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NNTP-Posting-Date: Tue, 03 Sep 2024 00:25:05 +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: Mon, 2 Sep 2024 17:25:05 -0700
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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_,
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