Re: The set of necessary FISONs

On 2/10/2025 5:00 AM, WM wrote:

On 09.02.2025 18:20, Jim Burns wrote:

On 2/9/2025 5:59 AM, WM wrote:

On 2/8/2025 4:54 PM, WM wrote:

Therefore U(F(n)) = ℕ ==>
U{ } = { } = ℕ.

If U(F(n)) = ℕ, then
F(1) can be omitted without changing
the result.

If F(k) can be omitted,
then F(k+1) can be omitted too.

The set of FISONs which can be omitted
is an inductive set, i.e., all FISONs.

However,
the FISONs are inductive.
No FISON ends the FISONs.

>
Induction holds
for all natural numbers and
for all FISONs of the infinite set.

Do you accept
∀ᴺj′:∀ᴺi′:
  ∃ᴺk′ = max{i′,j′+1}
?
----
The set {F} of
  FISONs
The set {F:after.F′} of
  FISONs after F′
The set {F:non.omissible} of
  FISONs which cannot be omitted.
  {F:non.omissible} = {F:{F₂:after.F}={}}
The set {F:omissible} of
  FISONs which can be omitted.
  {F:omissible} = {F:{F₂:after.F}≠{}}
∀ᴺj′:∀ᴺi′:
  ∃ᴺk′ = max{i′,j′+1}
∀F′ ∈ {F}:
  ∃F″ ∈ {F}: F″ after F′
∀F′ ∈ {F}
  {F:after.F′} /= {}
¬∃F′ ∈ {F}:
  {F:after.F′} = {}
{F:omissible} = {F}
{F:non.omissible} = {}

Therefore U(F(n)) = ℕ ==>
U{ } = { } = ℕ.

Any FISON F′ such that
  {F:after.F′} = {}
is in {F:non.omissible}
{F:non.omissible} = {}
∀ᴺj′:∀ᴺi′:
  ∃ᴺk′ = max{i′,j′+1}