Re: universal quantification, because g⤨(g⁻¹(x)) = g(y) [1/2] Re: how

On 05/05/2024 03:02 PM, Jim Burns wrote:

On 5/5/2024 2:30 PM, Ross Finlayson wrote:

On 05/05/2024 10:59 AM, Jim Burns wrote:

>

The meaning of (1.) depends upon
'n' NOT being
the true name of any number which CAN be counted.to,
the way that "Rumpelstiltskin" is the true name of
a certain straw.into.gold.spinner.
>
An analogy better than "name" is "pronoun"
'n' is more like a pronoun than a name.
"It is a natural number", etc.
>
Variable.names are a big improvement over pronouns
because, in every natural language I'm aware of,
there are no more than a handful of pronouns,
used with many handfuls of referents, and
their distinct referents are kept distinct
by context, AKA, figuring.it.out.
Even if the figuring.out doesn't fail, a lot of work.
>
The expression
| x < y and y < z implies x < z
|
is a big improvement in clarity over
a paragraph of muddle with three pronouns.
x y z act like pronouns, though.

>

How about disambiguating quantifiers so that
something like the universal quantifier
gets disambiguated to reflect
a for-any/for-each/for-every/for-all
when it's so that
things like the Sorites/Heap or transfer principle
apply.
>
Similarly
the existential quantifier is often to be disambiguated
"exists", "exists-unique", "exists-plural",
these kinds of things.
>
English as a language has a rich variety of copulas.

>
I think that your wished.for supplements of
standard.issue quantifiers
can be defined given
standard.issue quantifiers.
>
For my wish,
I would like everyone to be clear on what
standard.issue quantifiers and variables
mean.
>
I think that,
way off in that glorious future,
both you and I will be able to be
satisfactorily understood.
>
And what more could there be
to wish for?
>
>

>
Well, one might aver that extra-ordinary
universal quantifiers are merely syntactic sugar,
yet there's that in the low- and high- orders,
or the first and final, that what they would
reflect of the _effects_ of quantification,
something like
>
for-any A?
for-each A+
for-every A*
for-all A$
>
that it is so that the sputniks or extras
of the quantification in the extra-ordinary,
have that a quantifier disambiguation:
is in the syntax.
>
>
>
Then for the rest of it, like our discussions
on continuous domains and continuous topologies,
i.e. the topology that's initial and final itself,
then these line-reals field-reals signal-reals,
about the integer continuum linear continuum
long-line continuum, ubiquitous ordinals and
extra-ordinary theory, is that these are objects
of the universe of mathematics in the
Hilbert's Infinite Living Museum, of Mathematics.
>
When considering someone like Paul do Bois-Reymond,
who came up with the diagonal argument and the long-line,
and Mirimanoff, who came up with the axiom of regularity
and also the extra-ordinary, and for example Peano,
with his integers and infinitesimals, then one may well
aver that today's standard is a tenuous sort of course,
that is much more fully enriched by the first sort of
nonstandard function like the Dirac Delta, then into
the greater realm of the superclassical law(s) of large
numbers, and more replete three definitions of
continuous domains, and the Cantor space(s).
>
>