Re: The set of necessary FISONs

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.

>
A claim about an indefinite element of Z₀ = ⋂𝒫ⁱⁿᵈ(Z)
cannot have a counter.example outside of Z₀
That is the source of a matheologian's certainty.

>
It has, namely ω, ω/2, etc.

ω is outside Z₀
ω cannot be a counter.example to
a claim about an indefinite element of Z₀
ω/2 is outside Z₀ and outside the ordinals.
ω/2 cannot be a counter.example to
a claim about an indefinite element of Z₀, or to
a claim about an indefinite ordinal.

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
⎛ By Dedekind completness,
⎝ at least one such point exists.
Two such decimal.splitting.points √2 and √2′
don't exist.
⎛ Assume otherwise.
⎜ Assume √2 /= √2′ both split decimals the same.
⎜ |√2 - √2′| = δ > 0
⎜
⎜ However,
⎜ there are finite decimal representations x₊ and x₋
⎜ nearer than δ but on either side of √2 and √2′.
⎜ √2 and √2′ cannot be δ apart.
⎝ Contradiction.
Therefore,
two such decimal.splitting.points √2 and √2′
don't exist.