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Subject: Re: Replacement of Cardinality (infinite middle)
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From: Ross Finlayson <ross.a.finlayson@gmail.com>
Date: Sun, 29 Dec 2024 18:11:45 -0800
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On 09/05/2024 01:14 PM, Ross Finlayson wrote:
> 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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