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

On 5/7/2024 4:22 PM, Ross Finlayson wrote:

On 05/06/2024 12:36 PM, Ross Finlayson wrote:

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

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$

My guess is that 'A' is the ASCIIfication of '∀'
Thus
for-any ∀?
for-each ∀+
for-every ∀*
for-all ∀$
Please use each of ∀? ∀+ ∀* ∀$ in a sentence.
I am familiar with wildcard characters in
their programming.language context. In that context,
they look to me more like variables ranging over
standard.issue.all[1] of a set of characters or
a set of sequences of characters.
Maybe if you used ∀? ∀+ ∀* ∀$
I would see better what you mean.
[1]
| ∀x:B(x) ⇒ B(t)
| ∀x:(B⇒C(x)) ⇒ (B⇒∀x:C(x))
| B(x)  ⊢  ∀x:B(x)
| ∃x:B(x) ⇔ ¬∀x:¬B(x)

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).

>
That's what I'm talking about.

I think that you are over.estimating
how clear you've been.
I don't see how standard.issue.quantifiers are
enriched by the examples you give.
Standard.issue quantifiers are already rich enough
to describe them, so, huh?
I have had a bit of fun here in sci.math
brewing up non.standard notation for
non.standard quantifiers, among other things.
My non.standard notations are abbreviations for
expressions with perfectly standard quantification.
For example,
the ordinals are well.ordered.
If exists any γ: B(γ)
then, exists first β: B(β)
∃γ:B(γ) ⇒ ∃₁β:B(β)
That's an abbreviation.
∃₁β:B(β)  ⇔
∃β:(B(β) ∧ ¬∃α<β:B(α))
I am fond of abbreviating.
But all the interesting stuff is found
as a consequence of the disabbreviated forms.
I once spent an entire course studying
∫ᵟᶿω = ∫ᶿdω
the generalized Stokes theorem
https://en.wikipedia.org/wiki/Generalized_Stokes_theorem
It was a very full course.
A lot is crammed into ∫ᵟᶿω = ∫ᶿdω
Are ∀? ∀+ ∀* ∀$ abbreviations of
standard.issue.quantified expressions?