Re: The set of necessary FISONs

Am Wed, 05 Mar 2025 10:22:06 +0100 schrieb WM:

On 04.03.2025 23:05, Jim Burns wrote:

On 3/4/2025 2:56 PM, WM wrote:

On 04.03.2025 17:43, Jim Burns wrote:

On 3/4/2025 5:49 AM, WM wrote:

 

Here is only *one* argument standing for a long while.

The union ⋃{F} of FISONs and the intersection ⋂𝒫ⁱⁿᵈ of inductive
subsets are the same set.

Correct.

Proper supersets

We talk about subsets!
 

of ⋂𝒫ⁱⁿᵈ contain extra elements.
You (WM) have previously said that your (WM's) ℕ doesn't have extra
elements.

ℕ is a proper superset of ⋃F. It contains all the dark natural numbers.
⋃F contains only defined natural numbers.

No. N *is* UF.

You need not the intersection however because Z₀ can also be defined
by { } ∈ Z₀, and if {{{...{{{ }}}...}}} with n curly brackets ∈ Z₀
then {{{...{{{ }}}...}}} with n+1 curly brackets ∈ Z₀.

No.

Ha. Caught red handed. Yes there is no mathematical possibility to
contradict my claim.

Please specify?

For Z, not Z₀, ∀a: a ∈ Z ⇒ {a} ∈ Z Z₀ is the simplest inductive subset
of Z.

What does Z₀ contain that is not in my inductive description?

Gee, I wonder.

But you say "n+1 curly brackets".

You can also say: 0 and if n ∈ ℕ then n+1 ∈ ℕ. That is only another
notation of number n.
 

Plus, "finitely.many" eliminates your darkᵂᴹ numbers.

a then {a} means precisely as many as n then n+1.
>

ω/2 ω/10 and ω/20 follow ω?

No they follow Z₀ and UF.

And N.

Every induction surpasses every finite number but never all finite
numbers. It is always a finite number.
Cantor's ℕ however is larger:

 
⎛ "By an A.-U. (that abbreviates actual infinity), on the other hand,
we understand ⎜ a quantum that is on the one hand not variable,
⎜ but rather fixed and determined in all its parts,
⎜ a true constant,
⎜ but at the same time ⎜ on the other hand exceeds in size ⎜ any finite
quantity of the same kind.
⎜ As an example, I cite ⎜ the totality, the sum of all finite whole
positive numbers;
⎜ this set is a thing in itself and, quite apart from ⎜ the natural
sequence of the numbers belonging to it,
⎜ forms a quantum that is fixed and determined in all its parts,
⎝ ... which can obviously be called larger than any finite number.
-- google

>

Therefore ℕ is more than UF.
By induction like Zermelo we find:
∀n ∈ U(F): |ℕ \ {1, 2, 3, ..., n}| = ℵo.

∀n ∈ ⋃{F}: |⋃{F}\{1,2,3,...,n}| = ℵ₀

No. ⋃{F} is wrong because F is the set of FISONs. ⋃{F} = F.
UF does not contain an n that is larger than all n by ℵ₀.

Neither does N.

⎛ Assume ⋃{F} is finite.

UF is (potentially in-) finite. It is constructed by induction which is
variable and therefore has no largest number but never reaches a
completion.

Is it finite or not?

--
Am Sat, 20 Jul 2024 12:35:31 +0000 schrieb WM in sci.math:
It is not guaranteed that n+1 exists for every n.