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From: Ross Finlayson <ross.a.finlayson@gmail.com>
Date: Sun, 19 May 2024 07:07:14 -0700
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On 05/18/2024 01:11 PM, Ross Finlayson wrote:
> On 05/18/2024 11:16 AM, Jim Burns wrote:
>> On 5/18/2024 12:09 PM, Ross Finlayson wrote:
>>> On 05/16/2024 09:50 AM, Jim Burns wrote:
>>
>>>> [...]
>>>
>>> I think that "correct", in context, is the entire
>>> context, which is exactly what deductive inference
>>> contains, explaining when inductive inference either
>>> must complete, or meets its juxtaposition, with
>>> regards to any two forces that balance and align
>>> in symmetry.
>>>
>>> So, what you are claiming is that inductive inference
>>> is invincibly ignorant,
>>
>> I am claiming that inductive inference
>> is invincibly modest.
>> Post.inference, we only assert claims about
>> whatever.it.is we described pre.inference.
>>
>> Perhaps that doesn't seem modest,
>> because whatever.it.is is infinitely.many,
>> but induction holds for infinitely.many cisfinite ordinals
>> in the same way that geometry holds for infinitely.many
>> right triangles. Completely.
>>
>>> A given schema for induction has no more correctness,
>>> in its own vacuum, than any other,
>>
>> Induction on the cisfinite ordinals
>> ⎛ those countable.back.to.0 after only
>> ⎜ those countable.back.to.0
>> ⎝ and also 0
>> is a theorem.
>> Theorems are not optional.
>>
>>> and when they're put together and don't
>>> agree, then either they don't, and don't, or
>>> don't, and do.
>>>
>>> "Not.ultimately.untrue", ....
>>
>> Induction on the cisfinite ordinals
>> is not.first.false in a finite sequence of
>> only not.first.false claims
>> which begins "A cisfinite ordinal is ... ".
>>
>>> One can contrive simple inductive arguments
>>> that _nothing_ is so.
>>
>> An example of such an argument would be clarifying here.
>>
>>> So, I'd say your definition of "correct", isn't,
>>> and is simply a declaration of "relative" and "blind".
>>>
>>> No offense meant, of course, it's so that paradoxes
>>> are to be resolved, not obviated.
>>
>>
>
> Says nothing, says nothing, says nothing, says nothing, ....
>
> See, just saying so doesn't make it so,
> something that _goes_ has a _place_
> to go.
>
> Consider for example the mathematical limit,
> when, and if, all the terms are related there's
> only one rule need follow, and an arbitrary
> competing rule, shares no terms, so is altogether
> not relevant, we can say that the limit exists
> and we can get close enough to establish
> vanishing differences.
>
> Yet, the "infinite limit" is already stronger,
> close enough isn't good enough, when competing
> conditions would otherwise result its nullity,
> in effect.
>
> Then, the "continuum limit" is an idea of
> even a strong sort of setting, and about
> strong enough for purpose of a continuous
> milieu altogether, all relevant, all book-kept,
> like the natural/unit equivalency function,
> or most any other continuum limit you might
> come across in all sorts of mathematical
> treatments of probability and physics.
>
> For mathematical terms to maintain a relevance
> to each other, there's established as of the
> "relephant", as it were of each other. "Relephants:
> never forget".
>
> In case of fire: break glass.
>

Since we've all learned about Russell's antinomy,
one can consider that in the outlook of:

Says nothing, says nothing, says nothing, says nothing, ...

is nothing, yet in the comprehension of

{infinitely-many terms}

there results a:  something.

The difference between projection and perspective,
or perspective and projection, or projective and
perspection, perspection and projective, becomes
a object/subject distinction, and contradistinction,
and juxtaposition, making for a fuller dialectic.

We're not the most naive logicians in what one
may hope would be a sufficient modesty, of what
is covered and uncovered, by our ignorance: lack thereof.

This uncovering or the knackbaut, turning over
the field, the a-letheia, disclosing, or otherwise
these usual terms about the limits of knowledge,
can help a lot to establish that the inductive inference
is absurd, unless it's infinite.

So, of course this is widely explored since antiquity,
even moreso why today that the giants who already
trod on each other, left room for any of us to stand
in any their place.

The idea that ZF "provides" an inductive set, is
a great simplification, that there's at least one
case where it's justified, in the comprehension
of a regular, rulial, ordinary comprehension.
Yet, in the fuller dialectic, the more "correct"
of what's the fundamental concern of the
foundations, is the fundamental question of
metaphysics, the dialectic between nothing
and something.

One does not simply break lines into points,
nor push points into lines, yet deductive inference
does have division of a whole into parts, and,
the invertability, one does not simply break lines
into points and push points into lines without both.


Then these variegated universal quantifiers are
a just sort of trivial annotation, affecting to reflect
what it is, what they are, what it is.