Re: Replacement of Cardinality (infinite middle)

On 08/19/2024 05:33 PM, Jim Burns wrote:

On 8/19/2024 8:08 PM, Ross Finlayson wrote:

On 08/19/2024 04:18 PM, Jim Burns wrote:

>

[...]

>
Then, about that
the class of ordinal is an ordinal

>
True because of what we mean by 'ordinal'
⎛ which leaves open the other question about
⎜ whether that class or finite ordinals or
⎝ inaccessible cardinal or ... _exist_
>

and needn't be given by axiom or relation to an axiom,
yet instead as a matter of comprehension over the class,

>
...if the class exists.
How we know that a class exists is by axiom.
It's an abstract object.
What other way could we know?
>
This need for some axiom to start off the existing
is harder to paper over in a formal language.
But, with either a formal or natural language,
it's inherent in exploring Plato's realm of Forms.
How else do we enter that realm?
>
>

Keep in mind that starting from either end,
and either making increments from nothing or
parts from a whole,
items increment ... heap whole
whole heap ... increment items
that either and both ways result the Sorites/Heap
paradox in definition, about individuation, of
continua, here Integer Continua, where it's so
that "a mere infinity of integers" makes for all
the properties resulting of an "Integer Continuum".
So, people most usually have arithmetic and algebra
together from the get-go, the complete ordered field,
which we're familiar with and not merely because of
our Physics and Mathematics degrees in Science and
Philosophy, yet because it's introduced as elementary
arithmetic and algebra from the most entire primary
curriculum.
Yet, one may deconstruct the field into two separate
operations, "increment", and "halving", then that
these fundamentally define "addition" and "division",
while keeping them altogether separate.
Then, from "zero" there's most usually this increment,
then repetitions of that. Someone eventually introduces
division then to keep that closed the ratios result as
a simple matter of un-reducible expression and in the
ordering theory and the result systems of equalities
and inequalities, because they can't be the numbers
that there already are. Or can they, ..., when they're
merely rounded, for example, to the nearest.
In much the same way, from "infinity" as a whole,
there's dividing it yet the "increments" do no exist
until such time as iota-values define the "increments"
as of "least increment", that "integers" are "least
increments" of an infinity of numbers while "iota-value"
are "least increment" of a whole of a number.
Of course, this is totally how most people see
what is a continuum between zero and one,
as entirely equi-distributed, nowhere any more
concentrated than any other, and entirely
self-similar all ranges of all regions of all scales.
Then, numbers as just, one of those, is not only
simple and built neatly formally as just mentioned
for all standard arithmetic, then also it's rather
sort of required for at least some facts of the
objects of the language of arithmetic.