Re: The set of necessary FISONs

On 2/17/2025 2:36 PM, WM wrote:

On 16.02.2025 23:43, Jim Burns wrote:

On 2/15/2025 9:51 AM, WM wrote:

On 14.02.2025 19:06, Jim Burns wrote:

The set of all natural numbers is not.in
the only inductive subset of the set of
all natural numbers.

>
The set of all natural numbers is constructed
by induction.

>
By axiom "infinity",
an inductive set exists.

>
The axiom applies induction.

A proof by induction proves the inductivity of
a set which is subset to
an inductive set with an only.inductive.subset.
"Infinity" is used to prove that
a set of that description exists.

If the set with an only.inductive.subset
was an element, we'd prove the set has P.
But the set isn't an element.
We don't prove the set has P.

>
Sometimes this is right, sometimes it is wrong.

In the case of
an inductive set with an only.inductive.subset,
we never prove
the inductive set with an only.inductive.subset
has P
only that each of its elements has P.
About any other thing,
the proof is silent both for and against P

When all elements of a set are subject to induction
then the set is an inductive set.

Better:
When all elements of a set are subject to induction,
then the set is inductive with an only.inductive.subset.
⎛ A proof by induction shows that
⎜ a set is inductive which is subset to
⎜ an inductive set with an only.inductive.subset.
⎜
⎜ There is only one set which
⎜ the proved.to.be.inductive set can be:
⎜ the inductive set with an only.inductive.subset.
⎜ That's the proof, at least, its key step.
⎝ Its essence is:  x ∈ {a}  ⇒  x = a

When all elements of a set are removed,
then the set is removed.

In an inductive set with an only.inductive.subset,
there is no element which,
upon the removal of it and its priors,
all the elements -- or even _almost_ all --
have been removed.
Consider
Aᶠˡⁱᵍ(k) == "finitely.many < k < infinitely many"
0 ∈ {i∈ℕ:Aᶠˡⁱᵍ(i)}
k ∈ {i∈ℕ:Aᶠˡⁱᵍ(i)}  ⇒  k+1 ∈ {i∈ℕ:Aᶠˡⁱᵍ(i)}
inductive {i∈ℕ:Aᶠˡⁱᵍ(i)}
{S⊆ℕ:inductive.S} = {ℕ}
{i∈ℕ:Aᶠˡⁱᵍ(i)} ∈ {S⊆ℕ:inductive.S}
{i∈ℕ:Aᶠˡⁱᵍ(i)} ∈ {ℕ}
{i∈ℕ:Aᶠˡⁱᵍ(i)} = ℕ
∀k ∈ {i∈ℕ:Aᶠˡⁱᵍ(i)}: Aᶠˡⁱᵍ(k)
{i∈ℕ:Aᶠˡⁱᵍ(i)} = ℕ
∀k ∈ ℕ: Aᶠˡⁱᵍ(k)
∀k ∈ ℕ:  finitely.many < k < infinitely many
That is an example of a proof by induction.

Example:
If every human has an end,
then the human race need not have an end.
If every human has ended,
then the human race has ended.

If each human ends
but, a day after they end, another has not ended,
then the human race does not end.
If there is a baton (named 'Bob') such that
each human receiving Bob passes it to
a human who ends a day or more after they end,
then,
after all passes of the baton,
Bob is not held by any human who has received it,
nor is Bob held other than by a human who has received it.
After all passes, Bob isn't.
Infinity isn't finite.
It isn't even almost finite.