Re: The set of necessary FISONs

On 2/18/2025 4:14 AM, WM wrote:

On 17.02.2025 23:03, Jim Burns wrote:

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.

>
Therefore such proofs are done by induction.
They cover all elements.

"All elements" covers only all _existing_ elements.
Again:
An element which,
upon the removal of it and its priors,
all the elements -- or even _almost_ all --
have been removed,
is not an existing element,
is not here referred.to by "all".

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 every human has ended, then there is no other one.

If no human is last,
then the human race has no 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,

>
Bob passes with the last one.

No.
Each human passes Bob to another.
With humans, there is another by assumption,
but, with FISONs, there is another, provably.
⎛ If set A is smaller than set B, then
⎜⎛ fuller.by.one Aᐡᵃ is smaller than
⎜⎝ fuller.by.one Bᐡᵇ
⎜ #A < #B  ⇒  #Aᐡᵃ < #Bᐡᵇ
⎜
⎜ Let A = emptier.by.one Bᐠᵃ
⎜ Then,
⎜ #Bᐠᵃ < #B  ⇒  #B < #Bᐡᵇ
⎜
⎜ Also,
⎜ #Bᐠᵃ < #B  ⇐  #B < #Bᐡᵇ
⎜ Thus,
⎝ #Bᐠᵃ < #B  ⇔  #B < #Bᐡᵇ
For each FISON F, there is
fuller.by.one Fᐡꟳ
Because F is a (finite) FISON,
emptier.by.one #Fᐠ⁰ < #F
Because #Bᐠᵃ < #B  ⇔  #B < #Bᐡᵇ and #Fᐠ⁰ < #F
#F < #Fᐡꟳ
and
fuller.by.one Fᐡꟳ is a larger (finite) FISON.

After all passes, Bob isn't.
Infinity isn't finite.
It isn't even almost finite.

>
Nevertheless:
If all elements of a set have been omitted,
then the set has been omitted.

A claim that
each FISON and its sub.FISONs, omitted,
leave FISONs which have an unchanged union
is
a claim that
all FISONs, omitted,
leave FISONs which have an unchanged union
only if
there is a FISON which, with its sub.FISONs,
are all FISONs.
There is no such (last) FISON.
⎛ #A < #B  ⇒  #Aᐡᵃ < #Bᐡᵇ
⎜
⎜ #Bᐠᵃ < #B  ⇔  #B < #Bᐡᵇ
⎜
⎝ #Fᐠ⁰ < #F  ⇔  #F < #Fᐡꟳ