Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)

On 09.01.2025 20:16, Jim Burns wrote:

On 1/9/2025 11:51 AM, WM wrote:

It proves that the numbers of
the first and second number class
form a linear system.

 Apparently,
'linear' is yet.another term which
means something other than
what you (WM) want it to mean.

It means simply that no parallel scales are involved. One comes after the other. For every pair of elements we trichotomy.

 Apparently,
what you (WM) think you have concluded is that
the ordinal.line has the Archimedeanⁿᵒᵗᐧᵂᴹ property

No, I simply know that all natural numbers exist and then comes ω. All points of the ordinal axis are in trichotomy.

The ordinal.line is linearⁿᵒᵗᐧᵂᴹ and
its linearityⁿᵒᵗᐧᵂᴹ does follow from its well.order,
but that's not the Archimedeanⁿᵒᵗᐧᵂᴹ property.

Is Archimedes your strawman?

 ⎛ Because ⦃α,β⦄ holds a first ordinal,
⎜ α≠β  ⇒  α<β ∨ α>β

but not both.

⎜ and '<' is connected
⎜
⎜ Because ⦃α,β,γ⦄ holds a first ordinal,
⎜ α<β ∧ β<γ  ⇒  α<γ
⎜ and '<' is transitive.
⎜
⎜ By definition, ¬(α<α)
⎜ and '<' is irreflexive.
⎜
⎜ A connected, transitive, irreflexive order
⎝ is a linear order.

In short: All elements are in trichotomy.
Regards, WM