Re: The set of necessary FISONs

On 2/23/2025 9:43 AM, WM wrote:

On 23.02.2025 14:55, Jim Burns wrote:

On 2/23/2025 5:34 AM, WM wrote:

On 22.02.2025 21:29, Jim Burns wrote:

On 2/22/2025 5:17 AM, WM wrote:

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>

>

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{{}} ≠  {}

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Wrong.

{} ∈ {{}}
{} ∉ {}
{{}} ≠ {}
Your (WM) would find your posts greatly improved
by using 'set', 'element', 'in', 'not in', and so on
the same way they are in
what you (apparently) intend to criticize.
----

We can remove all numbers from Z_0
and produce the empty set.

>
We can remove all numbers from {137}
and produce the empty set.
>
{137} ≠ 137

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If we remove all numbers from {137}
we produce { },
if we remove all numbers from 137
nothing remains.
It is a matter of taste
whether the empty set is nothing.

>
Nothing is what is in {}.
{{}} ≠ {}
Nothing is not what is in {{}}.
>
The set of all FISONs with
each omissible FISON omitted
is {F}\{F} = {}

>
Right. UF = ℕ ==> Ø = ℕ.
>

The set of all FISONs with
the set of omissible FISONs omitted
is {F}\{{F}} = {F}

>
There is no reason to consider {{F}} at all.

There is reason, but
only for people wanting to be correct.
∀S ∈ {F}: ⋃{F:S⊆F} = ⋃{F}
∀S ∈ {{F}}: ~(⋃{F:S⊆F} = ⋃{F})
{F} ≠ {{F}}

We omit all F(n) which amounts to remove F.
Like all natural numbers amount to ℕ (not {ℕ})

∀x: x ∈ A\B  ⇔  x ∈ A ∧ x ∉ B
⎛ ∀x: x ∈ {F}\{F}  ⇔  x ∈ {F} ∧ x ∉ {F}
⎜
⎜ ∀x: x ∉ {F}\{F}
⎜
⎜ ∀x: x ∉ {}
⎜
⎝ {F}\{F} = {}
⎛ ∀x: x ∈ {F}\{{F}}  ⇔  x ∈ {F} ∧ x ∉ {{F}}
⎜
⎜ ∀x: x ∈ {F}\{{F}}  ⇔  x ∈ {F} ∧ x ≠ {F}
⎜
⎜⎛ ∀x: x ∈ {F} ⇒ x ≠ {F}
⎜⎜
⎜⎝ ~Ex: x ∈ {F} ∧ x = {F}
⎜
⎜ ∀x: x ∈ {F}\{{F}}  ⇔  x ∈ {F}
⎜
⎝ {F}\{{F}} = {F}
{F}\{F} ≠ {F}\{{F}}

Homework: Show the same for the set of FISONs.

Homework: Read
https://en.wikipedia.org/wiki/Set_(mathematics)
or
https://de.wikipedia.org/wiki/Menge_(Mathematik)