Re: The set of necessary FISONs

On 2/20/2025 4:09 AM, WM wrote:

On 19.02.2025 19:58, Jim Burns wrote:

Not proven for the set {F} of FISONs,
which is not a FISON.

>
1)
Induction covers all elements of
an infinite inductive set.

Induction covers all elements of
an inductive subset of
a set with only on inductive subset, itself.
There is only one subset it can be,
its (infinite) (inductive) superset.

F(1) ∈ F und F(n) ∈ F ==> F(n+1) ∈ F
describes the infinite inductive set F of FISONs.

This is also true:
If
F₁ ∈ S ⊆ {F} and Fₙ ∈ S ⊆ {F} ⇒ Fₙ₊₁ ∈ S ⊆ {F}
then
S = {F}

2) Subtraction all
FISONs {1, 2, 3, ..., n} satisfying
|ℕ \ {1, 2,  3, ..., n}| = ℵo
from F leaves the empty set.

Why do you think the set has the same property as
its elements?

These two well-established arguments prove my case:
UF = ℕ  ==> Ø = ℕ.

...if you also establish {F} ∈ {F:omissible}
which you can't establish.

The sum of any two natural numbers
is a natural number.
>
The "sum" of all the natural numbers,
for any reasonable definition of that,
will be larger than any natural number,
and not a natural number.

Something often is true of each element
and false of the set.
Proving one doesn't prove the other.

Like the product.

Like not.omissible {F:omissible}

Reason is the potential infinity of definable numbers.
>
Nevertheless:
Zermelo proves
the existence of an inductive *set*
by induction.

You keep using that word.
I do not think it means what you think it means.
⎛ 'Infinity' proves by 'infinity' that
⎜ an inductive set exists superset to
⎜ a set which is its only inductive subset,
⎝ a set for which inductive proofs are valid.