Re: The set of necessary FISONs

On 3/2/25 1:05 PM, WM wrote:

On 02.03.2025 13:22, Richard Damon wrote:

On 3/1/25 1:18 PM, WM wrote:

On 01.03.2025 17:25, Richard Damon wrote:

On 3/1/25 6:58 AM, WM wrote:

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∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo. Note for all! The limit cannot exist without bridging this infinite gap.
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But what is "n" in the above definition?

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It is a natural number that we can represent in principle by making as many strokes.

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SO, Which Natural Number can it not be that makes N_def different than N?

 They cannot be expressed, neither by stroks nor by digits. Their existence can only be proved: UF = ℕ ==> Ø = ℕ.

So, what numbers can't be expressed?
What is the highest expressable number?
If there isn't one, why not?
This is the flaw in your logic, you think there are two different classes of the infinite set of Natural Numbers, one "defined" that can't have a highest (as what keeps us from defining the next number) and then you need to invent something higher to hide the fact that your first set is actually an infinite set of all the Natural Numbers, that you Naive Logic just can't handle

Note, the value AT the limit, and the values appraching the limit of things can be different.

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The cannot differ by a fixed quantity like ℵo.

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Sure a limit can. What says it can't?

 If it differes then it is not a limit by a matheologial credo in absurdum.
 Regards, WM
  

And you are the one trying to take that limit based on your naive mathological operations.
Yes, when we talk about "in the limit of completing the set" it is a different sort of operation than the normal mathematics limit, as we get infinities that of course never change "value". That is why we don't write it as a normal mathematics limit. N is not limit n-> inf of F(n), but the set of all Natural Numbers with the appropriate set of properties.