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

On 05/15/2024 07:10 AM, Jim Burns wrote:

On 5/14/2024 4:15 PM, Ross Finlayson wrote:
>

They're not all quite so strong,
the many, many examples
of the balking and clamming,
the actually quite a few very many,
though, these are pretty good.

>
You don't want to talk about
what I want to talk about.
And there's nothing wrong with that.
Really.
>
However, it's just as true
in the other direction.
>
Date: Sat, 11 May 2024 19:47:38 -0400
Message-ID: <a4700775-be6c-46db-ad41-361eee6a3b67@att.net>
<JB<RF>>
>

The case is that induction goes through,
an inviolable law you call it:
does it go all the way through?
Does it complete?

>
It is complete.
There is no completing.activity,
so I wouldn't say it completes.
>
Compare to right triangles:
Are all the squares of two shorter sides
summed to the square of the longest side?
>
That's a tricky question to answer because
there is no summing done.
That relationship between the sides
is simply something true about right triangles.
>
And it is complete == it is true for each.
>
We don't typically ask the tricky question
about right triangles.
We ask the tricky question about cisfinite induction
because we imagine it as a process,
which we don't for right triangles.
>
Cisfinite induction is NOT a process.
Cisfinite induction is an argument,
completely correct or completely incorrect.
>
</JB<RF>>
>
>

You mean "not.ultimately.untrue"?
Just because ZF provides an inductive set,
doesn't mean one doesn't exist, courtesy
lesser means or "axiomless natural deduction".
If it's true it's true if it's not it doesn't exist -
only its fabricant simulant cromulant shell of a
prototype of a relation to the sole statement in
all of Comenius that isn't true: "the Cretan" (the Liar).
Ex Falso Nihilum, ....
So, about "refined" universal quantifiers, or
"variegated", say, universal quantifiers, yes I'd
be interested in making it syntactical, then though
I already sort of framed that we have to sort out
that a) terms, b) predicates, and c) relations are
three different things.
It's kind of like when Tarski wanted to "remove except
constants" in one of his cylindrical algebras, which
is sort of an attempt to geometrize what's otherwise
algebraical. Well, on the one hand, it's already sort
of built on a frame that won't support it, while on the
other, the really exists because there's a geometrical
model of things that like continuous domains are real,
or "exist".
About the "inductive impasse", again, then, is about
in this context of universal quantififiers, variegated,
when they do or respectively don't or positively affirmatively
or positively negatively or not-necessarily or necessarily-not
affirmatively or negatively, effect to reflect what's so,
where, yes indeed these become syntactical in a) terms,
b) predicates, and c) relations, where they "exist".