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From: Ross Finlayson <ross.a.finlayson@gmail.com>
Date: Sat, 11 May 2024 06:44:01 -0700
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On 05/11/2024 04:02 AM, FromTheRafters wrote:
> 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:
>>>
>>>>> [...]
>>>>
>>>> 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".
>>>>
>>>> I found a copy online from
>>>> the International Cognition and Culture Institute,
>>>> and about page 238.
>>>>
>>>> He explains that there _are_
>>>> non-standard models of integers.
>>>
>>> Is "When Intuition Outruns Axioms" concerned with
>>> other.than.standard.issue quantifiers?
>>> [1]
>>>
>>> If so,
>>> since you are currently holding the talking.stick,
>>> you could use the opportunity to expound on
>>> what Dehaene has to say.
>>>
>>> Maybe I should clarify:
>>>
>>> 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.
>>>
>>> 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.
>>>
>>> What I say is
>>> I don't know yet what sort of
>>> non.standard.quantification scheme
>>> you introduced at your "universal quantification"
>>> post.
>>>
>>> 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.
>>>
>>> Just saying.
>>>
>>>> Again, what correspondence of yours I see,
>>>> which is any in response to me, I've replied.
>>>
>>> If you have used ∀? ∀+ ∀* ∀$ in sentences,
>>> I have overlooked them.
>>>
>>> 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.
>>>
>>> Could you please do so again?
>>>
>>> [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)
>>>
>>>
>>
>> ">
>>
>> 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.
>>
>> Then, English affords "any, "each, "every, "all".
>>
>> The -any means for example that "it's always a fragment".
>> So in this sense the usual universal quantifier is for-each.
>>
>> Then, for-each, means usual comprehension, as if an enumeration,
>> or a choice function, each.
>>
>> 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.
>>
>> 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.
>>
>> 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.
>>
>> 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.

Well, at some point induction exhausts without completion,
while in some cases induction exhausts with completion,
in the infinite limit or continuum limit.

What's arrived at is deduction, completions both ways,
as what they must cross as meet in the middle,
the middle of what is to either way, to what's nowhere,
the middle of nowhere.


Consider for example "Cantor and counting backward from
infinity": to paraphrase ungently, "the idea arises
yet I, Cantor, discard it because I want the priority
of otherwise my induction that I read off from duBois-Reymond
the anti-digaonal then also with my mw-proof then that
nested intervals is just Archimedean, so much like my
idea about the Domain Principle that there exists a
universe of sets or Absolute, I'm willing to do without
it as I don't want Russell to leap on the first catch,
forget I said anything, as long as you remember I said
one thing".


The natural numbers: go to infinity.