Le 23/01/2024 à 14:12, Richard Damon a écrit :
> On 1/23/24 6:29 AM, WM wrote:

>> I do not switch, but you agree to the existence of ℵo unit fractions
>> which cannot be found. You call them infinitesimal. That is nonsense.
>> But you have recognized at least that invisible unit fractions exist.
>>
> I didn't say they could not be found,

The increase from 0 to ℵ cannot be discerned. Or can you distinguish the
first unit fractions? Note that never two sit at the same x.

Regards, WM

Le 23/01/2024 à 14:12, Richard Damon a écrit :
> On 1/23/24 6:32 AM, WM wrote:

>>
>> You don't know all of the numbers individually.
>
> Maybe not, since it is an infinite set, but I can know any of them, and
> know of all of them.

Not those the unit fractions of which lie between 0 and (0, 1], according
to your impression. How would you distinguish them?

Regards, WM

On Tuesday, January 23, 2024 at 5:12:46 AM UTC-8, Richard Damon wrote:
> On 1/23/24 6:29 AM, WM wrote:
> > Le 23/01/2024 à 05:00, Richard Damon a écrit :
> >> On 1/22/24 5:29 PM, WM wrote:
> >>> Le 22/01/2024 à 13:52, Richard Damon a écrit :
> >>>> ℵo number of unit fraction below any positive number, yes there are
> >>>
> >>> Even below all of your infinitesimals?
> >>>
> >> No, that should below any positive finite number, but then you keep on
> >> switching between excluding the transfinite and allowing them,
> >
> > I do not switch, but you agree to the existence of ℵo unit fractions
> > which cannot be found. You call them infinitesimal. That is nonsense.
> > But you have recognized at least that invisible unit fractions exist.
> >
> > Regards, WM
> >
> I didn't say they could not be found, you just think that is what I said
> as you don't seem to understand logical statement.
>
>
> There is nothing "nonsense" about infinitesimals, they are a fully
> developed number system. Now, your logic system and brain may not
> comprehend them, but that is YOUR problem, not theirs,
>
> Your definition of NUF(x) does not imply that NUF(x) needs to have any
> value other than 0 or Aleph0, and claiming that some unit fractions are
> "missing" where it has a finite, non-zero falue is just incorrect.
>
>
> There is no "smallest" unit fraction, but they are unboundedly small
>
> All Unit Fractions are "Defined" and "Visible" since that holds for all
> Natural Numbers too.
>
> Your logic is presuming the existance of something that doesn't exist,
> and thus has exploded in a sea of contradictions, and creates incorrect
> results.
>
> PERIOD.

Infinitesimals is a bunch of different number systems.

Peano, Dodgson, Aristotle, Veronese and Stolz,
Nelson, Robinso(h)n, all different.

It's kind of like Zdislav Kovaric's old "different kinds of infinity".

Anyways MW is a blockhead and your butting heads makes two.

WM wrote on 1/23/2024 :
> Le 23/01/2024 à 14:12, Richard Damon a écrit :
>> On 1/23/24 6:32 AM, WM wrote:
>
>>>
>>> You don't know all of the numbers individually.
>>
>> Maybe not, since it is an infinite set, but I can know any of them, and
>> know of all of them.
>
> Not those the unit fractions of which lie between 0 and (0, 1], according to
> your impression. How would you distinguish them?

There aren't any unit fractions between zero and the interval
mentioned.

On 1/22/2024 5:29 PM, WM wrote:
> Le 22/01/2024 à 13:52, Richard Damon a écrit :

>> ℵo number of unit fraction below
>> any positive number, yes there are
>
> Even below all of your infinitesimals?

After all final ordinals, not.final ordinals.

ℝ ∋ x > 0
ℕ ∋ mₓ < ⅟x ≤ mₓ⁺¹
⅟ℕₓ ∋ ⅟mₓ⁺¹ ≤ x

∀k ∈ ℕ₁:
∃uₖ ∈ ⅟ℕₓ:
uₖ=⅟mₓ⁺ᵏ ∧
¬∃k₂ ∈ ℕ₁:
k₂≠k ∧
⅟mₓ⁺ᵏ² = uₖ = ⅟mₓ⁺ᵏ

⅟ℕₓ ⇇ ℕ₁ 1.to.1
¬(|⅟ℕₓ| > |ℕ₁|)

∀u ∈ ⅟ℕₓ:
∃kᵤ ∈ ℕ₁:
kᵤ = (⅟u)-mₓ ∧
¬∃u₂ ∈ ⅟ℕₓ:
u₂≠u ∧
(⅟u₂)-mₓ = kᵤ = (⅟u)-mₓ

⅟ℕₓ ⇉ ℕ₁ 1.to.1
¬(|⅟ℕₓ| < |ℕ₁|)

¬(|⅟ℕₓ| > |ℕ₁|)
¬(|⅟ℕₓ| < |ℕ₁|)
|⅟ℕₓ| = |ℕ₁|

Le 23/01/2024 à 19:07, FromTheRafters a écrit :
> WM wrote on 1/23/2024 :
>> Le 23/01/2024 à 14:12, Richard Damon a écrit :
>>> On 1/23/24 6:32 AM, WM wrote:
>>
>>>>
>>>> You don't know all of the numbers individually.
>>>
>>> Maybe not, since it is an infinite set, but I can know any of them, and
>>> know of all of them.
>>
>> Not those the unit fractions of which lie between 0 and (0, 1], according to
>> your impression. How would you distinguish them?
>
> There aren't any unit fractions between zero and the interval
> mentioned.

Of course not. But then there are not ℵo unit fractions smaller than
every x > 0.

Assume the contrary
1. ∀ x ∈ (0, 1]: ∃^oo y ∈ {1/n : n e IN}: y < x
and
2. ∀ y ∈ {1/n : n e IN}: 0 < y .
Then
3. ∀ x ∈ (0, 1]: ∃^oo y ∈ {1/n : n e IN}: 0 < y < x
would follow. But that is wrong because
3'. ∀ x ∈ [0, 1]: ∃^oo y ∈ {1/n : n e IN}: 0 < y < x
is obviously wrong, but [0, 1] has only one point more than (0, 1].

Regards, WM

Le 23/01/2024 à 21:13, Jim Burns a écrit :
> On 1/22/2024 5:29 PM, WM wrote:
>> Le 22/01/2024 à 13:52, Richard Damon a écrit :
>
>>> ℵo number of unit fraction below
>>> any positive number, yes there are
>>
>> Even below all of your infinitesimals?
>
> After all final ordinals, not.final ordinals.
>
>
> ℝ ∋ x > 0
> ℕ ∋ mₓ < ⅟x ≤ mₓ⁺¹
> ⅟ℕₓ ∋ ⅟mₓ⁺¹ ≤ x
>
> ∀k ∈ ℕ₁:
> ∃uₖ ∈ ⅟ℕₓ:
> uₖ=⅟mₓ⁺ᵏ ∧
> ¬∃k₂ ∈ ℕ₁:
> k₂≠k ∧
> ⅟mₓ⁺ᵏ² = uₖ = ⅟mₓ⁺ᵏ
>
> ⅟ℕₓ ⇇ ℕ₁ 1.to.1
> ¬(|⅟ℕₓ| > |ℕ₁|)
>
> ∀u ∈ ⅟ℕₓ:
> ∃kᵤ ∈ ℕ₁:
> kᵤ = (⅟u)-mₓ ∧
> ¬∃u₂ ∈ ⅟ℕₓ:
> u₂≠u ∧
> (⅟u₂)-mₓ = kᵤ = (⅟u)-mₓ
>
> ⅟ℕₓ ⇉ ℕ₁ 1.to.1
> ¬(|⅟ℕₓ| < |ℕ₁|)
>
> ¬(|⅟ℕₓ| > |ℕ₁|)
> ¬(|⅟ℕₓ| < |ℕ₁|)
> |⅟ℕₓ| = |ℕ₁|

Contradiction

There are not ℵo unit fractions smaller than every x > 0.

Assume the contrary
1. ∀ x ∈ (0, 1]: ∃^oo y ∈ {1/n : n e IN}: y < x
and
2. ∀ y ∈ {1/n : n e IN}: 0 < y .
Then
3. ∀ x ∈ (0, 1]: ∃^oo y ∈ {1/n : n e IN}: 0 < y < x
would follow. But that is wrong because
3'. ∀ x ∈ [0, 1]: ∃^oo y ∈ {1/n : n e IN}: 0 < y < x
is obviously wrong, but [0, 1] has only one point more than (0, 1].

Regards, WM

On 1/23/2024 5:12 AM, Richard Damon wrote:
> On 1/23/24 6:32 AM, WM wrote:
>> Le 23/01/2024 à 05:00, Richard Damon a écrit :
>>> On 1/22/24 5:32 PM, WM wrote:
>>
>>>>>> ∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵ.
>>>>>
>>>>> But there is no "last" even temporarily.
>>>>
>>>> There is a last known prime number temporarily.
>>>> Same with the last visible natnumbers.
>>
>>> Again with "known".
>>>
>>> There may be a last expressed d Natural Number, you can't even get
>>> away with "Known" there, as we know all of the numbers exist, even if
>>> we don't know yet which are prime.
>>
>> You don't know all of the numbers individually.
>
> Maybe not, since it is an infinite set, but I can know any of them, and
> know of all of them.
>
>>>
>>> Knowledge of the properties of specific numbers is different then
>>> them having the properites.
>>
>> So it is.
>
> And thus, arguments about known numbers can' be used about the actual
> properties of numbers.
>
>>>
>>> All the Prime Numbers are Prime Nubers, we just can't name all of
>>> them since we don't know which ones they are out of the set of
>>> Natural Numbers.
>>
>> So it is.
>>>
>>> We do know which of the Natural Numbers are Natural Numbers, that is
>>> all of them, and thus all are visible.
>>
>> No. You know about the set, not about the individuals.
>
> I know they all exist and that I can know any one of them. That is enough.
>
> If you logic says you need to individually "know" every member of a set,
> then your logic system can not handle unbouded sets.
>
> And that seems to be where you are, using an inadiquite logic system on
> an unbounded set.

I am wondering if conversing with WM for extended periods of time might
cause some sort of "brain damage"? The parrot is dead! ;^)

On Tuesday, January 23, 2024 at 2:51:39 PM UTC-8, Chris M. Thomasson wrote:
> On 1/23/2024 5:12 AM, Richard Damon wrote:
> > On 1/23/24 6:32 AM, WM wrote:
> >> Le 23/01/2024 à 05:00, Richard Damon a écrit :
> >>> On 1/22/24 5:32 PM, WM wrote:
> >>
> >>>>>> ∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵ.
> >>>>>
> >>>>> But there is no "last" even temporarily.
> >>>>
> >>>> There is a last known prime number temporarily.
> >>>> Same with the last visible natnumbers.
> >>
> >>> Again with "known".
> >>>
> >>> There may be a last expressed d Natural Number, you can't even get
> >>> away with "Known" there, as we know all of the numbers exist, even if
> >>> we don't know yet which are prime.
> >>
> >> You don't know all of the numbers individually.
> >
> > Maybe not, since it is an infinite set, but I can know any of them, and
> > know of all of them.
> >
> >>>
> >>> Knowledge of the properties of specific numbers is different then
> >>> them having the properites.
> >>
> >> So it is.
> >
> > And thus, arguments about known numbers can' be used about the actual
> > properties of numbers.
> >
> >>>
> >>> All the Prime Numbers are Prime Nubers, we just can't name all of
> >>> them since we don't know which ones they are out of the set of
> >>> Natural Numbers.
> >>
> >> So it is.
> >>>
> >>> We do know which of the Natural Numbers are Natural Numbers, that is
> >>> all of them, and thus all are visible.
> >>
> >> No. You know about the set, not about the individuals.
> >
> > I know they all exist and that I can know any one of them. That is enough.
> >
> > If you logic says you need to individually "know" every member of a set,
> > then your logic system can not handle unbouded sets.
> >
> > And that seems to be where you are, using an inadiquite logic system on
> > an unbounded set.
> I am wondering if conversing with WM for extended periods of time might
> cause some sort of "brain damage"? The parrot is dead! ;^)

You mean dain bramage?

Yeah, I draeh of that.

What happens is people keep pounding in simple responses
that do nothing to enlighten the discussion, and only enforce the
dumb, reactionary bits.

Don't get me wrong, it's an exercise in itself,
to even read MW not ingenerously at all,
and point out any fault,
but it's not fixing his related rates problems,
and his asymptotic density problems,
and his nonstandard analysis problems,
and his problems about the rationals being huge

Which are things!

De Santillana describing Galileo the other day, "what he did, is that he
would take all the usual arguments, and state them very well, and indeed superbly.
Then, when he demolished them, it was all the more devastating."

For Virgil, it was "puke parrot".

So, well-order the rules. Ha! They already are.

Sounds like a recalled paper of Caicedo "the reals are projectively well-ordered".
Not sure whether subtle flaw or tastes like chicken.

On 1/23/24 11:47 AM, WM wrote:
> Le 23/01/2024 à 14:12, Richard Damon a écrit :
>> On 1/23/24 6:29 AM, WM wrote:
>
>>> I do not switch, but you agree to the existence of ℵo unit fractions
>>> which cannot be found. You call them infinitesimal. That is nonsense.
>>> But you have recognized at least that invisible unit fractions exist.
>>>
>> I didn't say they could not be found,
>
> The increase from 0 to ℵ cannot be discerned. Or can you distinguish the
> first unit fractions? Note that never two sit at the same x.
>
> Regards, WM
>

Of course you can't distinguish the first unit fraction form zero, as
there isn't a "first" unit fraction from zero!

Let me riddle you a few questions,

Do you accept the principle of Induction?

If so, do you accept that 0 is definable/visible?

(if not, why not)

Do you accept that if the number n is definable/visible, then so will be
n+1?

(if not, what number doesn't this hold for. since this is for a n that
IS definable/visible, you should be able to name it)

If you accept these, then you have to accept that ALL Natural Numbers
are definable/visible by the necessary consequence of these properties.

And thus also the Unit Fractions.

If you don't accept the principle of Induction, you can't be using ZFC,
as the principle of Induction is provable from the axioms of ZFC, and if
you don't have ZFC, how are you getting your Natural Numbers?

Also, if you don't hav ZFC, of course you can't complain that ZFC
doesn't answer some of your questions, as you are assuming ZFC isn't in
your logic system.

This puts a bit of a damper on your arguements.

On 1/23/24 11:55 AM, WM wrote:
> Le 23/01/2024 à 14:12, Richard Damon a écrit :
>> On 1/23/24 6:32 AM, WM wrote:
>
>>>
>>> You don't know all of the numbers individually.
>>
>> Maybe not, since it is an infinite set, but I can know any of them,
>> and know of all of them.
>
> Not those the unit fractions of which lie between 0 and (0, 1],
> according to your impression. How would you distinguish them?
>
> Regards, WM

There are NO unit fractions between 0 and (0,1], how could there be. (0,
1] includes ALL finite numbers x where 0 < x <= 1

So, why do I need to distinguish numbers that just don't exist?

Let me riddle you a few questions,

Do you accept the principle of Induction?

If so, do you accept that 0 is definable/visible?

(if not, why not)

Do you accept that if the number n is definable/visible, then so will be
n+1?

(if not, what number doesn't this hold for. since this is for a n that
IS definable/visible, you should be able to name it)

If you accept these, then you have to accept that ALL Natural Numbers
are definable/visible by the necessary consequence of these properties.

And thus also the Unit Fractions.

If you don't accept the principle of Induction, you can't be using ZFC,
as the principle of Induction is provable from the axioms of ZFC, and if
you don't have ZFC, how are you getting your Natural Numbers?

Also, if you don't hav ZFC, of course you can't complain that ZFC
doesn't answer some of your questions, as you are assuming ZFC isn't in
your logic system.

This puts a bit of a damper on your arguements.

On 1/23/24 5:27 PM, WM wrote:
> Le 23/01/2024 à 19:07, FromTheRafters a écrit :
>> WM wrote on 1/23/2024 :
>>> Le 23/01/2024 à 14:12, Richard Damon a écrit :
>>>> On 1/23/24 6:32 AM, WM wrote:
>>>
>>>>>
>>>>> You don't know all of the numbers individually.
>>>>
>>>> Maybe not, since it is an infinite set, but I can know any of them,
>>>> and know of all of them.
>>>
>>> Not those the unit fractions of which lie between 0 and (0, 1],
>>> according to your impression. How would you distinguish them?
>>
>> There aren't any unit fractions between zero and the interval mentioned.
>
> Of course not. But then there are not ℵo unit fractions smaller than
> every x > 0.
>
> Assume the contrary
> 1. ∀ x ∈ (0, 1]: ∃^oo y ∈ {1/n : n e IN}: y < x
> and
> 2. ∀ y ∈ {1/n : n e IN}: 0 < y .
> Then
> 3. ∀ x ∈ (0, 1]: ∃^oo y ∈ {1/n : n e IN}: 0 < y < x would follow. But
> that is wrong because
> 3'. ∀ x ∈ [0, 1]: ∃^oo y ∈ {1/n : n e IN}: 0 < y < x
> is obviously wrong, but [0, 1] has only one point more than (0, 1].
>
> Regards, WM
>

Nope.

You can't talk about "1" point out of an infinite number.

Both sets (0, 1] and [0, 1] have the same number of points, aleph0

Yes, the "difference" of the sets is just one point, but that doesn't
mean there can't be an infinite number of points less that every number
in (0, 1].

You are just using incorrect logic. It may SEEM logical to you, but it
fails.

Try to actually PROVE that statement with logic that works for unbounded
sets.

Let me riddle you a few questions,

Do you accept the principle of Induction?

If so, do you accept that 0 is definable/visible?

(if not, why not)

Do you accept that if the number n is definable/visible, then so will be
n+1?

(if not, what number doesn't this hold for. since this is for a n that
IS definable/visible, you should be able to name it)

If you accept these, then you have to accept that ALL Natural Numbers
are definable/visible by the necessary consequence of these properties.

And thus also the Unit Fractions.

If you don't accept the principle of Induction, you can't be using ZFC,
as the principle of Induction is provable from the axioms of ZFC, and if
you don't have ZFC, how are you getting your Natural Numbers?

Also, if you don't hav ZFC, of course you can't complain that ZFC
doesn't answer some of your questions, as you are assuming ZFC isn't in
your logic system.

This puts a bit of a damper on your arguements.

On 1/23/2024 7:02 PM, Richard Damon wrote:
> On 1/23/24 11:55 AM, WM wrote:
>> Le 23/01/2024 à 14:12, Richard Damon a écrit :
>>> On 1/23/24 6:32 AM, WM wrote:
>>
>>>>
>>>> You don't know all of the numbers individually.
>>>
>>> Maybe not, since it is an infinite set, but I can know any of them,
>>> and know of all of them.
>>
>> Not those the unit fractions of which lie between 0 and (0, 1],
>> according to your impression. How would you distinguish them?
>>
>> Regards, WM
>
> There are NO unit fractions between 0 and (0,1], how could there be. (0,
> 1] includes ALL finite numbers x where 0 < x <= 1
>
> So, why do I need to distinguish numbers that just don't exist?

I must be misunderstanding you? :

0---->1/2---->1

1/2-->3/4---->1

?

>
>
> Let me riddle you a few questions,
>
> Do you accept the principle of Induction?
>
> If so, do you accept that 0 is definable/visible?
>
> (if not, why not)
>
> Do you accept that if the number n is definable/visible, then so will be
> n+1?
>
> (if not, what number doesn't this hold for. since this is for a n that
> IS definable/visible, you should be able to name it)
>
> If you accept these, then you have to accept that ALL Natural Numbers
> are definable/visible by the necessary consequence of these properties.
>
> And thus also the Unit Fractions.
>
> If you don't accept the principle of Induction, you can't be using ZFC,
> as the principle of Induction is provable from the axioms of ZFC, and if
> you don't have ZFC, how are you getting your Natural Numbers?
>
> Also, if you don't hav ZFC, of course you can't complain that ZFC
> doesn't answer some of your questions, as you are assuming ZFC isn't in
> your logic system.
>
> This puts a bit of a damper on your arguements.

It happens that Chris M. Thomasson formulated :
> On 1/23/2024 7:02 PM, Richard Damon wrote:
>> On 1/23/24 11:55 AM, WM wrote:
>>> Le 23/01/2024 à 14:12, Richard Damon a écrit :
>>>> On 1/23/24 6:32 AM, WM wrote:
>>>
>>>>>
>>>>> You don't know all of the numbers individually.
>>>>
>>>> Maybe not, since it is an infinite set, but I can know any of them, and
>>>> know of all of them.
>>>
>>> Not those the unit fractions of which lie between 0 and (0, 1], according
>>> to your impression. How would you distinguish them?
>>>
>>> Regards, WM
>>
>> There are NO unit fractions between 0 and (0,1], how could there be. (0, 1]
>> includes ALL finite numbers x where 0 < x <= 1
>>
>> So, why do I need to distinguish numbers that just don't exist?
>
> I must be misunderstanding you? :
>
> 0---->1/2---->1
>
> 1/2-->3/4---->1
>
> ?

---->1/2---->1

Zero is *NOT* in the interval.

Le 24/01/2024 à 04:02, Richard Damon a écrit :
> On 1/23/24 5:27 PM, WM wrote:
>> Le 23/01/2024 à 19:07, FromTheRafters a écrit :
>>> WM wrote on 1/23/2024 :
>>>> Le 23/01/2024 à 14:12, Richard Damon a écrit :
>>>>> On 1/23/24 6:32 AM, WM wrote:
>>>>
>>>>>>
>>>>>> You don't know all of the numbers individually.
>>>>>
>>>>> Maybe not, since it is an infinite set, but I can know any of them,
>>>>> and know of all of them.
>>>>
>>>> Not those the unit fractions of which lie between 0 and (0, 1],
>>>> according to your impression. How would you distinguish them?
>>>
>>> There aren't any unit fractions between zero and the interval mentioned.
>>
>> Of course not. But then there are not ℵo unit fractions smaller than
>> every x > 0.
>>
>> Assume the contrary
>> 1. ∀ x ∈ (0, 1]: ∃^oo y ∈ {1/n : n e IN}: y < x
>> and
>> 2. ∀ y ∈ {1/n : n e IN}: 0 < y .
>> Then
>> 3. ∀ x ∈ (0, 1]: ∃^oo y ∈ {1/n : n e IN}: 0 < y < x would follow. But
>> that is wrong because
>> 3'. ∀ x ∈ [0, 1]: ∃^oo y ∈ {1/n : n e IN}: 0 < y < x
>> is obviously wrong, but [0, 1] has only one point more than (0, 1].

> Nope.
>
> You can't talk about "1" point out of an infinite number.

Of course I can unless they are dark. But 0 is not dark.
>
> Both sets (0, 1] and [0, 1] have the same number of points, aleph0

Yes, infinitely many.
>
> Yes, the "difference" of the sets is just one point, but that doesn't
> mean there can't be an infinite number of points less that every number
> in (0, 1].

But it means that one point cannot exorcize an infinity of points which
fit into (0, 1] when 0 is added.
>
> You are just using incorrect logic. It may SEEM logical to you, but it
> fails.

It is logic of mathematics. What fails is ZF.
>
> Try to actually PROVE that statement with logic that works for unbounded
> sets.

There is only one correct logic. It is applied above and works for
unbounded sets. ZF does not work for unbounded sets.

Regards, WM

Le 24/01/2024 à 04:02, Richard Damon a écrit :

> Of course you can't distinguish the first unit fraction form zero, as
> there isn't a "first" unit fraction from zero!

NUF(0) = 0 and NUF(1) = greater. Hence there is a point where the first
unit fraction must sit or where more than one must sit. That kind of logic
cannot be defeated by the fools of matheology.

Regards, WM

On 1/24/2024 2:08 AM, FromTheRafters wrote:
> It happens that Chris M. Thomasson formulated :
>> On 1/23/2024 7:02 PM, Richard Damon wrote:
>>> On 1/23/24 11:55 AM, WM wrote:
>>>> Le 23/01/2024 à 14:12, Richard Damon a écrit :
>>>>> On 1/23/24 6:32 AM, WM wrote:
>>>>
>>>>>>
>>>>>> You don't know all of the numbers individually.
>>>>>
>>>>> Maybe not, since it is an infinite set, but I can know any of them,
>>>>> and know of all of them.
>>>>
>>>> Not those the unit fractions of which lie between 0 and (0, 1],
>>>> according to your impression. How would you distinguish them?
>>>>
>>>> Regards, WM
>>>
>>> There are NO unit fractions between 0 and (0,1], how could there be.
>>> (0, 1] includes ALL finite numbers x where 0 < x <= 1
>>>
>>> So, why do I need to distinguish numbers that just don't exist?
>>
>> I must be misunderstanding you? :
>>
>> 0---->1/2---->1
>>
>> 1/2-->3/4---->1
>>
>> ?
>
>  ---->1/2---->1
>
> Zero is *NOT* in the interval.

Why does WM seem to think there is a smallest unit fraction next to
zero? In my example:

0---->1/2----->1

Well, what about:

0---->1/4----->1/2

WM seems to think that there is a smallest unit fraction right after
zero. I am wondering why... Perhaps, he is a moron?

On 1/23/2024 10:22 PM, Chris M. Thomasson wrote:
> On 1/23/2024 7:02 PM, Richard Damon wrote:
>> On 1/23/24 11:55 AM, WM wrote:

>>> Not those the unit fractions of which
>>> lie between 0 and (0, 1],
>>> according to your impression.
>>> How would you distinguish them?

>> There are NO unit fractions between 0 and (0,1],
>> how could there be.
>> (0, 1] includes ALL
>> finite numbers x where 0 < x <= 1
>> So, why do I need to distinguish numbers that
>> just don't exist?
>
> I must be misunderstanding you? :
> 0---->1/2---->1
> 1/2-->3/4---->1
> ?

It's a nonsense claim, so
"That can't be what is meant"
would be an understandable take.

d is between 0 and (0,1] means
for each x e (0,1]: 0 < d < x

d not.exists
Ax e (0,1]: 0 < d < x
~Ex e (0,1]: 0 < x =< d
Nonsense.

On 1/24/2024 2:35 PM, Chris M. Thomasson wrote:
> On 1/24/2024 2:08 AM, FromTheRafters wrote:

> Why does WM seem to think
> there is a smallest unit fraction next to zero?
> In my example:
> 0---->1/2----->1
> Well, what about:
> 0---->1/4----->1/2
>
> WM seems to think that
> there is a smallest unit fraction right after zero.
> I am wondering why...
> Perhaps, he is a moron?

Perhaps he is the the grip of a mania.

Perhaps, one day, long, long ago,
before he had learned how quantifiers work,
he spotted what he thought was a mistake
on the part of Big Mathematics (imaginary),
and,
ever since then,
he has devoted his life to proving
his obvious genius (imaginary).

I have come to the opinion that
the most important skill taught in math classes,
the most difficult to learn,
the rarest outside of Big Mathematics,
is to say
"Oops. I was wrong."

WM seems to have been out sick
on the day, long, long ago, they taught that.

Chris M. Thomasson schrieb:
> On 1/24/2024 2:08 AM, FromTheRafters wrote:
>> It happens that Chris M. Thomasson formulated :
>>> On 1/23/2024 7:02 PM, Richard Damon wrote:
>>>> On 1/23/24 11:55 AM, WM wrote:
>>>>> Le 23/01/2024 à 14:12, Richard Damon a écrit :
>>>>>> On 1/23/24 6:32 AM, WM wrote:
>>>>>
>>>>>>>
>>>>>>> You don't know all of the numbers individually.
>>>>>>
>>>>>> Maybe not, since it is an infinite set, but I can know any of
>>>>>> them, and know of all of them.
>>>>>
>>>>> Not those the unit fractions of which lie between 0 and (0, 1],
>>>>> according to your impression. How would you distinguish them?
>>>>>
>>>>> Regards, WM
>>>>
>>>> There are NO unit fractions between 0 and (0,1], how could there be.
>>>> (0, 1] includes ALL finite numbers x where 0 < x <= 1
>>>>
>>>> So, why do I need to distinguish numbers that just don't exist?
>>>
>>> I must be misunderstanding you? :
>>>
>>> 0---->1/2---->1
>>>
>>> 1/2-->3/4---->1
>>>
>>> ?
>>
>>   ---->1/2---->1
>>
>> Zero is *NOT* in the interval.
>
> Why does WM seem to think there is a smallest unit fraction next to
> zero?

His "argument" is:

1. the distance of two points on the real axis, representing two
successive unit fractions

d_n := 1/n - 1/(n+1) = 1/(n*(n+1))

is always greater zero:

∀ n∈ℕ : d_n > 0

2. starting at zero the unit fractions are "linear ordered".

So WM's conclusion is: there must be a "first unit fraction".

> WM seems to think that there is a smallest unit fraction right after
> zero. I am wondering why... Perhaps, he is a moron?

That's right. He even can't understand, that for every natural number n
there is a successor n + 1, so for every unit fraction there is a
successor 1/(n+1).

His problem is: He can't deal with infinite sets. So he always
introduces a "cut" - i.e. he divides the infinite set in two parts:
a finite one and a "dark part", consisting of "not nameable (or
distinguishable) numbers".
For the set of unit fractions he uses as described the introduction of a
"first unit fraction" after zero to divide the infinite set of unit
fractions in the "dark part" and the finite part, which starts with the
"first unit fraction".

Shortly spoken: He talks nothing but rubbish.

Dieter Heidorn

On Wednesday, January 24, 2024 at 11:56:30 AM UTC-8, Jim Burns wrote:
> On 1/24/2024 2:35 PM, Chris M. Thomasson wrote:
> > On 1/24/2024 2:08 AM, FromTheRafters wrote:
>
> > Why does WM seem to think
> > there is a smallest unit fraction next to zero?
> > In my example:
> > 0---->1/2----->1
> > Well, what about:
> > 0---->1/4----->1/2
> >
> > WM seems to think that
> > there is a smallest unit fraction right after zero.
> > I am wondering why...
> > Perhaps, he is a moron?
> Perhaps he is the the grip of a mania.
>
> Perhaps, one day, long, long ago,
> before he had learned how quantifiers work,
> he spotted what he thought was a mistake
> on the part of Big Mathematics (imaginary),
> and,
> ever since then,
> he has devoted his life to proving
> his obvious genius (imaginary).
>
> I have come to the opinion that
> the most important skill taught in math classes,
> the most difficult to learn,
> the rarest outside of Big Mathematics,
> is to say
> "Oops. I was wrong."
>
> WM seems to have been out sick
> on the day, long, long ago, they taught that.

If you can't split a line into points,
and can't draw points into a line,
what makes you think that axiomatizing the gaplessness of the complete ordered field
is a good idea? Or is it just wishful thinking, of the contradictory sort?

The Least Upper Bound property of the standard complete ordered field,
is a non-logical/proper axiom.

It sort of works out better that line-reals and signal-reals have field-reals in the middle.

Lots better.

See also Jordan measure and Dirichlet problem, line-reals and signal-reals.

"Reke thine own rede." -- Virgil

On 1/24/2024 12:16 PM, Dieter Heidorn wrote:
> Chris M. Thomasson schrieb:
>> On 1/24/2024 2:08 AM, FromTheRafters wrote:
>>> It happens that Chris M. Thomasson formulated :
>>>> On 1/23/2024 7:02 PM, Richard Damon wrote:
>>>>> On 1/23/24 11:55 AM, WM wrote:
>>>>>> Le 23/01/2024 à 14:12, Richard Damon a écrit :
>>>>>>> On 1/23/24 6:32 AM, WM wrote:
>>>>>>
>>>>>>>>
>>>>>>>> You don't know all of the numbers individually.
>>>>>>>
>>>>>>> Maybe not, since it is an infinite set, but I can know any of
>>>>>>> them, and know of all of them.
>>>>>>
>>>>>> Not those the unit fractions of which lie between 0 and (0, 1],
>>>>>> according to your impression. How would you distinguish them?
>>>>>>
>>>>>> Regards, WM
>>>>>
>>>>> There are NO unit fractions between 0 and (0,1], how could there
>>>>> be. (0, 1] includes ALL finite numbers x where 0 < x <= 1
>>>>>
>>>>> So, why do I need to distinguish numbers that just don't exist?
>>>>
>>>> I must be misunderstanding you? :
>>>>
>>>> 0---->1/2---->1
>>>>
>>>> 1/2-->3/4---->1
>>>>
>>>> ?
>>>
>>>   ---->1/2---->1
>>>
>>> Zero is *NOT* in the interval.
>>
>> Why does WM seem to think there is a smallest unit fraction next to zero?
>
> His "argument" is:
>
> 1. the distance of two points on the real axis, representing two
>    successive unit fractions
>
>    d_n := 1/n - 1/(n+1) = 1/(n*(n+1))
>
>    is always greater zero:
>
>    ∀ n∈ℕ : d_n > 0
>
> 2. starting at zero the unit fractions are "linear ordered".
>
> So WM's conclusion is: there must be a "first unit fraction".
>
>> WM seems to think that there is a smallest unit fraction right after
>> zero. I am wondering why... Perhaps, he is a moron?
>
> That's right. He even can't understand, that for every natural number n
> there is a successor n + 1, so for every unit fraction there is a
> successor 1/(n+1).
>
> His problem is: He can't deal with infinite sets. So he always
> introduces a "cut" - i.e. he divides the infinite set in two parts:
> a finite one and a "dark part", consisting of "not nameable (or
> distinguishable) numbers".
> For the set of unit fractions he uses as described the introduction of a
> "first unit fraction" after zero to divide the infinite set of unit
> fractions in the "dark part" and the finite part, which starts with the
> "first unit fraction".
>
> Shortly spoken: He talks nothing but rubbish.

Agreed. Wrt my "number lines", it would be funny if WM says this is the
smallest:

0---->1/4---->1/2

Then we say, what about:

0---->1/8---->1/4

? Turtles all the way down wrt this "zoom" on a number line...

;^)

On 1/23/2024 5:30 PM, WM wrote:
> Le 23/01/2024 à 21:13, Jim Burns a écrit :
>> On 1/22/2024 5:29 PM, WM wrote:
>>> Le 22/01/2024 à 13:52, Richard Damon a écrit :

>>>> ℵo number of unit fraction below
>>>> any positive number, yes there are
>>>
>>> Even below all of your infinitesimals?
>>
>> After all final ordinals, not.final ordinals.
>>
>>
>> ℝ ∋ x > 0
>> ℕ ∋ mₓ < ⅟x ≤ mₓ⁺¹
>> ⅟ℕₓ ∋ ⅟mₓ⁺¹ ≤ x
>>
>> ∀k ∈ ℕ₁:
>> ∃uₖ ∈ ⅟ℕₓ:
>>   uₖ=⅟mₓ⁺ᵏ  ∧
>> ¬∃k₂ ∈ ℕ₁:
>>    k₂≠k ∧
>>    ⅟mₓ⁺ᵏ² = uₖ = ⅟mₓ⁺ᵏ
>>
>> ⅟ℕₓ ⇇ ℕ₁ 1.to.1
>> ¬(|⅟ℕₓ| > |ℕ₁|)
>>
>> ∀u ∈ ⅟ℕₓ:
>> ∃kᵤ ∈ ℕ₁:
>>   kᵤ = (⅟u)-mₓ  ∧
>> ¬∃u₂ ∈ ⅟ℕₓ:
>>    u₂≠u ∧
>>    (⅟u₂)-mₓ = kᵤ = (⅟u)-mₓ
>>
>> ⅟ℕₓ ⇉ ℕ₁ 1.to.1
>> ¬(|⅟ℕₓ| < |ℕ₁|)
>>
>> ¬(|⅟ℕₓ| > |ℕ₁|)
>> ¬(|⅟ℕₓ| < |ℕ₁|)
>> |⅟ℕₓ| = |ℕ₁|
>
> Contradiction

> There are not
> ℵo unit fractions smaller than every x > 0.
>
> Assume the contrary
> 1. ∀ x ∈ (0, 1]:
> ∃^oo y ∈ {1/n : n e IN}:
> y < x

For each positive point x
infinitely.many unit.fractions are between 0 and x
Yes.
∀x ∈ (0,1]: |⅟ℕₓ| = |ℕ₁|

> and
> 2. ∀ y ∈ {1/n : n e IN}:
> 0 < y .

Yes.

> Then
> 3. ∀ x ∈ (0, 1]:
> ∃^oo y ∈ {1/n : n e IN}:
> 0 < y < x
> would follow.

Yes. It follows.

> But that is wrong because
> 3'. ∀ x ∈ [0, 1]:
> ∃^oo y ∈ {1/n : n e IN}:
> 0 < y < x
> is obviously wrong,

(3') is the conjunction of (3) and (3₀)
| 3₀
| ∃^oo y ∈ {1/n : n e IN}:
| 0 < y < 0

(3₀) is obviously false,
which makes their conjunction false,
whether the other conjunct is true or false.

How would that work,
for false (3)∧(3₀) to make (3) false?
Suppose, instead, that (3) is (1+1=2)
Would false (1+1=2)∧(3₀)
also make (1+1=2) false?

> but [0, 1] has only one point more than (0, 1].

Why does (3₀) false make (3) false?

On 1/24/2024 12:17 PM, Ross Finlayson wrote:
> On Wednesday, January 24, 2024 at 11:56:30 AM UTC-8, Jim Burns wrote:
>> On 1/24/2024 2:35 PM, Chris M. Thomasson wrote:
>>> On 1/24/2024 2:08 AM, FromTheRafters wrote:
>>
>>> Why does WM seem to think
>>> there is a smallest unit fraction next to zero?
>>> In my example:
>>> 0---->1/2----->1
>>> Well, what about:
>>> 0---->1/4----->1/2
>>>
>>> WM seems to think that
>>> there is a smallest unit fraction right after zero.
>>> I am wondering why...
>>> Perhaps, he is a moron?
>> Perhaps he is the the grip of a mania.
>>
>> Perhaps, one day, long, long ago,
>> before he had learned how quantifiers work,
>> he spotted what he thought was a mistake
>> on the part of Big Mathematics (imaginary),
>> and,
>> ever since then,
>> he has devoted his life to proving
>> his obvious genius (imaginary).
>>
>> I have come to the opinion that
>> the most important skill taught in math classes,
>> the most difficult to learn,
>> the rarest outside of Big Mathematics,
>> is to say
>> "Oops. I was wrong."
>>
>> WM seems to have been out sick
>> on the day, long, long ago, they taught that.
>
>
>
>
> If you can't split a line into points,
> and can't draw points into a line,
> what makes you think that axiomatizing the gaplessness of the complete ordered field
> is a good idea? Or is it just wishful thinking, of the contradictory sort?
>
> The Least Upper Bound property of the standard complete ordered field,
> is a non-logical/proper axiom.
>
> It sort of works out better that line-reals and signal-reals have field-reals in the middle.
>
> Lots better.
>
> See also Jordan measure and Dirichlet problem, line-reals and signal-reals.
>
> "Reke thine own rede." -- Virgil

Think of creating a von Koch curve from two n-ary points alone. Well, to
get the mid point:

3-ary here:

p0 = {-1, 1, 0 };
p1 = {1, -.5, 2 };

// get the difference normal for the line itself

dif = p1 - p0;

Here is a mid point:

mid = p0 + dif / 2;

Here is 1/3 along the way:

one_third_point = p0 + dif / 3;

We can plot these points.

On 1/24/2024 3:17 PM, Ross Finlayson wrote:
> On Wednesday, January 24, 2024
> at 11:56:30 AM UTC-8, Jim Burns wrote:

> If you can't split a line into points,
> and can't draw points into a line,

Can't I?
Okay, then I won't do that.

Instead,
I will learn about lines and points by
describing what we mean by "lines and points"
and then
augmenting our description with
claims not.first.false about lines and points.

Some claims we can _see_ are not.first.false
For example, claim Q is not.first.false in
⟨ Q∨¬P P Q ⟩
   1   1 1
   1   0 1
   0   1 0
   1   0 0

Some claim.sequences are only not.first.false.

A finite only.first.false claim.sequence
is only true.

Whatever I can or can't do with lines and points,
I can make claims and see they're not.first.false.

> If you can't split a line into points,
> and can't draw points into a line,
> what makes you think that
> axiomatizing the gaplessness of
> the complete ordered field
> is a good idea?

It is a useful idea.
When I look at how much it's used,
I feel pretty confident of that judgment.

> Or is it just wishful thinking,
> of the contradictory sort?

If there exists anything satisfying our description,
then it is not of the contradictory sort. (theorem)

That is what constructions are for.
A construction doesn't build mathematical objects
like we build skyscrapers.
It proves something exists satisfying a description.

Suppose that we have already agreed somehow that
the power set 𝒫(ℚ) of the rationals
is not of the contradictory sort.

The restriction of 𝒫(ℚ) to edgeless.foresplits of ℚ
is something which satisfies our description of ℝ
(theorem)

In particular, the union of a bounded non.empty set
of edgeless.foresplits is an edgeless.foresplit, and
is a bound of that set, and is less than any other bound.
Edgeless.foresplits have the Least Upper Bound Property.

Thus, accepting the existence of the set of
edgeless foresplits makes necessary accepting
the non.contradictory nature of the complete line,
_even if_ edgelss foresplits are nothing like
what we thought we were talking about.

> The Least Upper Bound property of
> the standard complete ordered field,
> is a non-logical/proper axiom.

The Three Corner Property of a right triangle
is a non.logical/proper axiom,
one which we'll probably need to assert
if we plan to reason about right triangles.

I might be misunderstanding.
Are you objecting to the assertion
"This is what we are talking about"?
If you are, what is your objection?