Re: Replacement of Cardinality (infinite middle)

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.