Re: Replacement of Cardinality (infinite middle)

On 09/01/2024 11:44 AM, Ross Finlayson wrote:

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:

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Regularity of _difference_, and,
regularity of _dispersion_,
both _increment_, and _modularity_,
are examples of two various kinds of regularity,

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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.
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Consider an ordinal as the set of ordinals before it.
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Each ordinal ξ except the first, 0 = {},
holds the first, {} ∈ ξ
Each ordinal ξ is disjoint from {}
Each ordinal ξ is a regular set.
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A regular set is not an element of itself.
An ordinal is not an element of itself.
ω is not an element of itself.
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The reals actually give a well-ordering, though,
it's their normal ordering as via a model of line-reals.

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No.
The normal ordering of the reals is not a well.ordering.
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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.
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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.
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Well.ordered things which aren't ℝ
does not change that
the normal ordering of ℝ is not a well.order.
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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_.

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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.
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In that sense, there is only one well.order.
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Man means to perfect me....
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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.
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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".
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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).
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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", 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.
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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.
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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.
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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.
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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.
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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.
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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.
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That, "logic is what it is", among other such things, here "first order".
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How about, in evaluation, between "not.first.false", and,
"not.false.yet.null" or "not.false.yet.not", introducing "not"
as both a conjunction, and, a value, or representative of template
of a value, with "yet" being a free conjunction, then that it results
for whatever reason, that the Sorites or Heap that results introduced
is further, for example, "once.false.then.false", or
"once.false.then.forgotten",
or "false.why" in terms of, for example, things otherwise true.
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I.e. the point is that "not.first.false" can never break, it is
"undefined behavior", "not.first.false", is that all the expressions
in all the relations associated with it, at all, are formally completely
void, as if all assertions so held together, were struck together,
as concomitant terms, and the stroke of negation instead of the
stroke of assertion.
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Then, here is making a simple apologetics for "not.first.false",
that it's not the words' "fault" of ending an entire theory
and all what feeds into it, and all what comes out of it,
it's simply an idea case for an inference that's never wrong,
as there's of course the implicit total modality: that
_nothing ever goes wrong_, that reflects for itself though
that the "implicit total modality", doesn't exist, it's not
conditioned in any way except in terms of that its domain
terms, are either "logical" themselves or "properly logical" as it's
called or "non-logical", what exists in "all" the theory.
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So, I don't much have to axiomatize and defend or define
"not.ultimately.untrue", because in whatever greater
implicit modality of whatever theory it is, as wherever
the independence of perceived and attributed fact arises,
in terms of the terms, "not.ultimately.untrue" and "not.first.false"
have the same identity in deciding what's decided by completion,
here what these terms declare to decide.
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I.e. that's their "values", to be sure.
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