Re: how

On 6/19/2024 2:37 PM, WM wrote:

Le 18/06/2024 à 23:06, Jim Burns a écrit :

For subsets of the minimal inductive meta.set
{i:∀₃Xᴬ⤾⁺¹₀:X∋i}
only the empty set does not hold a first element.

If the set of numbers.remaining
  does not hold a first element,
then the set of numbers.remaining
  is the empty set.

>
That is your big mistake!

Xᴬ⤾⁺¹₀ "X is inductive"  ⇔
∀₂j∈X: X∋j⁺¹ ∧ X∋0
Under proposal 3
because predicate ∀₃Xᴬ⤾⁺¹₀:X∋i
has truth.values for all existing₂ sets,
meta.set {i:∀₃Xᴬ⤾⁺¹₀:X∋i} exists₃ such that
j e {i:∀₃Xᴬ⤾⁺¹₀:X∋i}  ⇔  ∀₃Xᴬ⤾⁺¹₀:X∋j
{i:∀₃Xᴬ⤾⁺¹₀:X∋i} is the minimal inductive
For convenience,
⋂{Xᴬ⤾⁺¹₀}  =  {i:∀₃Xᴬ⤾⁺¹₀:X∋i}
The minimal inductive is inductive.
Each inductive meta.set superset the minimal inductive.
(⋂{Xᴬ⤾⁺¹₀})ᴬ⤾⁺¹₀
Yᴬ⤾⁺¹₀  ⇒  Y ⊇ ⋂{Xᴬ⤾⁺¹₀}
⟨0…n⟩ᶠⁱˢᵒⁿ "⟨0…n⟩ is a FISON"  ⇔
0 ≤ᴬ∈ ⟨0…n⟩ ∋ᴬ≤ n  ∧
∀₃F ⊆ ⟨0…n⟩: ∅ ≠ F ᴬ<ᴬ ⟨0…n⟩\F ≠ ∅  ⇒
  ∃₂i ≥ᴬ∈ F:
  ∃₂j ≤ᴬ∈ ⟨0…n⟩\F:
  i⁺¹ = j
Under proposal 3
because predicate ∃₃⟨0…n⟩ᶠⁱˢᵒⁿ∋i
has truth values for all existing₂ sets i
meta.set {i:∃₃⟨0…n⟩ᶠⁱˢᵒⁿ∋i} exists₃ such that
j ∈ {i:∃₃⟨0…n⟩ᶠⁱˢᵒⁿ∋i}  ⇔  ∃₃⟨0…n⟩ᶠⁱˢᵒⁿ∋j
{i:∃₃⟨0…n⟩ᶠⁱˢᵒⁿ∋i} is the union of FISONs
For convenience,
⋃{⟨0…n⟩}  =  {i:∃₃⟨0…n⟩ᶠⁱˢᵒⁿ∋i}
⋃{⟨0…n⟩} is inductive.
⋃{⟨0…n⟩} is a superset of ⋂{Xᴬ⤾⁺¹₀}
Therefore,
minimal inductive ⋂{Xᴬ⤾⁺¹₀} holds only
FISON.end elements of ⋃{⟨0…n⟩}

For subsets of the minimal inductive meta.set
{i:∀₃Xᴬ⤾⁺¹₀:X∋i}
only the empty set does not hold a first element.

If the set of numbers.remaining [in ⋂{Xᴬ⤾⁺¹₀}]
  does not hold a first element [in ⋂{Xᴬ⤾⁺¹₀}],
then the set of numbers.remaining [in ⋂{Xᴬ⤾⁺¹₀}]
  is the empty set.

>
That is your big mistake!

For subsets of the union ⋃{⟨0…n⟩} of FISONs
only the empty set does not hold a first element.
[1]
Subsets of the minimal inductive ⋂{Xᴬ⤾⁺¹₀} are
subsets of the union ⋃{⟨0…n⟩} of FISONs of which
only the empty set does not hold a first element.
[1]
| Assume S ⊆ ⋃{⟨0…n⟩} is nonempty
|
| kₛ is in S
| and in ⋃{⟨0…n⟩}
| and in FISON ⟨0…kₛ⟩
| ⟨0…kₛ⟩∩S ≠ ∅
|
| In FISON ⟨0…kₛ⟩
| exists first jₛ such that
| ⟨0…jₛ⟩∩S ≠ ∅  &  ⟨0…jₛ⁻¹⟩∩S = ∅
|
| jₛ is first in S
Therefore,
for subsets of the union ⋃{⟨0…n⟩} of FISONs
only the empty set does not hold a first element,
and,
for subsets of minimal inductive ⋂{Xᴬ⤾⁺¹₀}
only the empty set does not hold a first element.

Start to count, continue, continue, continue,.. .
What you can determine that you can count.
The set of not counted numbers remains infinite.
But you cannot determine a first element.
All your following waffle is worthless,
because it violates this fundamental truth.
Simply try it instead of "proving"
counterfactual nonense.

We do not count infinitely.many.
We do not even count Avogadroᴬᵛᵒᵍᵃᵈʳᵒ.many,
not in our 13.7×10⁹.year.old universe, and
Avogadroᴬᵛᵒᵍᵃᵈʳᵒ is barely a start on infinity.
Instead,
we make or find or learn of
finite claim.sequences of only not.first.false.
which we know _can only hold_
true claims about Avogadroᴬᵛᵒᵍᵃᵈʳᵒ
which we do not count.to,  and
true claims about infinity
which we do not count.to.
Therefore,
because we can make or find or learn of
finite claim.sequences of only not.first.false
which hold the claim
⎛ for subsets of minimal inductive ⋂{Xᴬ⤾⁺¹₀}
⎝ only the empty set does not hold a first element.
we know that claim can only be true.
We can learn this in our 13.7×10⁹.year.old universe.