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NNTP-Posting-Date: Thu, 05 Sep 2024 20:14:07 +0000
Subject: Re: Replacement of Cardinality (infinite middle)
Newsgroups: sci.logic,sci.math
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From: Ross Finlayson <ross.a.finlayson@gmail.com>
Date: Thu, 5 Sep 2024 13:14:11 -0700
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On 09/05/2024 12:57 PM, Ross Finlayson wrote:
> On 09/03/2024 01:50 PM, Jim Burns wrote:
>> On 9/2/2024 8:25 PM, Ross Finlayson wrote:
>>> On 09/02/2024 02:46 PM, Jim Burns wrote:
>>>> On 9/1/2024 2:44 PM, Ross Finlayson wrote:
>>
>>>>> Then the point that induction lets out is
>>>>> at the Sorites or heap,
>>>>> for that Burns' "not.first.false", means
>>>>> "never failing induction first thus
>>>>> being disqualified arbitrarily forever",
>>>>
>>>> Not.first.false is about formulas which
>>>> are not necessarily about induction.
>>>>
>>>> A first.false formula is false _and_
>>>> all (of these totally ordered formulas)
>>>> preceding formulas are true.
>>>>
>>>> A not.first.false formula is not.that.
>>>>
>>>> not.first.false Fₖ  ⇔
>>>> ¬(¬Fₖ ∧ ∀j<k:Fⱼ)  ⇔
>>>> Fₖ ∨ ∃j<k:¬Fⱼ  ⇔
>>>> ∀j<k:Fⱼ ⇒ Fₖ
>>>>
>>>> A finite formula.sequence S = {Fᵢ:i∈⟨1…n⟩} has
>>>> a possibly.empty sub.sequence {Fᵢ:i∈⟨1…n⟩∧¬Fᵢ}
>>>> of false formulas.
>>>>
>>>> If {Fᵢ:i∈⟨1…n⟩∧¬Fᵢ} is not empty,
>>>> it holds a first false formula,
>>>> because {Fᵢ:i∈⟨1…n⟩} is finite.
>>>>
>>>> If each Fₖ ∈ {Fᵢ:i∈⟨1…n⟩} is not.first.false,
>>>> {Fᵢ:i∈⟨1…n⟩∧¬Fᵢ} does not hold a first.false, and
>>>> {Fᵢ:i∈⟨1…n⟩∧¬Fᵢ} is empty, and
>>>> each formula in {Fᵢ:i∈⟨1…n⟩} is true.
>>>>
>>>> And that is why I go on about not.first.false.
>>
>>> Then about not.first.false
>>> thanks for writing that up a bit more,
>>> then that also you can see what I make of it.
>>
>> What I find poetic about not.first.false and all that
>> is that our finiteness isn't only _permitted_
>> It is _incorporated into_ our logic. _Required_
>>
>> A finite linear order _must be_ well.ordered
>> (must be, both ways)
>> ∀γ:T(γ) ⇐ ∀β:(T(β) ⇐ ∀α<β:T(α))
>> ∀α:T(α) ⇐ ∀β:(T(β) ⇐ ∀γ>β:T(γ))
>>
>> We are finite.
>> The formulas we write are finitely.many.
>> In a linear order, they must be in a well.order.
>>
>> In a well.order,
>> if each formula Φ[β] is not.first.false
>> ∀β:¬(¬T(Φ[β] ∧ ∀α<β:T(Φ[α])
>> ∀β:(T(Φ[β]) ⇐ ∀α<β:T(Φ[α]))
>> then each formula is not.false.
>> ∀γ:T(Φ[γ])
>>
>> ...because well.order (because finite).
>> ∀γ:T(Φ[γ]) ⇐ ∀β:(T(Φ[β]) ⇐ ∀α<β:T(Φ[α]))
>>
>>> Not.ultimately.untrue, ..., has that
>>> F, bears the value for all F_alpha parameterized by ordinals
>>> (which suffice, large enough, to totally order things),
>>> of true, and that,
>>> there are classes of formulas F,
>>> for example self-referential or differential formulas,
>>> defined for example according to
>>> "when F_alpha is not also as for an ordinal less than omega",
>>> at least making a trivial clear example of
>>> a definition that is for classes of these sorts formulas
>>> where "not.ultimately.untrue" is not held by all classes
>>> for formulas "not.first.false".
>>
>> "Not.ultimately.untrue" sounds to me vaguely like "ω-consistent".
>> But I don't really know what you are talking about.
>> I usually don't know what you are talking about.
>> It is what it is.
>>
>>
>
> That "points do not make lines" and "lines do not make points"
> yet "any two points define a line" and "any two intersecting lines
> define a point", are of course quite fundamental and elementary
> since for most of time that Euclid's Elements is the second-most
> published book in the world.
>
> (Euclid is a panel.)
>
> https://en.wikipedia.org/wiki/Transfer_principle
>
> I have pretty much no use for the hyper-reals as merely
> a "conservative" (i.e., saying nothing) extension of the
> usual Archimedean field, while, something like Nelson's
> Internal Set Theory and that it's co-consistent with ZFC,
> with regards to either "both or neither", much like the
> "both or neither" of "the anti-diagonal and the only-diagonal",
> have that there are "conservative non-standard" extensions
> saying _nothing_ and "non-conservative non-standard" extensions
> saying _something_.
>
> When Hilbert _added_ a postulate of continuity to Euclid's axioms,
> so to establish that a point-set topology could be a thing at
> all, it's quite a non-conservative non-standard axiom, as it were,
> itself, though of course for "axiomless geometry" it already
> exists from there being a prototype continuum as elementary
> in a theory, co-consistent this theory of geometry "points
> and spaces" with the usual theory of words (algebra's,
> set theory's, ...), that, more-than-less you might as
> well start reading the most-published book in the world,
> or just the first few items "in the beginning ..." there
> was space then from the middle "in the beginning ..." there
> was the word, of an example of a necessary sort of ontological
> commitment with regards to nominalism, and its weaker forms
> fictionalism, fallibilism, and anti-realism.
>
> I.e., as a strong mathematical platonist with a stronger
> logicist positivism, my model philosopher's model physicist's
> model philosophy's model physics, easily encompasses the
> tiny, weaker, hereditarily-finite fragment what's conservative
> off ZFC.
>
> https://en.wikipedia.org/wiki/Eleatics
>
> "The Eleatics have traditionally been seen as advocating a strict
> metaphysical view of monism in response to the materialist monism
> advocated by their predecessors, the Ionian school."
>
>
>
>
> It certainly is what it is, ....
>
>
>
>

Back in the 80's and 90's it was Nelson's Internal Set Theory
where it was figured that the avenue toward true non-standard
real analysis was to result.

I.e., not-a-real-functions with real analytical character,
like Dirac's delta function or here for example the Natural/Unit
Equivalency Function, it is expected that "foundations" _does_
formalize them, and that what doesn't, simply, isn't, respectively.

Not saying much, ....


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