Re: Replacement of Cardinality (infinite middle)

On 08/10/2024 04:37 PM, Moebius wrote:

Am 11.08.2024 um 00:32 schrieb Chris M. Thomasson:

On 8/8/2024 5:26 PM, Ross Finlayson wrote:

>

0 ... ( ... infinitely-many ... ) ... infinity

>
Sure. Think of two points, and draw a line between them.

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It's hard to conceive a "continuous" line between 0 and omega. :-P
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Hint: Of course we may "imagine" the real line:
>
...|-----|-----|-----|-----|--..
    0     1     2     3     4
>
But omega is not a point on this line. :-P

Aristotle has two different definitions of continuous domain,
one's Archimedes' field and another's contiguous points.
One way to distinguish these is points "in" a line and
points "on" a line, when the line is just "going" or
"drawing" the line, line-drawing, "in" the line, instead
of being range, found, picked, or bounded, "on" the line,
then there's also a third definition of continuity,
definition of gaplessness, "about" or "around", the
line, that gets into the formalism of Fourier and
is quite a bit later than the Aristotlean, since when
at least two definitions of continuity that aren't the
same exist, and since Fourier-style, since when a third
was established by Fejer and Dirichlet to be around,
then that these days there's Jordan _measure_ again,
where for a while it had to be content being "content",
and Dirac's delta is a _function_ again, when for a while
it had to be content being a distribution or functional,
that "remedial line-drawing" is a thing again, as for example
the necessity that Hilbert found of a postulate of continuity,
for geometry.
So, "line-reals", "field-reals", and "signal-reals", are
three _different_ definitions of completeness as gaplessness,
in the _repleteness_, of definition, _required_ due their
recognition, for conscientious formalists.