Re: Replacement of Cardinality (infinite middle)

On 08/13/2024 08:37 PM, Jim Burns wrote:

On 8/13/2024 9:03 PM, Ross Finlayson wrote:

On 08/12/2024 09:25 PM, Jim Burns wrote:

On 8/12/2024 8:28 PM, Ross Finlayson wrote:

>

It's like yesterday,
in this thread with the subject of it
talking about
"infinite in the middle and
always with both ends",

>
I have just realized that
I have been overlooking your "always".
>
"ALWAYS with both ends" is finite.

>

If it's infinite in the middle

>
If
it's infinite in the middle and
its non.{} subsets always have both ends,
then
it's not infinite in the middle.
>

then, the middle acts as the fixed-point,
thus augmenting automatically yon definition,
and suffering not this.
>
Or, for example, it's a counter-example.

>
Definitions do not have counter.examples.
A four.sided triangle is not a counter.example.
It is an incorrectly.identified non.triangle.
>
>

Hm. Thanks for your reply.
So, you seem to imply that the integers by your definition,
by simply assigning 1 and -1 to the beginning, then interleaving
them, and filling in as like a Pascal's Triangle of sorts,
or pyramidal numbers, that that's, not, infinite?
Or, the rationals in the usual assignment of assigning them
integer values and all the criss-crossing and from either
end, building in the middle, not, infinite?
(This was "fill out the rationals".)
The ideas of potential, practical, effective, actual, or,
potential, effective, practical, actual, infinity, show up
a lot for example in theories of atomism, like physics.
("Democritan".)
Consider one of the ideas of Cantor, or wherever he got it,
the idea that omega is a sort of fixed-point, and you know
it's a limit ordinal so has no immediate predecessor yet it's
an eventual successor of all the ordinals before it and in at
least one model contains them, so his idea was "counting backwards".
You just define omega as an initial ordinal and then all its
successors are just less, reversing the ordering, then zero
is a limit ordinal again, yet a fixed-point instead.
Putting those together, all of a sudden it's just another
example of a counterexample. Something you omitted for convenience
instead of clarity.