Re: The non-existence of "dark numbers"

On 16.03.2025 18:19, Jim Burns wrote:

On 3/16/2025 12:17 PM, WM wrote:

On 16.03.2025 13:17, Alan Mackenzie wrote:

WM <wolfgang.mueckenheim@tha.de> wrote:

 

ℵo numbers have no FISONs.

>
You haven't said what you mean by F.

>
I did in the discussion with JB:
F is the set of FISONs.

 And a FISON is
a Finite Initial Segment Of Naturals.
However,
if you (WM) have _non.circularly_ said
what 'finite' means, I have missed it.

Finite means a natural number. If you don't know try to learn what they are. 1 and if n then n+1.

 

All
natural numbers "have" a FISON

>
Then all natural numbers would be in FISONs.
But because of
∀n ∈ U(F): |ℕ \ {1, 2, 3, ..., n}| = ℵo
all FISONs fail to contain all natural numbers.

 You (WM) use an unreliable quantifier swap
to arrive at that conclusion.

No it is very reliable to say that when all FISONs have infinite distance from ω, then no FISON comes closer. Only real cranks can claim that this is wrong. But that is not the question. The question is this:
All infinitely many lossless exchanges have finite indices as you understand them. Which index, as you understand it, has the first lossless exchange where a loss happens (by the way one of infinitely many which are required to empty the matrix of O's)?
Regards, WM