On 04/03/2024 08:59 AM, FromTheRafters wrote:
> WM presented the following explanation :
>> Le 02/04/2024 à 17:51, Jim Burns a écrit :
>>> On 4/2/2024 3:36 AM, WM wrote:
>>
>>> If your assumption leads to "no bijection",
>>> but there is a bijection,
>>> then your assumption is wrong.
>>
>> My trick proves that there is no bijection.
>> Or could you explain why first bijecting n and n/1 should destroy an
>> existing bijection?
>
> Your 'trick' only fails to demonstrate a bijection. Failing to
> demonstrate a bijection does not mean that there is no bijection, only
> that your 'trick' doesn't work to that end.

The only luck he's going to have is with something
like the Equivalency Function, the Natural/Unit Equivalency Function,
which only exists as the continuum limit of a very least amount
of numerical resources that involve the integer continuum,
and that as a function, isn't a Cartesian function in the usual
sense of being re-orderable, so that Cantor's proofs about the
existence and lack thereof of functions, become refined to
specifically being about Cartesian functions, quite simply.

The definition of function, while it deserves its own theory
altogether as for matters of relation, is among the most fluid
of the mathematical concepts, while that the special and extra-
ordinary formalism like EF, N/U EF, sweep, for matters of functions,
makes for that it's a nice result in axiomatic set theory's usual
descriptive set theory's usual world.

What do you think about that?

On 04/03/2024 07:43 PM, Ross Finlayson wrote:
> On 04/03/2024 08:59 AM, FromTheRafters wrote:
>> WM presented the following explanation :
>>> Le 02/04/2024 à 17:51, Jim Burns a écrit :
>>>> On 4/2/2024 3:36 AM, WM wrote:
>>>
>>>> If your assumption leads to "no bijection",
>>>> but there is a bijection,
>>>> then your assumption is wrong.
>>>
>>> My trick proves that there is no bijection.
>>> Or could you explain why first bijecting n and n/1 should destroy an
>>> existing bijection?
>>
>> Your 'trick' only fails to demonstrate a bijection. Failing to
>> demonstrate a bijection does not mean that there is no bijection, only
>> that your 'trick' doesn't work to that end.
>
> The only luck he's going to have is with something
> like the Equivalency Function, the Natural/Unit Equivalency Function,
> which only exists as the continuum limit of a very least amount
> of numerical resources that involve the integer continuum,
> and that as a function, isn't a Cartesian function in the usual
> sense of being re-orderable, so that Cantor's proofs about the
> existence and lack thereof of functions, become refined to
> specifically being about Cartesian functions, quite simply.
>
> The definition of function, while it deserves its own theory
> altogether as for matters of relation, is among the most fluid
> of the mathematical concepts, while that the special and extra-
> ordinary formalism like EF, N/U EF, sweep, for matters of functions,
> makes for that it's a nice result in axiomatic set theory's usual
> descriptive set theory's usual world.
>
> What do you think about that?
>
>

One might wonder "wouldn't that require some axiom or definition
about functions besides the usual expansion of comprehension of
the definition of functions as being any subset as from the
Cartesian product of two domains the any of the one or many
to one or many?" Yet, it's just sort of demonstrated and
just following the definition of infinite limit as is quite
usual already, then that it also arrives at a model of a
continuous domain [0,1] is also quite a spectacle.

Especially that it furthermore falls out of otherwise
the arguments for uncountability of the complete ordered
field without being contradicted, of course is required.

Le 03/04/2024 à 15:59, FromTheRafters a écrit :
> WM presented the following explanation :
>> Le 02/04/2024 à 17:51, Jim Burns a écrit :
>>> On 4/2/2024 3:36 AM, WM wrote:
>>
>>> If your assumption leads to "no bijection",
>>> but there is a bijection,
>>> then your assumption is wrong.
>>
>> My trick proves that there is no bijection.
>> Or could you explain why first bijecting n and n/1 should destroy an existing
>> bijection?
>
> Your 'trick' only fails to demonstrate a bijection. Failing to
> demonstrate a bijection does not mean that there is no bijection, only
> that your 'trick' doesn't work to that end.

Explain why first bijecting n and n/1 should destroy an existing
bijection!

Regards, WM

Le 03/04/2024 à 23:48, Jim Burns a écrit :
> On 4/3/2024 9:32 AM, WM wrote:
>> Le 02/04/2024 à 17:51, Jim Burns a écrit :
>
>>> If your assumption leads to "no bijection",
>>> but there is a bijection,
>>> then your assumption is wrong.
>>
>> My trick proves that there is no bijection.
>
> See below,
> for each k ∈ ℕ its own iₖ/jₖ ∈ ℚᶠʳᵃᶜ
> for each i/j ∈ ℚᶠʳᵃᶜ its own kᵢⱼ ∈ ℕ
>
> Your trick gives incorrect results.

My trick unveils that there are mor fractions than indices.

> Bijecting n and n/1 do not "destroy"
> bijections between ℕ and ℚᶠʳᵃᶜ

Mapping n/1 in the fractions shows that no bijection is possible.

Regards, WM

On 4/4/24 5:33 AM, WM wrote:
> Le 03/04/2024 à 15:59, FromTheRafters a écrit :
>> WM presented the following explanation :
>>> Le 02/04/2024 à 17:51, Jim Burns a écrit :
>>>> On 4/2/2024 3:36 AM, WM wrote:
>>>
>>>> If your assumption leads to "no bijection",
>>>> but there is a bijection,
>>>> then your assumption is wrong.
>>>
>>> My trick proves that there is no bijection.
>>> Or could you explain why first bijecting n and n/1 should destroy an
>>> existing bijection?
>>
>> Your 'trick' only fails to demonstrate a bijection. Failing to
>> demonstrate a bijection does not mean that there is no bijection, only
>> that your 'trick' doesn't work to that end.
>
> Explain why first bijecting n and n/1 should destroy an existing bijection!
>
> Regards, WM

It doesn't, Bijections are always between two DISTINCT sets, not a set
and a piece of itself thought of as a set.

Not following directions breaks a lot of things.

Le 04/04/2024 à 14:19, Richard Damon a écrit :
> On 4/4/24 5:33 AM, WM wrote:
>> Le 03/04/2024 à 15:59, FromTheRafters a écrit :
>>> WM presented the following explanation :
>>>> Le 02/04/2024 à 17:51, Jim Burns a écrit :
>>>>> On 4/2/2024 3:36 AM, WM wrote:
>>>>
>>>>> If your assumption leads to "no bijection",
>>>>> but there is a bijection,
>>>>> then your assumption is wrong.
>>>>
>>>> My trick proves that there is no bijection.
>>>> Or could you explain why first bijecting n and n/1 should destroy an
>>>> existing bijection?
>>>
>>> Your 'trick' only fails to demonstrate a bijection. Failing to
>>> demonstrate a bijection does not mean that there is no bijection, only
>>> that your 'trick' doesn't work to that end.
>>
>> Explain why first bijecting n and n/1 should destroy an existing bijection!
>>
> It doesn't, Bijections are always between two DISTINCT sets, not a set
> and a piece of itself thought of as a set.
>
"In mathematics, a set A is Dedekind-infinite (named after the German
mathematician Richard Dedekind) if some proper subset B of A is
equinumerous to A. Explicitly, this means that there exists a bijective
function from A onto some proper subset B of A." Wikipedia.

Regards, WM

On 4/4/24 9:07 AM, WM wrote:
> Le 04/04/2024 à 14:19, Richard Damon a écrit :
>> On 4/4/24 5:33 AM, WM wrote:
>>> Le 03/04/2024 à 15:59, FromTheRafters a écrit :
>>>> WM presented the following explanation :
>>>>> Le 02/04/2024 à 17:51, Jim Burns a écrit :
>>>>>> On 4/2/2024 3:36 AM, WM wrote:
>>>>>
>>>>>> If your assumption leads to "no bijection",
>>>>>> but there is a bijection,
>>>>>> then your assumption is wrong.
>>>>>
>>>>> My trick proves that there is no bijection.
>>>>> Or could you explain why first bijecting n and n/1 should destroy
>>>>> an existing bijection?
>>>>
>>>> Your 'trick' only fails to demonstrate a bijection. Failing to
>>>> demonstrate a bijection does not mean that there is no bijection,
>>>> only that your 'trick' doesn't work to that end.
>>>
>>> Explain why first bijecting n and n/1 should destroy an existing
>>> bijection!
>>>
>> It doesn't, Bijections are always between two DISTINCT sets, not a set
>> and a piece of itself thought of as a set.
>>
> "In mathematics, a set A is Dedekind-infinite (named after the German
> mathematician Richard Dedekind) if some proper subset B of A is
> equinumerous to A. Explicitly, this means that there exists a bijective
> function from A onto some proper subset B of A." Wikipedia.
>
> Regards, WM
>

Right, but that "Proper Subset" is considered as an independent item,
not as just pieces of the original set.

You don't seem to understand what a "Set" is.

Looking at the set of numbers: {1, 2, 3, 4, 5, 6, 7, 8, ...}

Just "marking" every other one as:

{ 1, 2*, 3, 4*. 5, 6*, 7, 8*, ...}

The marked numbers are not a Subset (yet) but just a piece of a set.

You can make them a set, and thus a subset by creating the set as:

{2, 4, 6, 8, ...}

Your sloppiness might work in finite sets, and you only seem to think in
finte terms, but it doesn't work for infinte sets, because it causes
these sorts of issues.

By your logic, it would be IMPOSSIBLE for the "subset" to be
equinumerous, as you have shown.

This doesn't mean that statement is false, just that it uses definitions
that differ from what you seem to be willing to use, so you have
excluded yourself from those parts of logic.

The clown WM drivels:

> Bijections are always between two DISTINCT sets,

In finite sets.

> not a set and a piece of itself thought of as a set.

This is even /the condition/ for infinite sets.

On 4/4/2024 5:43 AM, WM wrote:
> Le 03/04/2024 à 23:48, Jim Burns a écrit :

>> Bijecting n and n/1 do not "destroy"
>> bijections between ℕ and ℚᶠʳᵃᶜ
>
> Mapping n/1 in the fractions
> shows that no bijection is possible.

You have assumed that ℕ and ℚᶠʳᵃᶜ are Mückenheim sets.

I define a Mückenheim set such that,
for any set, either
all injections are onto, or
all injections are not.onto.

For a Mückenheim set,
any not.onto.injection implies no onto.injection,
which "destroys" the onto.injection I showed you.

A Mückenheim set is a finiteⁿᵒᵗᐧᵂᴹ set.

For each Mückenheim (finiteⁿᵒᵗᐧᵂᴹ) set M
there is a Mückenheim (finiteⁿᵒᵗᐧᵂᴹ) ordinal ⟦0,m⟧
such that not.exists an injection from ⟦0,m⟧ to M
such that ⟦0,m⟧ ⇉| M
such that ⟦0,m⟧ is bigger than M

Never( not.Mückenheim ⟦0,ζ⟧ < Mückenheim ⟦0,m⟧ )

Each not.Mückenheim ⟦0,ζ⟧ is
an upper.bound of {Mückenheim ⟦0,n⟧}

Never( not.Mückenheim ⟦0,ζ+1⟧ and Mückenheim ⟦0,ζ⟧ )
Never( Mückenheim ⟦0,m⟧ and not.Mückenheim ⟦0,m+1⟧ )

Each Mückenheim ⟦0,m⟧ is
not an upper.bound of {Mückenheim ⟦0,n⟧}

Define ⟦0,ω⟧ as least.upper.bound of {Mückenheim ⟦0,n⟧}
⟦0,ω⟧ := lub{Mückenheim ⟦0,n⟧}

For ⟦0,m⟧ < ⟦0,ω⟧ < ⟦0,ζ⟧
Mückenheim ⟦0,m⟧
not.Mückenheim ⟦0,ζ⟧

| Assume ⟦0,m+1⟧ = ⟦0,ω⟧
| | For ⟦0,m⟧ < ⟦0,ω⟧ < ⟦0,m+2⟧
| Mückenheim ⟦0,m⟧
| not.Mückenheim ⟦0,m+2⟧
| | However,
| only
| Mückenheim ⟦0,m⟧ and Mückenheim ⟦0,m+2⟧
| or
| not.Mückenheim ⟦0,m⟧ and not.Mückenheim ⟦0,m+2⟧
| Contradiction.

Therefore
always ⟦0,m+1⟧ ≠ ⟦0,ω⟧

On 4/3/2024 10:13 PM, Ross Finlayson wrote:
> On 04/03/2024 06:13 AM, Jim Burns wrote:

>> [...]
>
> Iota-values:
> the word "iota" means "smallest non-zero value".

That which you use iota to describe
is not the continuum.

....and you've been told this before,
but you think that those telling you are wrong.

I think the reason that you wrongly think we're wrong
is that what you think a limit is
is not what a limit is.

Consider an example we use when we explain
why an arc.length.integral is ∫ √​̅1​̅+​̅f​̅′​̅²​̅(​̅x​̅) 𝑑x
AKA
a proof that π = 4

Consider the n×n grid of points in [0,1]×[0,1]
and the unit.circle in [0,1]×[0,1]
x² + f²(x) = 1

The horizontal and vertical grid.to.grid segments
which intersect the unit circle form
a continuous but not.differentiable curve
from ⟨0,1⟩ to ⟨1,0⟩

The length of
those joined horizontal and vertical segments
is 2

As n ⟶ ∞ the length is 2
That limit is 2

As n ⟶ ∞
for dₙ = the maximum distance between the circle and
those joined horizontal and vertical segments
dₙ ⟶ 0

It's very reasonable to define 'limit' in such a way
that the limit of the horizontal and vertical segments
is the circle.

However,
unless π = 4
the arc.length of the limit (circle) is not
the limit of the arc.lengths (2)

----
> Iota-values:
> the word "iota" means "smallest non-zero value".

That which you use iota to describe
is not the continuum.

The continuum is not simply
points veryveryvery close.

> Real-values:
> all the values between negative infinity and infinity.

There are several ambiguities in that description.

How about instead
Real.values:
least.upper.bounds of
bounded.non.empty.sets of
differences.of.ratios of
ordinals not.fitting.predecessors.

On 04/04/2024 12:01 PM, Jim Burns wrote:
> On 4/3/2024 10:13 PM, Ross Finlayson wrote:
>> On 04/03/2024 06:13 AM, Jim Burns wrote:
>
>>> [...]
>>
>> Iota-values:
>> the word "iota" means "smallest non-zero value".
>
> That which you use iota to describe
> is not the continuum.
>
> ...and you've been told this before,
> but you think that those telling you are wrong.
>
> I think the reason that you wrongly think we're wrong
> is that what you think a limit is
> is not what a limit is.
>
> Consider an example we use when we explain
> why an arc.length.integral is ∫ √​̅1​̅+​̅f​̅′​̅²​̅(​̅x​̅) 𝑑x
> AKA
> a proof that π = 4
>
> Consider the n×n grid of points in [0,1]×[0,1]
> and the unit.circle in [0,1]×[0,1]
> x² + f²(x) = 1
>
> The horizontal and vertical grid.to.grid segments
> which intersect the unit circle form
> a continuous but not.differentiable curve
> from ⟨0,1⟩ to ⟨1,0⟩
>
> The length of
> those joined horizontal and vertical segments
> is 2
>
> As n ⟶ ∞ the length is 2
> That limit is 2
>
> As n ⟶ ∞
> for dₙ = the maximum distance between the circle and
> those joined horizontal and vertical segments
> dₙ ⟶ 0
>
> It's very reasonable to define 'limit' in such a way
> that the limit of the horizontal and vertical segments
> is the circle.
>
> However,
> unless π = 4
> the arc.length of the limit (circle) is not
> the limit of the arc.lengths (2)
>
> ----
>> Iota-values:
>> the word "iota" means "smallest non-zero value".
>
> That which you use iota to describe
> is not the continuum.
>
> The continuum is not simply
> points veryveryvery close.
>
>> Real-values:
>> all the values between negative infinity and infinity.
>
> There are several ambiguities in that description.
>
> How about instead
> Real.values:
> least.upper.bounds of
> bounded.non.empty.sets of
> differences.of.ratios of
> ordinals not.fitting.predecessors.
>
>

It's simple that the continuum limit, a limit of functions,
as we used to say modeling a function as a limit of a family
of functions, like for Dirac delta, these days it's often
called a "generalized distribution", such a function, with
its real analytical character, here the Equivalency Function
is unlike the Dirac delta in that it's not just an infinite
spike at the origin only under which is area one, while,
it is integrable, which is particularly unique for a function
from a discrete domain, and under it is area one, thus that
it's a generalized distribution if you will, while also it's
a continuum limit with the usual meaning of the words.

Then, that its range has extent, density, completeness, measure,
particularly completeness and measure, in [0,1], establishes
its range is a continuous domain, that instead of one or
the other of line-reals or field-reals, there are both.

About the line integral, and the catenary, and the quadrature,
and the small-angle approximation and about the quadrature,
or the arc.length.integral as you put it the line integral,
it's sort of contrived and there are issues with it.

The continuum, between the Integer Continuum and the Long-Line
Continuum, the Linear Continuum, is, any model of a continuous
domain, as so connecting the integers or other bounded regions.

It _is_, complete, it's gapless, as you may put it there are
no jumps. Then iota-values, live in real values, where the
definition of real-values, is the linear continuum's points, each.

The continuum is not necessarily points dense, in the numbers,
it is points as numbers complete, in the numbers.

Recall also that when we were talking about this years ago
on sci.logic, there was R^bar and R^dots and R^hat and R^umlaut,
R^crown, one of which is our usual R the complete ordered field,
and the others are other models of continuous domains, including,
the commingling of the models of continuous domains.

Then it was rather simplified "line-reals, field-reals, signal-reals".

The arc.length.integral is very closely associated with the
quadrature, and a usual first exercise in it is to compute
the catenary, which is not a parabola, which is a quadratic.

So, "iota-values" is the perfect way to describe as an infinitude
of them arrayed between zero and one, as of equi-partitioning,
a special and quite unique non-Cartesian function, domain discrete,
range continuous, integrable, its own anti-derivative, a pdf, and CDF.

Of course it helps that axiomless natural deduction arrives at
an axiomless geometry that formalized results, for example, Euclid's.

It's similar with axiomless natural deduction for usual comprehension
in set theory, about ordinals, to result ubiquitous ordinals,
because the theory's strongly platonist and strongly formalist, also.

Warmly, ...

I think it's great that the replete nature of the continuum
has that the limit in the limit of the natural integers
results building a continuous domain at all, then that
the complete ordered field's stipulated axioms of the
least-upper-bound property and measure 1.0, which they
don't otherwise have, seem justified, or in the sense of
axioms, true.

Le 04/04/2024 à 15:22, Richard Damon a écrit :
> On 4/4/24 9:07 AM, WM wrote:

>>> It doesn't, Bijections are always between two DISTINCT sets, not a set
>>> and a piece of itself thought of as a set.
>>>
>> "In mathematics, a set A is Dedekind-infinite (named after the German
>> mathematician Richard Dedekind) if some proper subset B of A is
>> equinumerous to A. Explicitly, this means that there exists a bijective
>> function from A onto some proper subset B of A." Wikipedia.
> Right, but that "Proper Subset" is considered as an independent item,
> not as just pieces of the original set.

Nevertheless it is a piece of the original set.

Regards, WM

Le 04/04/2024 à 17:03, Jim Burns a écrit :

> always ⟦0,m+1⟧ ≠ ⟦0,ω⟧

The difference between ⟦0,m+1⟧ and ⟦0,ω⟧ is how large?
Is it ω for every m? Then what are the ordinals between m and ω? They
are dark.
On the other hand no ordinal fits between ℕ and ω. Dark ordinals reach
till ω.
Agreed?

Regards, WM

On 4/5/24 4:56 AM, WM wrote:
> Le 04/04/2024 à 15:22, Richard Damon a écrit :
>> On 4/4/24 9:07 AM, WM wrote:
>
>>>> It doesn't, Bijections are always between two DISTINCT sets, not a
>>>> set and a piece of itself thought of as a set.
>>>>
>>> "In mathematics, a set A is Dedekind-infinite (named after the German
>>> mathematician Richard Dedekind) if some proper subset B of A is
>>> equinumerous to A. Explicitly, this means that there exists a
>>> bijective function from A onto some proper subset B of A." Wikipedia.
>
>> Right, but that "Proper Subset" is considered as an independent item,
>> not as just pieces of the original set.
>
> Nevertheless it is a piece of the original set.
>
> Regards, WM

No, its ELEMENTS are part of the original set.

The set of Natural Numbers does not have as a member of it, the set of
Even numbers, only all the Even numbers as members of it.

Ross Finlayson submitted this idea :
> On 04/03/2024 08:59 AM, FromTheRafters wrote:
>> WM presented the following explanation :
>>> Le 02/04/2024 à 17:51, Jim Burns a écrit :
>>>> On 4/2/2024 3:36 AM, WM wrote:
>>>
>>>> If your assumption leads to "no bijection",
>>>> but there is a bijection,
>>>> then your assumption is wrong.
>>>
>>> My trick proves that there is no bijection.
>>> Or could you explain why first bijecting n and n/1 should destroy an
>>> existing bijection?
>>
>> Your 'trick' only fails to demonstrate a bijection. Failing to
>> demonstrate a bijection does not mean that there is no bijection, only
>> that your 'trick' doesn't work to that end.
>
> The only luck he's going to have is with something
> like the Equivalency Function, the Natural/Unit Equivalency Function,
> which only exists as the continuum limit of a very least amount
> of numerical resources that involve the integer continuum,
> and that as a function, isn't a Cartesian function in the usual
> sense of being re-orderable, so that Cantor's proofs about the
> existence and lack thereof of functions, become refined to
> specifically being about Cartesian functions, quite simply.
>
> The definition of function, while it deserves its own theory
> altogether as for matters of relation, is among the most fluid
> of the mathematical concepts, while that the special and extra-
> ordinary formalism like EF, N/U EF, sweep, for matters of functions,
> makes for that it's a nice result in axiomatic set theory's usual
> descriptive set theory's usual world.
>
> What do you think about that?

Too deep. He need only realize that the ordering (ranking) is
irrelevant to the bijection. When an ordering that shows you having a
ranking of first, second, third, next, next, etcetera ad infinitum, and
that no element is left out logically by the step by step sequence
being generated, it is a countable set which can be bijected with the
naturals (also having a first second third etc.). Not finding that
sequence or finding some other sequence which doesn't work is not earth
shattering mathematical progress and using steps to make it clear as a
demonstration does not mean steps are necessarily part of the
structure.

WM explained on 4/4/2024 :
> Le 03/04/2024 à 15:59, FromTheRafters a écrit :
>> WM presented the following explanation :
>>> Le 02/04/2024 à 17:51, Jim Burns a écrit :
>>>> On 4/2/2024 3:36 AM, WM wrote:
>>>
>>>> If your assumption leads to "no bijection",
>>>> but there is a bijection,
>>>> then your assumption is wrong.
>>>
>>> My trick proves that there is no bijection.
>>> Or could you explain why first bijecting n and n/1 should destroy an
>>> existing bijection?
>>
>> Your 'trick' only fails to demonstrate a bijection. Failing to demonstrate
>> a bijection does not mean that there is no bijection, only that your
>> 'trick' doesn't work to that end.
>
> Explain why first bijecting n and n/1 should destroy an existing bijection!

You still seem to think that sets change. If you mean 'n' is an element
of the naturals then of course N bijects with the naturals as embedded
in Q. Also, the complement of the naturals over one in Q is the same
size as the proper subset you created. No sets (read also functions)
were destroyed.

On 4/4/2024 11:28 PM, Ross Finlayson wrote:
> On 04/04/2024 12:01 PM, Jim Burns wrote:
>> On 4/3/2024 10:13 PM, Ross Finlayson wrote:

>>> Iota-values:
>>> the word "iota" means "smallest non-zero value".
>>
>> That which you use iota to describe
>> is not the continuum.

>>> Real-values:
>>> all the values between negative infinity and infinity.
>>
>> There are several ambiguities in that description.
>>
>> How about instead
>> Real.values:
>> least.upper.bounds of
>> bounded.non.empty.sets of
>> differences.of.ratios of
>> ordinals not.fitting.predecessors.

> It's simple that the continuum limit, a limit of functions,
> as we used to say modeling a function as a limit of a family
> of functions, like for Dirac delta, these days it's often
> called a "generalized distribution", such a function, with
> its real analytical character, here the Equivalency Function
> is unlike the Dirac delta in that it's not just an infinite
> spike at the origin only under which is area one, while,
> it is integrable, which is particularly unique for a function
> from a discrete domain, and under it is area one, thus that
> it's a generalized distribution if you will, while also it's
> a continuum limit with the usual meaning of the words.
>
> Then, that its range has extent, density, completeness, measure,
> particularly completeness and measure, in [0,1], establishes
> its range is a continuous domain, that instead of one or
> the other of line-reals or field-reals, there are both.

If I recall correctly, your claim is that
lim[n→∞, n∈ℕ] {d/n: 0≤d≤n, d∈ℕ} =
[0,1]ᶜᵒⁿᵗⁱⁿᵘᵘᵐ = {x∈ℝ: 0≤x≤1}

Define
[0,1]ᴿꟳ =
lim[n→∞, n∈ℕ] {d/n: 0≤d≤n, d∈ℕ}

How do we know ⅟√​̅2 ∈ [0,1]ᴿꟳ ?

Using my upthread definition of ℝ
define
[0,1]ⁿᵒᵗᐧᴿꟳ =
{ x = lub bnes ⊆ ℚ: 0≤x≤1}

lub least.upper.bound
bnes bounded.non.empty.set

How we know ⅟√​̅2 ∈ [0,1]ⁿᵒᵗᐧᴿꟳ is that
⅟√​̅2 = lub bnes {p: p² < ⅟2 } ⊆ ℚ

By the upthread definition,
[0,1]ⁿᵒᵗᐧᴿꟳ = [0,1]ᶜᵒⁿᵗⁱⁿᵘᵘᵐ

Define
[0,1]ˡⁱᵐᐧˢᵘᵖ =
⋂{ ⋃{{d/q: 0≤d≤q, d∈ℕ}: q≥n, q∈ℕ}: n∈ℕ}

For each n ∈ ℕ,
each s ∈ ℕ⁺ has a multiple q > n
∀n∈ℕ, ∀s∈ℕ⁺, ∃q∈ℕ: s | q > n

∀n∈ℕ:
⋃{{d/q: 0≤d≤q, d∈ℕ}: q≥n, q∈ℕ} =
{d/s: 0≤(d/s)≤1, (d/s)∈ℚ} =
[0,1]ʳᵃᵗⁱᵒⁿᵃˡ

[0,1]ˡⁱᵐᐧˢᵘᵖ =
⋂{[0,1]ʳᵃᵗⁱᵒⁿᵃˡ: n∈ℕ} =
[0,1]ʳᵃᵗⁱᵒⁿᵃˡ

We know ⅟√​̅2 ∉ [0,1]ʳᵃᵗⁱᵒⁿᵃˡ = [0,1]ˡⁱᵐᐧˢᵘᵖ
Thus, [0,1]ˡⁱᵐᐧˢᵘᵖ ≠ [0,1]ᶜᵒⁿᵗⁱⁿᵘᵘᵐ

However,
[0,1]ˡⁱᵐᐧˢᵘᵖ uses a very.widely.used, very.sensible
definition of a limit of a set.sequence.
https://en.wikipedia.org/wiki/Set-theoretic_limit

Your definition
lim[n→∞, n∈ℕ] {d/n: 0≤d≤n, d∈ℕ}
would need some other definition,
a definition which I have not seen you (RF) give.

What is the definition of limit which you use?
Or, is it instead that
[0,1]ᴿꟳ ≠ [0,1]ᶜᵒⁿᵗⁱⁿᵘᵘᵐ
?

On 4/5/2024 5:06 AM, WM wrote:
> Le 04/04/2024 à 17:03, Jim Burns a écrit :

>> always ⟦0,m+1⟧ ≠ ⟦0,ω⟧

I define a Mückenheim set such that,
for any set, either
all injections are onto, or
all injections are not.onto.

Mückenheim.set M ⟺
∀S: ∀f:M⇉S=f(M) ∨ ∀g:M⇉S≠g(M)

not.Mückenheim.set M ⟺
∃S: ∃f:M⇉S≠f(M) ∧ ∃g:M⇉S=g(M)

Define ⟦0,ω⦆ = lub{Mückenheim ⟦0,m⟧}

> The difference between ⟦0,m+1⟧ and ⟦0,ω⟧ is
> how large?

Larger than any Mückenheim ⟦0,m⟧
thus
not a Mückenheim.set
thus
the same as
between ⟦0,m⟧ and ⟦0,ω⟧ and
between ⟦0,m+2⟧ and ⟦0,ω⟧.

> Is it ω for every m?

∀k,m ∈ ⟦0,ω⦆: k+m ∈ ⟦0,ω⦆
So, yes.

> Then what are the ordinals between m and ω?

They are the elements of ⦅m,ω⦆

> They are dark.

Visibleᵂᴹ or darkᵂᴹ, always ⟦0,m+1⟧ ≠ ⟦0,ω⟧

> On the other hand
> no ordinal fits between ℕ and ω.

ℕ = ⟦0,ω⦆
So, we agree on something.

> Dark ordinals reach till ω.
> Agreed?

⟦0,ξ⟧ which reaches 'til ω both
is a Mückenheim.set and is not a Mückenheim.set.
Agreed?

Am 29.03.2024 um 15:09 schrieb Tom Bola:
> The clown WM drivels:
>
>> The set {1, 2, 3, 4, 5, ..., ω} is infinite, although there is a
>> last element.
>
> One may construct such a set "N-with-omega".

Right. We may define IN' = IN u {ω}, where IN = {1, 2, 3, ...}.

Hint@Mückenheim: ω !e {1, 2, 3, ...}.

>> Under f(x) = 2x we get the image {2, 4, 6, 8, 10, ..., 2ω}.

Indeed!

Hint@Mückenheim: {2, 4, 6, 8, 10, ..., 2ω} = {2n : n e IN} u {2ω}.

Another Hint: {2n : n e IN} c IN and 2ω !e {2n : n e IN}.

> One may also define such a function, which domain contains both sort of ordinals,
> i.e. natural numbers and also a limit ordinal like ω.

Right. Since natural numbers are ordinal numbers (too). Hence we may use
the multiplication defined on ordinals (<= ω) here. This multiplication
will work for all elements in IN' = IN u {ω}.

>> The mapping restricted to the natural numbers shows less evens than naturals.

Errr...

> The above sentence is very idiotic nonsense, [...]

Right.

> You are way too dense for even simplest thinking and math...

Right. Hey, it's Mückenheim!

On 04/05/2024 04:32 AM, Richard Damon wrote:
> On 4/5/24 4:56 AM, WM wrote:
>> Le 04/04/2024 à 15:22, Richard Damon a écrit :
>>> On 4/4/24 9:07 AM, WM wrote:
>>
>>>>> It doesn't, Bijections are always between two DISTINCT sets, not a
>>>>> set and a piece of itself thought of as a set.
>>>>>
>>>> "In mathematics, a set A is Dedekind-infinite (named after the
>>>> German mathematician Richard Dedekind) if some proper subset B of A
>>>> is equinumerous to A. Explicitly, this means that there exists a
>>>> bijective function from A onto some proper subset B of A." Wikipedia.
>>
>>> Right, but that "Proper Subset" is considered as an independent item,
>>> not as just pieces of the original set.
>>
>> Nevertheless it is a piece of the original set.
>>
>> Regards, WM
>
> No, its ELEMENTS are part of the original set.
>
> The set of Natural Numbers does not have as a member of it, the set of
> Even numbers, only all the Even numbers as members of it.

One can contrive a model of the modularity properties of integers,
wherein according to the model of modularity, variously the parts
can have models building them according to being sets: that do
define membership as whatever relation there is, because set theory
has only one relation: and it's Elt, not Members.

Of course that's contrived, and the default most simple model
of integers is pretty simple, and the even integers are a subset
of it.

Set theory has only one relation and it's "Elt", "element-of".
The whole point of it being fundamental is that there's only
one relation, and then usually it's ordinary regular well-founded,
keeping things simple, then models of other things built on that,
implemented in their relations according to set theory's relation,
then labelling those and calling that "descriptive set theory".

Otherwise most usual sorts of objects are more proper in their
own theory where they're primary in their own theory, and for
example, model things of other relations by the relations they have.

Number theory, geometry, part theory, set/class distinction,
part/particle distinction, category theory, type theory, ...,
are a bunch of different theories with different primary objects
logically, with the idea that set theory is so simple that it's
fundamental, non-logically then those in set theory.

Function theory, ....

On 04/05/2024 11:04 AM, Jim Burns wrote:
> On 4/4/2024 11:28 PM, Ross Finlayson wrote:
>> On 04/04/2024 12:01 PM, Jim Burns wrote:
>>> On 4/3/2024 10:13 PM, Ross Finlayson wrote:
>
>>>> Iota-values:
>>>> the word "iota" means "smallest non-zero value".
>>>
>>> That which you use iota to describe
>>> is not the continuum.
>
>>>> Real-values:
>>>> all the values between negative infinity and infinity.
>>>
>>> There are several ambiguities in that description.
>>>
>>> How about instead
>>> Real.values:
>>> least.upper.bounds of
>>> bounded.non.empty.sets of
>>> differences.of.ratios of
>>> ordinals not.fitting.predecessors.
>
>> It's simple that the continuum limit, a limit of functions,
>> as we used to say modeling a function as a limit of a family
>> of functions, like for Dirac delta, these days it's often
>> called a "generalized distribution", such a function, with
>> its real analytical character, here the Equivalency Function
>> is unlike the Dirac delta in that it's not just an infinite
>> spike at the origin only under which is area one, while,
>> it is integrable, which is particularly unique for a function
>> from a discrete domain, and under it is area one, thus that
>> it's a generalized distribution if you will, while also it's
>> a continuum limit with the usual meaning of the words.
>>
>> Then, that its range has extent, density, completeness, measure,
>> particularly completeness and measure, in [0,1], establishes
>> its range is a continuous domain, that instead of one or
>> the other of line-reals or field-reals, there are both.
>
> If I recall correctly, your claim is that
> lim[n→∞, n∈ℕ] {d/n: 0≤d≤n, d∈ℕ} =
> [0,1]ᶜᵒⁿᵗⁱⁿᵘᵘᵐ = {x∈ℝ: 0≤x≤1}
>
> Define
> [0,1]ᴿꟳ =
> lim[n→∞, n∈ℕ] {d/n: 0≤d≤n, d∈ℕ}
>
> How do we know ⅟√​̅2 ∈ [0,1]ᴿꟳ ?
>
>
> Using my upthread definition of ℝ
> define
> [0,1]ⁿᵒᵗᐧᴿꟳ =
> { x = lub bnes ⊆ ℚ: 0≤x≤1}
>
> lub least.upper.bound
> bnes bounded.non.empty.set
>
> How we know ⅟√​̅2 ∈ [0,1]ⁿᵒᵗᐧᴿꟳ is that
> ⅟√​̅2 = lub bnes {p: p² < ⅟2 } ⊆ ℚ
>
> By the upthread definition,
> [0,1]ⁿᵒᵗᐧᴿꟳ = [0,1]ᶜᵒⁿᵗⁱⁿᵘᵘᵐ
>
>
> Define
> [0,1]ˡⁱᵐᐧˢᵘᵖ =
> ⋂{ ⋃{{d/q: 0≤d≤q, d∈ℕ}: q≥n, q∈ℕ}: n∈ℕ}
>
> For each n ∈ ℕ,
> each s ∈ ℕ⁺ has a multiple q > n
> ∀n∈ℕ, ∀s∈ℕ⁺, ∃q∈ℕ: s | q > n
>
> ∀n∈ℕ:
> ⋃{{d/q: 0≤d≤q, d∈ℕ}: q≥n, q∈ℕ} =
> {d/s: 0≤(d/s)≤1, (d/s)∈ℚ} =
> [0,1]ʳᵃᵗⁱᵒⁿᵃˡ
>
> [0,1]ˡⁱᵐᐧˢᵘᵖ =
> ⋂{[0,1]ʳᵃᵗⁱᵒⁿᵃˡ: n∈ℕ} =
> [0,1]ʳᵃᵗⁱᵒⁿᵃˡ
>
> We know ⅟√​̅2 ∉ [0,1]ʳᵃᵗⁱᵒⁿᵃˡ = [0,1]ˡⁱᵐᐧˢᵘᵖ
> Thus, [0,1]ˡⁱᵐᐧˢᵘᵖ ≠ [0,1]ᶜᵒⁿᵗⁱⁿᵘᵘᵐ
>
> However,
> [0,1]ˡⁱᵐᐧˢᵘᵖ uses a very.widely.used, very.sensible
> definition of a limit of a set.sequence.
> https://en.wikipedia.org/wiki/Set-theoretic_limit
>
> Your definition
> lim[n→∞, n∈ℕ] {d/n: 0≤d≤n, d∈ℕ}
> would need some other definition,
> a definition which I have not seen you (RF) give.
>
> What is the definition of limit which you use?
> Or, is it instead that
> [0,1]ᴿꟳ ≠ [0,1]ᶜᵒⁿᵗⁱⁿᵘᵘᵐ
> ?
>
>

I usually write that n/d with n as "numerator" and
d as "denominator".

It's not identified which f(n) is 1/root2, only
that it exists. There's that for each r in [0,1],
exists n s.t. f(n) = r, and f^-1(r) = n, mostly infinite.

There's not much said except that d goes to infinity,
and n goes to d. (Thus that it's not just zero.)

Of course it helps that it falls out of the proofs of
uncountability uncontradicted, not being a real function
and all, not being a Cartesian function, and having
a discrete domain and what's a continuous domain [0,1].

There's f(0) = 0, otherwise it's just that the
infinitely-many values of n, as go to the
not-an-upper-bound d, infinity, have that
lim_n->d (n/d) = 1.

That and the 10,000's posts, and, there's about a
dozen posts between sci.math and sci.logic, which
pretty much establish the needful that ran(f) = [0,1].

Of course, it's pretty direct to summarize it,
it really does fit in a few pages.

Le 05/04/2024 à 12:57, FromTheRafters a écrit :
> WM explained on 4/4/2024 :

>> Explain why first bijecting n and n/1 should destroy an existing bijection!
>
> You still seem to think that sets change. If you mean 'n' is an element
> of the naturals then of course N bijects with the naturals as embedded
> in Q.

Of course. But if someone doubts it, I could directly map the naturals n/1
to the fractions with the result that there is no bijection.

> Also, the complement of the naturals over one in Q is the same
> size as the proper subset you created.

No, that is disproved by the remaining Os.

Regards, WM

On 4/5/2024 10:37 PM, Ross Finlayson wrote:
> On 04/05/2024 11:04 AM, Jim Burns wrote:

>> [...]
>
> I usually write that n/d with
> n as "numerator" and d as "denominator".

Of course you do. I stand corrected.

> It's not identified which f(n) is 1/root2,
> only that it exists.
> There's that for each r in [0,1],
> exists n s.t. f(n) = r,
> and f^-1(r) = n, mostly infinite.

I don't see how ⅟√​̅2 is known to exist in [0,1]ᴿꟳ
from the definition
[0,1]ᴿꟳ =
lim[d→∞, d∈ℕ⁺] lim[n→d, n∈ℕ] n/d =
lim[d→∞, d∈ℕ⁺] {n/d: 0≤n≤d, n∈ℕ}

Or, more generally, what supports a claim that
[0,1]ᴿꟳ = [0,1]ᶜᵒⁿᵗⁱⁿᵘᵘᵐ

I gave an example of what would support
such a claim, from a different definition.
[0,1]ⁿᵒᵗᐧᴿꟳ =
{ x = lub bnes ⊆ ℚ: 0≤x≤1}

lub least.upper.bound
bnes bounded.non.empty.set

I'm not telling you to do it my way, but,
if not my way, then how?

> There's not much said except that
> d goes to infinity, and n goes to d.
> (Thus that it's not just zero.)

I think it's essential to your project
that more be said.

I have inserted what I can, as best I can,
of what you might mean.

For example, for some reason, you really like
to call a connected domain "continuous",
an adjective I expect to see applied to
a function. Not a deal.breaker.

You have the opportunity to correct my insertions
where I have misunderstood you.

I read "goes to" as "ranges up to"
which is why I write
lim[d→∞, d∈ℕ] lim[n→d, n∈ℕ] n/d =
lim[d→∞, d∈ℕ] {n/d: 0≤n≤d, n∈ℕ}

And there's the rub.

I can expand the limit in an often.used way
lim[d→∞, d∈ℕ] {d/n: 0≤n≤d, n∈ℕ} =
⋂[d∈ℕ] ⋃[d′≥d:d′∈ℕ] {n/d′: 0≤n≤d′, n∈ℕ}

However,
⋂[d∈ℕ+] ⋃[d′≥d:d′∈ℕ] {n/d′: 0≤n≤d′, n∈ℕ} =
[0,1]ʳᵃᵗⁱᵒⁿᵃˡ

[0,1]ʳᵃᵗⁱᵒⁿᵃˡ ≠ [0,1]ᶜᵒⁿᵗⁱⁿᵘᵘᵐ
[0,1]ʳᵃᵗⁱᵒⁿᵃˡ ∌ ⅟√​̅2

But
[0,1]ᴿꟳ =
lim[d→∞, d∈ℕ⁺] lim[n→d, n∈ℕ] n/d =
lim[d→∞, d∈ℕ⁺] {n/d: 0≤n≤d, n∈ℕ} =
⋂[d∈ℕ+] ⋃[d′≥d:d′∈ℕ] {n/d′: 0≤n≤d′, n∈ℕ} =
[0,1]ʳᵃᵗⁱᵒⁿᵃˡ

Either
that's how "limit" is defined here and
[0,1]ᴿꟳ ≠ [0,1]ᶜᵒⁿᵗⁱⁿᵘᵘᵐ
or
"limit" is defined some other way and
you should say what that way is.

On 4/6/24 9:26 AM, WM wrote:
> Le 05/04/2024 à 12:57, FromTheRafters a écrit :
>> WM explained on 4/4/2024 :
>
>>> Explain why first bijecting n and n/1 should destroy an existing
>>> bijection!
>>
>> You still seem to think that sets change. If you mean 'n' is an
>> element of the naturals then of course N bijects with the naturals as
>> embedded in Q.
>
> Of course. But if someone doubts it, I could directly map the naturals
> n/1 to the fractions with the result that there is no bijection.

No, not "No Bijection", but that mapping isn't a bijection.

Showing one attempted mapping doesn't form a bijection doesn't show that
no bijection exists when working with infinite sets. You are just stuck
in your finite thinking.

>
>> Also, the complement of the naturals over one in Q is the same size as
>> the proper subset you created.
>
> No, that is disproved by the remaining Os.

Which only shows that this one mapping doesn't work.

And, when you try it within one set, as opposed to between two sets,
that you don't understand how it is supposed to work.

>
> Regards, WM

Le 05/04/2024 à 19:03, Jim Burns a écrit :
> On 4/5/2024 5:06 AM, WM wrote:

>> The difference between ⟦0,m+1⟧ and ⟦0,ω⟧ is
>> how large?
>
>> Is it ω for every m?
>
> ∀k,m ∈ ⟦0,ω⦆: k+m ∈ ⟦0,ω⦆
> So, yes.

Then it is variable, not a fixed number. Actually infinite sets are
constant. Potentially infinite collections are variable.

>> no ordinal fits between ℕ and ω.
>
> ℕ = ⟦0,ω⦆
> So, we agree on something.
>
>> Dark ordinals reach till ω.
>> Agreed?
>
> ⟦0,ξ⟧ which reaches 'til ω both
> is a Mückenheim.set and is not a Mückenheim.set.
> Agreed?

There are no Mückenheim sets. But we can use the ordinal axis as Cantor
has described it
0, 1, 2, 3, ..., ω, ω + 1, ..., ω + k, ..., ω + ω (= ω2), ω2 + 1,
...
and multgiply 0, 1, 2, 3, ..., ω, by 2. What is the fate of the distance
between ℕ and ω? Does it grow to the complete interval ω + 1, ..., ω
+ k, ..., ω + ω?

Regards, WM