Re: Replacement of Cardinality (infinite middle)

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:

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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",

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Not.first.false is about formulas which
are not necessarily about induction.
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A first.false formula is false _and_
all (of these totally ordered formulas)
preceding formulas are true.
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A not.first.false formula is not.that.
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not.first.false Fₖ  ⇔
¬(¬Fₖ ∧ ∀j<k:Fⱼ)  ⇔
Fₖ ∨ ∃j<k:¬Fⱼ  ⇔
∀j<k:Fⱼ ⇒ Fₖ
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A finite formula.sequence S = {Fᵢ:i∈⟨1…n⟩} has
a possibly.empty sub.sequence {Fᵢ:i∈⟨1…n⟩∧¬Fᵢ}
of false formulas.
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If {Fᵢ:i∈⟨1…n⟩∧¬Fᵢ} is not empty,
it holds a first false formula,
because {Fᵢ:i∈⟨1…n⟩} is finite.
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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.
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And that is why I go on about not.first.false.

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Then about not.first.false
thanks for writing that up a bit more,
then that also you can see what I make of it.

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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_
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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))
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We are finite.
The formulas we write are finitely.many.
In a linear order, they must be in a well.order.
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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])
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...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.
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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.
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(Euclid is a panel.)
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https://en.wikipedia.org/wiki/Transfer_principle
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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_.
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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.
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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.
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https://en.wikipedia.org/wiki/Eleatics
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"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."
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It certainly is what it is, ....
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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, ....
That quote about the Eleatics from the Wikipedia is great, though,
that constructivism and intuitionism go around and round and that
these days is for the re-Vitali-ization of measure theory in the
realm of modern axiomatics and for re-attaching a metaphysics and
teleology for a proper philosophy beyond the classically computable.
Then this "infinite middle" is just about the simplest
"non-Archimedean" that there is, and in fact even simpler,
than for example axiomatizing "0" and "omega" with an
infinite-middle pretty much exactly like ZF does, except
symmmetric about the middle instead of non-inductive yet
declared fiat (stipulated).