Re: The set of necessary FISONs

On 3/3/2025 3:57 AM, WM wrote:

On 02.03.2025 21:28, Jim Burns wrote:

On 3/2/2025 1:52 PM, WM wrote:

On 02.03.2025 18:32, Jim Burns wrote:

On 3/2/2025 4:48 AM, WM wrote:

Induction abbreviates a supertask.
If 1 then 2, if 2 then 3, and so on.
But supertasks will never pass through
the dark numbers.

They can only
extend the defined numbers without end,
never crossing the infinitely larger domain of
dark numbers - if such exist at all!

>
Even if dark numbers exist, √2 remains irrational.

>
Yes.
But it has no decimal representation.

>
√2 splits all finite decimal representations.
Each is < √2 or > √2

>
Yes.
>

[...]
>
Therefore,
two such decimal.splitting.points √2 and √2′
don't exist.

>
Yes, but the topic is this:
In exactly the same way as Z₀
is constructed by its elements,
the set of removable FISONs
is constructed by its elements,
namely by induction.

The real numbers are too many
to be constructed[made] by induction,
even if construct[make]ing.by.induction
were a thing.
Nevertheless,
we can construct[know] ℝ the real numbers as
some of the subsets of ℚ the rationals.
ℝ ⇄ {S⊆ℚ:(ℚ\S)ᵉᵃᶜʰ>ᵉᵃᶜʰSᵉᵃᶜʰ<ₛₒₘₑS}\{{},ℚ}
Our participation in that construction[proof] is finite:
making finitely.many finite.length claims,
(finitely) verifying they're each true.or.not.first.false
-- and thus verifying that each is true.