Re: The set of necessary FISONs

On 2/14/2025 8:02 AM, WM wrote:

On 13.02.2025 20:47, Jim Burns wrote:

On 2/13/2025 7:04 AM, WM wrote:

On 13.02.2025 00:49, Jim Burns wrote:

Learn that
by A(1) and if A(n) then A(n+1)
an infinite set is created

>
described
>

which has no successors.

>
Each FISON has a successor in the set of FISONs.
The set of FISONs isn't any FISON.

 The set of all FISONs is created.

described

Learn that if there were successors,
they would have a fixed first element.

>
Since induction on the set of FISONs concerns
a predicate on FISONs, not the set of FISONs,
presence or absence of successors for
the set of FISONS and anything else not.in
the set of FISONs
is irrelevant.

>
No, induction concerns
alle the natural numbers n of the FISONs A(n).
>

Learn that
{1}
{2, 1}
{3, 2, 1}
...
is not longer than broad.

>
For each set A which has
emptier.by.one subsets A\{a} smaller than A
there is a FISON larger than A

>
Irrelevant.
All are finite and will never become infinite.

For each set W which has, for each FISON F′,
a subset S′ ⊆ W larger than F′
W is not smaller than ⋃{F}
A subset is not larger than its superset.

The set of rows and the set of columns
are both the size of ⋃{F} and each other.

>
Yes, that is potentially infinite.

The set of rows, the set of columns, and ⋃{F}
do not change.
Nothing changes from in to not.in, or not.in to in.
They do not have any FISON larger.
They do not have emptier.by.one subsets smaller.
They have swaps such that,
wherever there is a swap.in,
there is a later swap.out;
swaps such that, after all swaps,
Bob is not anywhere he has swapped to.