Re: Does the number of nines increase?

On 7/1/2024 12:13 PM, WM wrote:

Le 30/06/2024 à 22:59, Jim Burns a écrit :

On 6/27/2024 8:15 AM, WM wrote:

I assume that sets are complete.

>
I (JB) think that what you (WM) call 'logic' is that
each nonempty set has a trichotomous order such that
each nonempty subset holds a first and a last,
whether its first or last is visibleᵂᴹ or darkᵂᴹ
>
Is that what you're saying?

>
Yes, but
only under the pr4condition of completed infinity.

Completed.infinityᵂᴹ is finiteⁿᵒᵗᐧᵂᴹ.
https://en.wikipedia.org/wiki/Finite_set
| Necessary and sufficient conditions for finiteness
| (Paul Stäckel)
| S can be given a total ordering which is
| well-ordered both forwards and backwards.
| That is, every non-empty subset of S has
| both a least and a greatest element in the subset.
The set ℕ of all finiteⁿᵒᵗᐧᵂᴹ cardinals
cannot be given
a total ordering which is
well-ordered both forwards and backwards.
ℕ isn't a finiteⁿᵒᵗᐧᵂᴹ set.
For each finiteⁿᵒᵗᐧᵂᴹ cardinal j ∈ ℕ
j ends an initial segment {0,1,2,...,j} of ℕ
which can be given a total ordering which is
well-ordered both forwards and backwards.
{0,1,2,...,j} is a finiteⁿᵒᵗᐧᵂᴹ set.
For each finiteⁿᵒᵗᐧᵂᴹ cardinal j ∈ ℕ
j begins an end segment {j,j+1,...} of ℕ
which cannot be given a total ordering which is
well-ordered both forwards and backwards.
{j,j+1,...} isn't a finiteⁿᵒᵗᐧᵂᴹ set.