Re: Replacement of Cardinality (infinite middle)

On 09/02/2024 02:46 PM, Jim Burns wrote:

On 9/1/2024 2:44 PM, Ross Finlayson wrote:

On 08/30/2024 02:41 PM, Jim Burns wrote:

On 8/30/2024 4:00 PM, Ross Finlayson wrote:

>

The reals actually give a well-ordering, though,
it's their normal ordering as via a model of line-reals.

>
No.
The normal ordering of the reals
is not a well.ordering.
In a well.ordering,
each nonempty subset holds a minimum.
In the normal ordering of ℝ,
(0,1] does not hold a minimum.
The normal ordering of ℝ is not a well.ordering.

>

Then, here is the great example of examples
from well-ordering the reals,
because
they're given an axiom to provide least-upper-bound,

>
Greatest.lower.bound property of standard ⟨ℝ,<⟩
For each bounded non.{} S ⊆ ℝ
exists greatest.lower.bound.S ∈ ℝ   🖘🖘🖘
>
Well.order property of standard ⟨ℕ,<⟩
For each (bounded) non.{} S ⊆ ℕ
exists greatest.lower.bound.S ∈ S   🖘🖘🖘
>
glb.S ∈ S = min.S
I threw in '(bounded)' for symmetry.
Each S ∈ ℕ is bounded by glb.ℕ = 0
>

"out of induction's sake",
then on giving for the axiom a well-ordering,
what sort of makes for a total ordering in any
what's called a space,
there are these continuity criteria where
thusly,
given a well-ordering of the reals,

>
If we are granted the Axiom of Choice,
then we can prove that
a well.ordering ⟨ℝ,◁⟩ of the reals exists.
>
That well.ordering ⟨ℝ,◁⟩ is NOT standard ⟨ℝ,<⟩
>

one provides various counterexamples
in least-upper-bound, and thus topology,
for example
the first counterexample from topology
"there is no smallest positive real number".

>
Ordered by standard order ⟨ℝ,<⟩
ℝ⁺ holds no smallest positive real number.
>
Ordered by well.order ⟨ℝ,◁⟩
ℝ⁺ holds a first positive real number.
>
They aren't counter.examples.
They are different orders.
>

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.
>

least-upper-bound, has that
that's been given as an axiom above or "in" ZFC,

>
No, least.upper.bound isn't an axiom above or in ZFC.
>

that the least-upper-bound property even exists
after the ordered field that is
"same as the rationals, models the rationals,
thus where it's the only model of the rationals
it's given the existence",

>
No, the complete ordered field isn't
a model of the rationals.
>

Here then this "infinite middle"
is just like "unbounded in the middle"
which is just like this
"the well-ordering of the reals up to
their least-upper-boundedness",

>
If the well.ordering of the reals exists,
it is not the standard order of the reals,
which has the least.upper.bound property,
but is not a well.order.
>
>

"... after the ordered field", the rationals,
just establishing we don't disagree for its own sake.
If a well-ordering exists, then, consider it as a bijective
function from ordinal O, and thus its "elements" or ordinals O,
to domain D. As a Cartesian function the usual way, that's
thusly a set of ordered pairs (o, d) which then via usual axioms
and schema of comprehension and the existence of choice, read
out in order the element (o_alpha, d).
So, a well-ordering of the reals, this function, takes any subset
of uncountably many elements (o_alpha, d, alpha). Now, what's so
is that only countably many of the d can be in their normal order,
that alpha < beta -> d_alpha < d_beta. This is because there are
rational numbers between any of those, and only countably many
of those.
Then, about the least-upper-bound actually being an axiom,
it sort of is, that Dedekind-Eudoxus-Cauchy or "there are
all the infinite sequences", as that there are "enough"
elements in Cantor space to fulfill least-upper-bound,
it's an axiom. So is "measure 1.0". In ZFC + these axioms,
or, ZFC descriptively what models standard real analysis.
Otherwise: it wouldn't need an axiom, which you intend
to equip it with "ordinary infinite induction" and "powerset".
Thus it results that it does, and non-logically, when modeling
real analysis in set theory, in as to whether it's _independent_,
that it's axiomatized.
Once upon a time there was a discussion here about well-ordering
and Zermelo's defense of it, it also sort of works out that most
usual things in infinite comprehension don't work out without it,
yet, then it does get involved that before restriction of
comprehension, infinite induction is extra-ordinary.
Or "unlimited", "unbounded" and also "unlimited", ....
Then about not.first.false thanks for writing that up a
bit more, then that also you can see what I make of it.
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".
Then this is usually about what's called "transfer principle",
that "what's true for each is true for all", these kinds of
things, and about limits and where it does or doesn't hold,
that's what it's called and that's what it is.
Thank you, good luck