Re: Replacement of Cardinality (infinite middle)

On 08/18/2024 09:56 PM, Jim Burns wrote:

On 8/18/2024 5:22 PM, Ross Finlayson wrote:

On 08/18/2024 10:50 AM, Jim Burns wrote:

On 8/18/2024 10:17 AM, Ross Finlayson wrote:

On 08/17/2024 02:12 PM, Jim Burns wrote:

>

Lemma 1.
⎛ No set B has both
⎝ finiteᵖᵍˢˢ order ⟨B,<⟩ and infiniteᵖᵍˢˢ order ⟨B,⩹⟩.
>
Definition.
⎛ An order ⟨B,<⟩ of B is finiteᵖᵍˢˢ  iff
⎜ each non.empty subset S ⊆ B holds
⎝ both min[<].S and max[<].S
>
A finiteᵖᵍˢˢ set has a finiteᵖᵍˢˢ order.
An infiniteᵖᵍˢˢ set doesn't have a finiteᵖᵍˢˢ order.
>
ℕ ℤ ℚ and ℝ each have infiniteᵖᵍˢˢ orders.
In the standard order,
ℕ ℤ ℚ and ℝ are subsets of ℕ ℤ ℚ and ℝ with
0 or 1 ends.
Thus, the standard order is infiniteᵖᵍˢˢ.
Thus, by lemma 1, no non.standard order is finiteᵖᵍˢˢ.
>
They do not have any finiteᵖᵍˢˢ order.
Whatever non.standard order you propose,
you are proposing an infiniteᵖᵍˢˢ order;
you are proposing an order with
some _subset_ with 0 or 1 ends.

>

Robinson arithmetic has non.standard models
with infinite naturals.
For example, {0}×ℕ ∪ ℚ⁺×ℤ
⎛ ⟨p,j⟩ <ꟴ ⟨q,k⟩  ⇔
⎝ p < q ∨ (p = q ∧ j < k)
>
⎛ Numbers ⟨p,j⟩ and ⟨q,k⟩ with p<q are
⎝ infinitely.far apart.
⎛ There are splits between ⟨p,j⟩ and ⟨q,k⟩
⎝ with no step from foresplit to hindsplit.
( ⟨p,j⟩ is not countable.to ⟨q,k⟩
( Not all subsets are 2.ended.

>

I'm really beginning to warm up to this idea of
"finite" and "all orderings are well-orderings"
being a thing.

>
If you're referring to the idea of
⎛ for finite,
⎝ all orderings are well.ordered both ways
then I'm pleased to hear
that you're warming to the idea.
I wish you much future warming.
>

[...] that they're not "immediate" successors,
thus it's delineated that they're "deferred" successors.

>
Standardly, "successor" is "immediate successor".
>
We have other ways to say "deferred successor".
For example, "after".
>
Other than an opportunity to enmurken,
what does the use of "deferred successor" offer?
>

So, ordinals less than a limit ordinal are predecessors,

>
To review:
>

So, with "infinite in the middle", it's just
that the natural order
0, infinity - 0,
1, infinity - 1,
...
has pretty simply two constants "0", "infinity",
then successors,
and it has all the models where infinity equates to
one of 0's successors, and they're finite,
and a model where it doesn't, that it's infinite.

>
This model in which infinity isn't a successor of 0
by which you mean infinity doesn't come after 0
how would infinity not coming after 0 work, exactly?
>
>

>
I mean it's a great definition that finite has that
there exists a normal ordering that's a well-ordering
and that all the orderings of the set are well-orderings.
>
That's a great definition of finite and now it stands
for itself in enduring mathematical definition in defense.
>
Why is it you think that Stackel's definition of finite
and "not Dedekind's definition of countably infinite"
don't agree?
>
The entire idea here that there's a particular _regularity_
due dispersion and modularity only courtesy division down
from a fixed-point, that "Peano's axioms" don't give integers,
they only give increments, i.e. not necessarily constant increments,
that there's more than one _regularity_, REQUIRED, is another
little fact of mathematics missing from your neat little hedgerow.
>
>