Re: The set of necessary FISONs

On 3/3/25 4:10 AM, WM wrote:

On 03.03.2025 01:42, Richard Damon wrote:

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

 

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

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

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So, what numbers can't be expressed?

 Dark numbers.

And where is the line?
It seems your N_def includes the infinite set of Natural Numbers, as there is no upper limit to them, and thus there is no need for your dark numbers.
They are just an artifact that your logic blew up when N_def became infinite, blinding you to the truth.

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What is the highest expressable number?

 That does not exist because with n also n+1 is expressable.

So, why isn't N_Def the same as N?

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If there isn't one, why not?

 We call that phenomenon potential infinity.

WHich is just infinity,

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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,

 It can be proved. All infiite sets as produced by Peano, Zermelo, or v. Neumann are potentially infinite, namely produced by inductive reasoning with subsequent quantifier exchange. In exactly the same way as Z₀ is constructed by its elements, the set of removable FISONs is constructed by its elements. The result is Z₀ = ℕ  ==> Ø = ℕ.

That is NOT "inductive Reasoning", at least not as the word is used in English. It is an infinitely iterative method.

 

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

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They cannot differ by a fixed quantity like ℵo.

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

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There are dark numbers collected.

Where?

Yes, when we talk about "in the limit of completing the set" it is a different sort of operation than the normal mathematics limit,

 It adds the dark numbers to the inductive set.

So where are they? What Natural Number wasn't in N_def?

 

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),

 Right. You try to collect the dark numbers without mentioning it.

No, I just collect all the defined numbers. Only numbers that are one more than anther defined number, the full infinite set of them.
Your darkness is only a glitch in your logic, because it can't handle the infinite and blows your system to smithereen leaving the darkness behind.

 Regards, WM