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

Ross Finlayson submitted this idea :

On 05/10/2024 03:26 PM, Jim Burns wrote:

On 5/10/2024 4:12 PM, Ross Finlayson wrote:

On 05/10/2024 03:59 AM, Jim Burns wrote:

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[...]

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I'd like to suggest a reading from
Dehaene's "The Number Sense",
in Chapter 9 "Of Neurons and Numbers",
in the section
"When Intuition Outruns Axioms".
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I found a copy online from
the International Cognition and Culture Institute,
and about page 238.
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He explains that there _are_
non-standard models of integers.

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Is "When Intuition Outruns Axioms" concerned with
other.than.standard.issue quantifiers?
[1]
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If so,
since you are currently holding the talking.stick,
you could use the opportunity to expound on
what Dehaene has to say.
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Maybe I should clarify:
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I don't say that there aren't non.standard objects.
That's not the same as saying that
there aren't non.standard quantifiers.
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I also don't say that there aren't
non.standard quantifiers.
There might not be.
Quantifiers are live near the roots of our logic.
There might be.
Mathematicians are smart.
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What I say is
I don't know yet what sort of
non.standard.quantification scheme
you introduced at your "universal quantification"
post.
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If you wish I had more to say about your posts
(a big IF, not everyone does)
helping me to understand your posts seems like
an effective strategy for bringing that about.
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Just saying.
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Again, what correspondence of yours I see,
which is any in response to me, I've replied.

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If you have used ∀? ∀+ ∀* ∀$ in sentences,
I have overlooked them.
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Surely, it would only be a very small favor to me
to repeat those sentences.
By doing so, you would increase the chances
of me NOT balking and clamming up.
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Could you please do so again?
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[1]
By other.than.standard.issue quantifiers, I mean
other than those such that:
| ∀x:B(x) ⇒ B(t)
| ∀x:(B⇒C(x)) ⇒ (B⇒∀x:C(x))
| B(x)  ⊢  ∀x:B(x)
| ∃x:B(x) ⇔ ¬∀x:¬B(x)
>
>

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">
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Well, first of all, it's after pondering that there
is quantifier comprehension artifacts of the extra sort,
as of a set of all sets, order type of ordinals, a universe,
set of sets that don't contain themself, sets that contain
themselves, and so on.
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Then, English affords "any, "each, "every, "all".
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The -any means for example that "it's always a fragment".
So in this sense the usual universal quantifier is for-each.
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Then, for-each, means usual comprehension, as if an enumeration,
or a choice function, each.
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Then, for-every, means as a sort of comprehension, where it
so establishes itself again, any differently than -each,
when -each and -every implies both none missing and all gained.
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Then, "for-all", sort of is for that what is so "for-each"
and "for-every" is so, "for-all", as for the multitude as
for the individual.
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Then, I sort of ran out of words, "any", "each", "every", "all",
then that seems their sort of ordering, about comprehension,
in quantification, in the universals, of each particular.
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About sums it up, ...." -- Monday

To my way of thinking; any, each, and every natural number has one successor by axiom. One gets to *all* via induction. WM likes to include *all* (as in each element having infinitely many successors) to construct the naturals and then refuses to accept induction with the 'definable' numbers so that he can imagine aleph_zero sucessors for the naturals sequence being greater in length than his (potentially infinite) subsequence.