Re: The set of necessary FISONs

On 2/19/2025 9:47 AM, WM wrote:

Am 18.02.2025 um 18:14 schrieb Jim Burns:

For inductive sets with multiple.inductive.subsets,
it's inadequate to prove that a subset is inductive
in order to conclude that subset is the whole set.

>
The [set of all] FISONs [is]
an inductive set with only one inductive subset.

Yes.

The subset of {F} [holding all omissibles]
is inductive.
Showing that a subset is inductive
is called proof by induction.

>
That is what I do.

That is among the things you do.
You also make an unjustified leap.
⎛ 'extensionality':
⎜ ∀x: x ∈ S ⇔ P(x)
⎜⎛ 'x is S'
⎜⎜ implies and is implied by
⎜⎝ 'P(x)'
⎜
⎜ In set.builder notation,
⎜ S = {x:P(x)}
⎜
⎜ Without extensionality,
⎜ we do not have sets.as.we.mean.sets.
⎜ One way to say it is that
⎜ 'extensionality' is what '∈' means.
⎜
⎜⎛ In George Boolos' tiny set theory ST,
⎜⎜ the only axioms are
⎜⎜ {} and X∪{y} exist and
⎝⎝ extensionality is true.
You prove
the subset of omissible FISONs
is one of the inductive subsets of
the set of all FISONs (out of one).
What '∈' means is that
you prove
∀F′: x ∈ {F} ⇔ omissible F′
no less and no more than that.

{F} is the only.inductive.subset of {F}.
In the case of {F}, a proof by induction shows
that any subset of {F} with that property is {F},
because that subset can't be anything else..

>

That reasoning is silent about
whether the _set_ (not its elements) has A(k).

>
The set without any element is empty.

Nothing is in the empty set.
∀x: x ∈ {} ⇔ FALSE
The empty set isn't nothing.
{{}} holds {}
{{}} doesn't hold nothing.
{} ∈ {{}}

The elements are defined by induction
in order to guarantee the infinite set.
Um aber die Existenz "unendlicher" Mengen zu sichern,
bedürfen wir noch des folgenden ... Axioms.

AXIOM I 'extensionality' means
proving each FISON is omissible  is no
proving {F} is omissible.

[Zermelo: Untersuchungen über
die Grundlagen der Mengenlehre I, S. 266]