jim Burns wrote:
> On 1/26/2025 4:49 AM, WM wrote:

>> No swaps can complete actual infinity of ℕ.
> 
>> No actual infinity is represented by the first column.
> 
> No appendix 𝔻 such that ∀d ∈ 𝔻: f(d) = d
> completes potential.infinity to actual.infinity.

∀n ∈ U(F(n)): |ℕ \ {1, 2, 3, ..., n}| = ℵo
ℕ \ {1, 2, 3, ...} = { }	
>> No swaps can complete actual infinity of ℕ.
> 
> For each FISON, there is
> a fuller.by.one FISON which is larger.
> That is complete enough
> in order to allow Bob to disappear.

Not outside of all numbers.

All the FISONs are smaller than ℵo elements:

{1}
{2, 1}
{3, 2, 1}
....

which prove that the first column is smaller than ℵo elements too.

That means no actual infinity is involved in visible numbers. Bob may go 
further. Then he becomes invisible.

> The sets aren't larger and aren't smaller.
> They are the same size,
> which allows Bob to disappear by lossless swaps.

Yes, in the dark domain.

Regards, WM