Re: Replacement of Cardinality

On 8/28/2024 2:10 PM, Jim Burns wrote:

On 8/28/2024 3:04 PM, Jeff Barnett wrote:

On 8/28/2024 12:13 AM, Jim Burns wrote:

On 8/27/2024 3:11 PM, WM wrote:

Le 27/08/2024 à 02:06, Jim Burns a écrit :

On 8/25/2024 3:45 PM, WM wrote:

This function exists because
nothing contradicts its existence.

Except for the contradicting
consequences of its existence.

The function exists if
actual infinity exists.
The function does not exist if
only potential infinity exists.

¬∃ᴿx>0: NUF(x) = 1

Then NUF(x) does not exist
and infinity is not actual
and sets are not complete.

In a finiteⁿᵒᵗᐧᵂᴹ order ⟨B,<⟩
each non.empty S ⊆ B is 2.ended.

In a finiteᵂᴹ order ⟨B,<⟩
no one can say
what a setᵂᴹ is,
what an orderᵂᴹ is,
what finiteᵂᴹ is.
Perhaps, in 30 more years,
these question will have answers.

An infiniteⁿᵒᵗᐧᵂᴹ order ⟨B,◁⟩ is
trichotomous and not finiteⁿᵒᵗᐧᵂᴹ.

Trichotomous?
Did you actually mean to say "dense order",
"dense in itself", or something else akin?

I meant "trichotomous".

I could equally.well have said "total ⟨B,◁⟩"
but "trichotomous" seems less misunderstood by WM.

I didn't bother to say "trichotomous and finite ⟨B,◁⟩"
because trichotomy and transitivity follow from

each non.empty S ⊆ B is 2.ended.

I've proved these elsethread:

⎛ if
⎜ ⟨B,◁⟩: each S ⊆ B: {} or 2.ended and
⎜ ⟨B,<⟩: NOT each S ⊆ B: {} or 2.ended
⎜ then
⎜ sets S, {x} exist such that
⎜ ⟨S,<⟩: each T ⊆ S: {} or 2.ended
⎝ ⟨S∪{x},<⟩: NOT each T ⊆ S∪{x}: {} or 2.ended

⎛ no sets S, {x}, trichotomous '<' exist such that
⎜ ⟨S,<⟩: each T ⊆ S: {} or 2.ended
⎝ ⟨S∪{x},<⟩: NOT each T ⊆ S∪{x}: {} or 2.ended

"Trichotomous" avoids x not being comparable to
the contents of S, and {x} being not.2.ended.

That is how I prove:
because
ℕ has ONE order with ONE non.2.ended non.{} subset
ℕ has NO order WITHOUT one non.2.ended non.{} subset.

That's why any all.2.ended.non.{}.subsets ORDER works
as a way to distinguish a finite SET from infinite.

I think that some of those in this thread
are skeptical of that result.
Thank you for the excuse to repeat it.

Well "dense order" and "dense in itself" are well know terms in
elementary studies of ordering, point set topology (aka real analysis)
and so on. They only imply non finite if you have the right axioms and
definitions at hand. By using the terms I suggested, you automatically
drag in the rest of the necessary magic. "Trichotomous" generates the
right mental images, I must admit. But I'm not sure how one could show
that a relation R on S, where R is Trichotomous, implies that I can't
1-1 map a strict subset of R to R. [Actually you can't! Let R be "=" and
let S consist of a single element, then trichotomy is a property of R on
S but S is finite. [Humm! I wonder if the same example blows my
suggested terminology out of the water too?]]
--
Jeff Barnett