Re: Seven deadly sins of set theory

On Friday, January 26, 2024 at 1:59:37 PM UTC-8, Jim Burns wrote:

On 1/26/2024 3:41 PM, Ross Finlayson wrote:

On Friday, January 26, 2024
at 10:31:06 AM UTC-8, Jim Burns wrote:

On 1/25/2024 9:15 PM, Ross Finlayson wrote:

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So, how do you get
Least Upper Bound established again?

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ℝ := {F ⊆ ℚ| F ᣔ<ᘁ F ᣔ<ᣔ ℚ\F }\{∅,ℚ}
for bounded non.empty S, lub S = ⋃S

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After,
in descriptive set theory,
the existence of a complete ordered field,
which is a stipulation of existence in the standard
that the equivalence classes of sequences with
the property that they are Cauchy so model a thing,
_after_, then one might aver
that the real numbers each partition
the rational numbers,
partitions inequality related to given real number,
uncountable,
not "just the rationals", countable.

You are incorrect.
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One model of the real numbers is
the equivalence classes of
Cauchy sequences of rationals.
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Their existence is not _stipulated_ in set theory,
it is _proved_
"proved not stipulated" is what makes them
a construction of ℝ from/in set theory.
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That isn't the only construction of ℝ
https://en.wikipedia.org/wiki/Construction_of_the_real_numbers
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Any construction of ℝ proves that
the complete ordered field is not
a contradictory collection of requirements,
assuming it's true that
whatever the construction began with
exists.
Sets, for example.
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(Note that a two.set _partition_ of ℝ doesn't need to be
each of one < each of the other, merely disjoint.
That's why I refer to splits.)

0) the standard reals have cardinality c
1) the rationals are not gapless and
have cardinality < c

The power set 𝒫(ℚ) of the rationals
has cardinality 𝒫(ℕ) ≥ |ℝ| > |ℕ|
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The edgeless.ℚ.foresplits are elements of 𝒫(ℚ) not ℚ

2) the long-line of
all expressions of real-valued formula has
cardinality > c

I bet you (RF) aren't going to tell me
what you mean by that.

so it results that
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0) the rationals don't have enough partitions
because transfinite pigeonhole

There are splits of rationals which
don't have rationals last.before or first.after.
Almost all of them don't, actually.
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The splits are sets of rationals, not rationals.
There are more sets than rationals.
There is, in particular,
a union for each bounded non.empty collection of
edgeless.ℚ.foresplits,
which is the least upper bound of the collection,
whence "model".

1) the reals subsume having partitions about them,
these are individuals
I insist the Dedekind cuts are what they are,
not, what they are not,
which is that they are:
defined by the real numbers a complete ordered field:

Then give this some name other than "Dedekind cuts":
{F ⊆ ℚ| F ᣔ<ᘁ F ᣔ<ᣔ ℚ\F }\{∅,ℚ}
Whatever you name it,
it is provably a model of ℝ
Look!
The existence of ℝ is not a prerequisite for it.

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The space of all real-valued formulas is having a
greater cardinal than the set of real numbers that's
the complete ordered field:  I must imagine that you
know it's well-known that the cardinal of real-valued
functions is greater than the cardinal of the reals.
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Then the long line is basically putting those in order,
their real-valued formula for all their inputs,
the equivalence classes of those.  I mean if you tent out
rationals that fall back into line, then these are simply
above that.  Anyways though at least they're already
larger than the cardinal of the reals, so, then though
they all fit somewhere on the linear continuum.
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(This is generally called "the long-line of du Bois-Reymond".)
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About "the complete ordered field being complete",
and Eudoxus/Dedekind/Cauchy, and why they're insufficient,
and why Least Upper Bound property of the equivalence classes
of sequences that are Cauchy, that have the property of being Cauchy,
is a stipulation, not a derivation, after all the definable ones,
if you start with Dedekind cuts it is, because you forget,
because the _model_ in the _model theory_ must always embody
_all its relations_, and, the assignment to partitions of rationals,
by real numbers of course or else it wouldn't be suitable at all,
you will so assert Least Upper Bound to make gaplessness,
then it was just an artifice and a convenience, not a contrivance.
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About though why fully what I mean is "Eudoxus/Cauchy/Dedekind
is insufficient", that's what I mean, i.e., that's what I intend, and,
when I say "I mean", it means "I intend", and "no mean-ness" is implied.
Here though what it means is that continuity the property of a
continuous domain, has that field-reals are a model of a continuous
domain, but not sufficient in the repleteness, of the linear continuum,
when there are others that only themselves fully embody the relation.
Basically it's about the constructible and definable, about that
Eudoxus/Cauchy/Dedekind sequences have their assignment as
infinite and bit-string or often called "decimal", the representation.
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I.e. it's "well the complete ordered field is complete ordered field
meaning Least Upper Bound, after density, about that each
is having a representation as a sequence because it's after
that all do."
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"The Dedekind cut of pi":  like all others, defined by a real number.
(Yeah, I know transcendental numbers can't be written in the usual
algebraic way like "the rationals whose square is less than 2 for root 2".)
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Anyways if you like infinity then du Bois-Reymond is really famous with
the "Infinitarcalcul", and "long-line" where mostly of course historically
the study of infinitesimals and indivisibles is what's today called real analysis.
You'll want to know delta-epsilonics after the definition of limit
because everybody knows that, and then there's a very common
axiomatic set theory's descriptive set theory's milieu of fundamental
elements of sorts, called "standard", then though all the matters of
"logical paradox" somehow resolve themselves in "the universe of
logical objects", where "standard theory" sort of ignores there
being a "universe of mathematical objects", for example "universe of sets".
It makes a singularity in a multiplicity theory then they call it "classes".
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It's like "Here's a model of a copy of integer quotients a.k.a. the
rationals.  Illustrate what you need to show they can be partitioned
and what is the cardinal of their possible partitions, under what
ancillary data structures or stipulations, then as with all their consequences."
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As if we're all "naive".
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Heh, "the only construction of real numbers", ..., until there's a bigger one.
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I don't care to be mathematically incorrect and indeed my mathematical conscience
disallows it.  So, if I hadn't written just now how I am correct, then I hope it would
so follow that I would acknowledge such mistake of course, in a world where
there are no accidents, only errors.
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I.e. the usual tack of posters here of ignoring their errors or omissions,
is sort of considered dishonesty by omission.
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It's hind of like Quine define, reals before rationals, but I just have there
already exists a linear continuum, and the field-reals are a usual contrivance.
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So, rationals:  partition them by reals.
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Thanks for writing, I do appreciate your writing and style.
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