Re: Replacement of Cardinality (ubiquitous ordinals)

On 08/01/2024 04:23 AM, Jim Burns wrote:

On 7/31/2024 8:30 PM, Ross Finlayson wrote:

On 07/31/2024 01:21 PM, Jim Burns wrote:

>

If I remember correctly,  your (RF's) name for
not.talking about
what's outside the domain of discussion
is hypocrisyᴿꟳ.
>
That sounds like you're delivering a value.judgment:
that we _should not_ not.talk about
what's outside the domain of discussion,
that we _should not_ for example, not.talk about
_all_ triangles when we discuss whether
the square of its longest side equals
the sum of the squares of the two remaining sides.
>
However,
it is because we are hypocriticalᴿꟳ (in your sense?)
that such discussions produce results.
"Conclusions", if you like.
>
We make finite.length.statements which
we know are true in infinitely.many senses.
>
We can know they are so because
we have narrowed our attention to
those for which they are true without exception.
Stated once, finitely, for infinitely.many.
>
Non.hypocrisyᴿꟳ (sincerityᴿꟳ?) throws that away.
>

You're talking about a field,
I'm talking about foundations.

>
I doubt that
you and I are calling the same thing a field:
a set with addition, multiplication, identities, inverses
such that
a+(b+c)=(a+b)+c  a+b=b+a  a+0=a  a+(-a)=0
a⋅(b⋅c)=(a⋅b)⋅c    a⋅b=b⋅a   a⋅1=a  a≠0 ⇒ a⋅⅟a=1
a⋅(b+c)=(a⋅b)+(a⋅c)
?
>
fieldᴿꟳ == domainⁿᵒᵗᐧᴿꟳ ?
>
The counterpart of a variable is its domainⁿᵒᵗᐧᴿꟳ
== those to which the variable possibly refers.
>
 From what I can see,
both fieldsᴿꟳ and foundationsᴿꟳ are domainsⁿᵒᵗᐧᴿꟳ
>
I'm guessing that the distinction between
fieldsᴿꟳ and foundationsᴿꟳ is the distinction between
retail mathematics and wholesale mathematics,
issues of the day and grand unification.
>
In given circumstances, there may well be
excellent reasons to do retail mathematics or
to do wholesale mathematics.
I am skeptical about there ever being
logical reasons to choose one over the other.
>

You're talking about a field,
I'm talking about foundations.
... Of which there is one and a universe of it.

>
If a theory has any model of infinite cardinality,
it has models of each infinite cardinality.
>
That's a general result.
The empty theory (with no extralogical axioms)
has models of each infinite cardinality.
>

That sounds like you're delivering a value.judgment:
that we _should not_ not.talk about
what's outside the domain of discussion,
that we _should not_ for example, not.talk about
_all_ triangles when we discuss whether
the square of its longest side equals
the sum of the squares of the two remaining sides.

>

About triangles and right triangles,
and classes and sets in an ordinary theory
like ZFC with classes, now your theory has
classes that aren't sets.

>
Somewhere, in axioms or definitions,
there are statements we know are true
as long as they are referring to a right triangle.
>
I fully expect that
things other than right triangles exist.
Those other things' existence doesn't change
the truth of those statements
as long as they are referring to a right triangle.
>
That isn't a particularly difficult insight.
We know they're true because
we know what a right triangle is. Duh.
>
I think that I find myself repeating
that not.particularly.difficult insight
because it _sounds like_
teeny, tiny finite beings <waves at camera> are
somehow engaging in some sort of infinite activity.
>
We are not engaged in any sort of
infinite activity.
Making finitely.many finite.length statements
is not an infinite activity.
Yes, they are true _about_ infinitely.many, but
we do not "true" the statements infinitely.often
as though we're laying infinitely.many bricks.
>

Yeah, my mathematical conscience demands that
hypocrisy is bad.

>
Bad why?
>
>

"Wrong", ....
Yes we must disambiguate algebra's "fields" and physics' "fields"
and science's "fields" and logic's "fields, all one field, as it
were, in terms of "foundations", "the field of foundations".
Definition usually expands, when it doesn't, then there is
the conscientious disambiguation, for example as you rightly
note X^WM or X^RF definitions as you see fit to delineate in
the anaphora and cataphora, what's forward definition or for
the multi-pass parsing in definition, in terms of the "descriptive
dynamics" and "definitional dynamics".
For example, where "differential geometry" reduces the world of
functions and curves, they no longer are conscientiously free
in the wider field, encumbered instead with an ambiguity, which
is why then there's function^DG and curve^DG, for example, then
that what remains a restriction has its results attached.
The notion of function varies since the days of classical function,
since when it's delineated and disambiguated with regards to the
Cartesian and function, and that functions, for example, make
their own sort of fundamental theory.
Arithmetic, geometry, number theory, algebra, set theory, part theory,
topology, function theory: each has their own "theory" in their
own fundamental terms, thus resulting modeling each other or modeling
the disambiguation, here about "ubiquitous ordinals of a linear then
integer continuum". Each has their own "field", yet there's still
only one foundations, only one "meta-theory", all one theory.