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NNTP-Posting-Date: Sun, 01 Sep 2024 18:44:25 +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: Sun, 1 Sep 2024 11:44:43 -0700
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On 08/30/2024 02:41 PM, Jim Burns wrote:
> On 8/30/2024 4:00 PM, Ross Finlayson wrote:
>> On 08/29/2024 10:24 PM, Jim Burns wrote:
>
>>> [...]
>>
>> Regularity of _difference_, and,
>> regularity of _dispersion_,
>> both _increment_, and _modularity_,
>> are examples of two various kinds of regularity,
>
> There is yet another kind of regularity which is
> relevant to whether ω contains ω.
> ⎛ Axiom of Regularity.
> ⎜ Every non-empty set x contains a member y such that
> ⎝ x and y are disjoint sets.
>
> Consider an ordinal as the set of ordinals before it.
>
> Each ordinal ξ except the first, 0 = {},
> holds the first, {} ∈ ξ
> Each ordinal ξ is disjoint from {}
> Each ordinal ξ is a regular set.
>
> A regular set is not an element of itself.
> An ordinal is not an element of itself.
> ω is not an element of itself.
>
>> 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.
>
> If the non.hypocritical stance on that result is that,
> there are non.standard.this and non.standard.that
> _without_ that result,
> consider that triangles without right angles are
> not counter.examples to the Pythagorean theorem.
> They are _not relevant_ to the Pythagorean theorem.
>
> Well.ordered things which aren't ℝ
> does not change that
> the normal ordering of ℝ is not a well.order.
>
>> Of course
>> any other one you'd give would have
>> taking a subset of ordinals,
>> which of course are _always_ well-ordered,
>> with those being an uncountable subset's, of the reals,
>> _also in their normal ordering_.
>
> I find it interesting,
> though possibly not to.the.point,
> that,
> for any two well.ordered sets,
> one set is order.isomorphic to
> an initial segment of the other set.
>
> In that sense, there is only one well.order.
>
>

Man means to perfect me....

What I found was Cervantes where Sancho and Quixote,
Sancho says "focile" for "docile", as from a few pages
before where Sancho and Sancha had as for the meaning
of made-up words, the intended receipt of the text, and
the supposed and generous, receipt of the text.

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

It's called "the first counterexample in topology" it's the first example
in "countereexamples in topology", which of course in the style throughot
is given the direct case in terms of the standard open topology, as with
regards to it and its consequences the complete ordered field (in ZFC).



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", least-upper-bound,
has that that's been given as 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",
the "complete" ordered field, then has about "not.ultimately.untrue",
involves the sort opposite case what always _concludes_, whether
infinite induction was perfectly deterministic and informational,
or not, for whatever reason.

Thus, establising the "closure" and otherwise the "independence",
of the _models_, here is among reasons why there's well-ordering
and well-foundedness in regularity, being given and giving, well-orderings
of the reals and well-foundedness of sets, in terms of induction about
each other, the spaces and the spaces, of the reals and the sets.

Then, in usual small spaces, the independence of modularity with
cardinality and ordering, that increment and the field are given
("first order", given), is where closure and independence are free.

Here for example a case for infinite induction is
both shortest distance, and, straight line. It _is_,
in that space, it _is_, "not.first.false", and whatever
reason and all the reasons where it's "not.first.false",
compared to "possibly.first.false" or "false", as vis-a-vis,
"not.first.false", not.false", "not.was.false". I.e., in any
condition where "not.first.false", considering all "not.false"
or "true" i.e. "not falsified", those cases together form free
modules, courtesy their closures and independence.

It's always good to know if it's known, when, "not.ultimately.untrue",
would _win_, that "not.first.false" provides otherwise not information
to an otherwise free and complete and closed and independenct model,
of evaluating forward infinite induction, vis-a-vis, _defining_ forward
infinite induction.

Then, counting is much and totally the same way, for example "given",
as evaluating, "having a cardinal", cardinal numbers, in terms of finite
sets and evaluating sums or counts, in cardinals, make free sorts
algebras transfinite cardinals, either to the side, free, or over or under
other arithmetics like reals, components, according to modularity
that the relations of fields of cardinals plainly according to both
larger and smaller results natural truncated models, that are true,
insofar as that's the point of model theory as that then a proof is given.


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", in terms of however independent
or "co-consistent" the the models can be what result, what
aren't counterexamples so much as simply modular logic.

That, "logic is what it is", among other such things, here "first order".