Provably unprovable eliminates incompleteness

[peteolcott]

"This sentence is unprovable" can be proven to be unprovable
on the basis that its satisfaction derives a contradiction.

Proof that Wittgenstein is correct about Gödel
https://www.researchgate.net/publication/333907915_Proof_that_Wittgenstein_is_correct_about_Godel

--
Copyright 2019 Pete Olcott All rights reserved

"Great spirits have always encountered violent
opposition from mediocre minds." Albert Einstein

[Mr Flibble]

On 08/08/2019 18:15, peteolcott wrote:
> "This sentence is unprovable" can be proven to be unprovable
> on the basis that its satisfaction derives a contradiction.
>
> Proof that Wittgenstein is correct about Gödel
> https://www.researchgate.net/publication/333907915_Proof_that_Wittgenstein_is_correct_about_Godel

Modulo the Sausage Conjecture of course.

/Flibble

--
"Snakes didn't evolve, instead talking snakes with legs changed into
snakes." - Rick C. Hodgin

“You won’t burn in hell. But be nice anyway.” – Ricky Gervais

“I see Atheists are fighting and killing each other again, over who
doesn’t believe in any God the most. Oh, no..wait.. that never happens.” –
Ricky Gervais

"Suppose it's all true, and you walk up to the pearly gates, and are
confronted by God," Bryne asked on his show The Meaning of Life. "What
will Stephen Fry say to him, her, or it?"
"I'd say, bone cancer in children? What's that about?" Fry replied.
"How dare you? How dare you create a world to which there is such misery
that is not our fault. It's not right, it's utterly, utterly evil."
"Why should I respect a capricious, mean-minded, stupid God who creates a
world that is so full of injustice and pain. That's what I would say."

[Dhu on Gate]

On Thu, 08 Aug 2019 12:15:48 -0500, peteolcott wrote:

> "This sentence is unprovable" can be proven to be unprovable
> on the basis that its satisfaction derives a contradiction.
>
> Proof that Wittgenstein is correct about Gödel
> https://www.researchgate.net/publication/333907915_Proof_that_Wittgenstein_is_correct_about_Godel

Since this is unreadable by peons, I can only speculate,
but provably unprovable sounds like a useless tautology.

Dhu

--
Je suis Canadien. Ce n'est pas Francais ou Anglaise.
C'est une esp`ece de sauvage: ne obliviscaris, vix ea nostra voco;-)

http://babayaga.neotext.ca/PublicKeys/Duncan_Patton_a_Campbell_pubkey.txt

[Stuart Redmann]

Dhu on Gate <camp...@neotext.ca> wrote:
> On Thu, 08 Aug 2019 12:15:48 -0500, peteolcott wrote:
>
>> "This sentence is unprovable" can be proven to be unprovable
>> on the basis that its satisfaction derives a contradiction.
>>
>> Proof that Wittgenstein is correct about Gödel
>> https://www.researchgate.net/publication/333907915_Proof_that_Wittgenstein_is_correct_about_Godel
>
>
> Since this is unreadable by peons, I can only speculate,
> but provably unprovable sounds like a useless tautology.
>
> Dhu
>

It isn't. It rather shows that - like the speed of light in physics - there
are some rather unexpected limitations to what is possible in mathematics.
For many years mathematicians believed that, given a set of axioms, every
possible theorem in the given field can be proven to be either true or
false based on logical deduction from these axioms.

Goedel showed that this is not always the case (see the continuum
hypothesis). This implies that there are things in mathematics that cannot
be proven. This culminated in Goedels incompleteness theorem, which is
considered by many mathematicians as a severe throwback in maths.

Regards,
Stuart

[peteolcott]

On 9/8/2019 1:06 AM, Dhu on Gate wrote:
> On Thu, 08 Aug 2019 12:15:48 -0500, peteolcott wrote:
>
>> "This sentence is unprovable" can be proven to be unprovable
>> on the basis that its satisfaction derives a contradiction.
>>
>> Proof that Wittgenstein is correct about Gödel
>> https://www.researchgate.net/publication/333907915_Proof_that_Wittgenstein_is_correct_about_Godel
>
>
> Since this is unreadable by peons, I can only speculate,
> but provably unprovable sounds like a useless tautology.
>
> Dhu
>

The entire body of conceptual knowledge is specified as relations
between finite strings: {axioms, rules-of-inference, axiom schemata}.

Formal proofs to theorem consequences specify provable and true
concurrently.

[peteolcott]

https://en.wikipedia.org/wiki/Continuum_hypothesis
It turns out that the basis concept of cardinality is ill-formed.
All infinities are the same size.

The number of real numbers is the number of adjacent geometric
points on a number line, countable using integers.

[Keith Thompson]

peteolcott <Here@Home> writes:
> "This sentence is unprovable" can be proven to be unprovable
> on the basis that its satisfaction derives a contradiction.

[...]

This has nothing to do with Prolog, Lisp, or C++, and probably
nothing to do with comp.ai.philosophy. Please modify the
"Newsgroups:" header when posting followups -- or just don't post.

See comp.theory for the background of this discussion (it's probably
more than 90% of the content of that newsgroup).

--
Keith Thompson (The_Other_Keith) ks...@mib.org <http://www.ghoti.net/~kst>
Will write code for food.
void Void(void) { Void(); } /* The recursive call of the void */

[Mr Flibble]

On 09/09/2019 16:01, peteolcott wrote:
> All infinities are the same size.

Nonsense. Infinity doesn't have a size by definition.

/Flibble

--

[Louis Valence]

Stuart Redmann <DerT...@web.de> writes:

> Dhu on Gate <camp...@neotext.ca> wrote:

[...]

>> Since this is unreadable by peons, I can only speculate,
>> but provably unprovable sounds like a useless tautology.
>>
>> Dhu
>
> It isn't. It rather shows that - like the speed of light in physics - there
> are some rather unexpected limitations to what is possible in mathematics.
> For many years mathematicians believed that, given a set of axioms, every
> possible theorem in the given field can be proven to be either true or
> false based on logical deduction from these axioms.
>
> Goedel showed that this is not always the case (see the continuum
> hypothesis). This implies that there are things in mathematics that cannot
> be proven. This culminated in Goedels incompleteness theorem, which is
> considered by many mathematicians as a severe throwback in maths.

Despite my very limited understanding of logic, mathematics and
everything else, I can't agree with you. There are some limitations to
what is possible in formal logic; the same cannot today be said about
mathematics. There are things in some formal logic systems that cannot
be proven, but AFAIK there is no known result to the effect of what
mathematics can or cannot prove.

Peter Smith in ``An Introduction to Godel's Theorems (ISBN:
978-0-511-35096-2)'', on section 1.2, page, writes:

--8<---------------cut here---------------start------------->8---
Suppose we try to specify a suitable axiomatic theory T that seems to
capture the structure of the natural number sequence and pin down
addition and multiplication (and maybe a lot more besides). Then Godel
gives us a recipe for coming up with a corresponding sentence G(T),
couched in the language of basic arithmetic, such that (i) we can show
(on very modest assumptions, e.g. that T is consistent) that neither
G(T) nor ~G(T) can be derived in T, and yet (ii) we can also recognize
that, at least if T is consistent, G(T) will be true.
--8<---------------cut here---------------end--------------->8---

So, Godel's recipe itself shows T is incomplete, meaning T cannot prove
(as true or false) everything that can be stated in T. One could think
that, therefore, we don't know whether the statement G(T) is true or
false. Peter Smith adds that we can also recognize that, assuming T is
consistent, then G(T) will be true. This /recognition/ is not backed up
by T because T only asserts what it can --- and T cannot assert G(T).

We may say we have a precise definition of a system of logic is, but we
cannot say we have a precise definition of what mathematics is. What we
can say is that if mathematics can be perfectly equated to logic, then
we don't have any idea of what this system of logic would be to be
equated to what we think mathematics is.

[peteolcott]

On 9/9/2019 3:38 PM, Mr Flibble wrote:
> On 09/09/2019 16:01, peteolcott wrote:
>> All infinities are the same size.
>
> Nonsense. Infinity doesn't have a size by definition.
>
> /Flibble
>

Congratulations your answer is more precisely accurate than mine.
I did not provide the more precise answer because it would have
been to much for the target audience to handle.

So Cantor was not just wrong about differing cardinality
he was dead wrong.

One can count all of the (otherwise uncountable) real numbers by
merely counting all of the adjacent geometric points on a number line.

[David Brown]

On 09/09/2019 17:01, peteolcott wrote:

>
> https://en.wikipedia.org/wiki/Continuum_hypothesis
> It turns out that the basis concept of cardinality is ill-formed.

No, it is not. Cardinality is a mathematical concept with a
mathematical definition. It turns out that we can do useful and
interesting things depending on certain extra features that we can
choose to define, or not to define, but that does not make the
definition ill-formed in any sense.

> All infinities are the same size.

Absolute drivel.

>
> The number of real numbers is the number of adjacent geometric
> points on a number line, countable using integers.
>

The cardinality of the set of real numbers is /easily/ shown to be
bigger than the cardinality of the integers (countable infinity).
(Well, /easily/ if you know that kind of mathematics.)

Since you posted a link to the Wikipedia article on the Continuum
Hypothesis, did you actually attempt to read it? The second line sums
up the hypothesis:

"""
There is no set whose cardinality is strictly between that of the
integers and the real numbers.
"""

The implication from this is that you have at least two different sizes
of infinity, the cardinality of integers (aleph_0) and the cardinality
of real numbers (2 ^ aleph_0). Whether you accept the continuum
hypothesis or not, you can't get past that sentence and still think
those two cardinalities are the same size.

[peteolcott]

On 9/10/2019 2:01 AM, David Brown wrote:
> On 09/09/2019 17:01, peteolcott wrote:
>
>>
>> https://en.wikipedia.org/wiki/Continuum_hypothesis
>> It turns out that the basis concept of cardinality is ill-formed.
>
> No, it is not. Cardinality is a mathematical concept with a
> mathematical definition. It turns out that we can do useful and
> interesting things depending on certain extra features that we can
> choose to define, or not to define, but that does not make the
> definition ill-formed in any sense.
>
>> All infinities are the same size.
>
> Absolute drivel.
>
>>
>> The number of real numbers is the number of adjacent geometric
>> points on a number line, countable using integers.
>>
>
> The cardinality of the set of real numbers is /easily/ shown to be
> bigger than the cardinality of the integers (countable infinity).
> (Well, /easily/ if you know that kind of mathematics.)
>

Yes I am aware of diagonal argument.
You did not pay attention that I just refuted this:

>> The number of real numbers is the number of adjacent geometric
>> points on a number line, countable using integers.

The next real number after 3.0 is the geometric point on the number
line that is immediately adjacent to 3.0 on its right side. This geometric
point is an infinitesimally larger than 3.0: encoded as the first point in
the interval: (3,4]

https://en.wikipedia.org/wiki/Bracket_(mathematics)

> Since you posted a link to the Wikipedia article on the Continuum
> Hypothesis, did you actually attempt to read it? The second line sums
> up the hypothesis:

If the concept of cardinality is bogus, then any theorem based on
a bogus concept is also bogus.

>
> """
> There is no set whose cardinality is strictly between that of the
> integers and the real numbers.
> """
>
> The implication from this is that you have at least two different sizes
> of infinity, the cardinality of integers (aleph_0) and the cardinality
> of real numbers (2 ^ aleph_0). Whether you accept the continuum
> hypothesis or not, you can't get past that sentence and still think
> those two cardinalities are the same size.
>

[Reinhardt Behm]

On 9/10/19 11:38 PM, peteolcott wrote:
> The next real number after 3.0 is the geometric point on the number
> line that is immediately adjacent to 3.0 on its right side. This geometric
> point is an infinitesimally larger than 3.0: encoded as the first point in
> the interval: (3,4]

And between 3.0 and your "next" number (xnext) to the right is another
real number: (3.0+xnext) / 2.

[peteolcott]

I specified a whole Infinitesimal number system:
Real_Part[Infinitesimal_Part] where the Infinitesimal_Part is the number
of geometric points offset from the real part.

0.0[1,∞] specifies the entire set of geometric points with a value
greater than zero on the number line which self-evidently corresponds
to the entire set of positive reals.

Notice that this entire set is countable using the set of positive integers.

[Reinhardt Behm]

On 9/12/19 1:04 AM, peteolcott wrote:
> On 9/11/2019 2:31 AM, Reinhardt Behm wrote:
>> On 9/10/19 11:38 PM, peteolcott wrote:
>>> The next real number after 3.0 is the geometric point on the number
>>> line that is immediately adjacent to 3.0 on its right side. This
>>> geometric
>>> point is an infinitesimally larger than 3.0: encoded as the first
>>> point in
>>> the interval: (3,4]
>>
>> And between 3.0 and your "next" number (xnext) to the right is another
>> real number: (3.0+xnext) / 2.
>>
>
>
> I specified a whole Infinitesimal number system:
> Real_Part[Infinitesimal_Part] where the Infinitesimal_Part is the number
> of geometric points offset from the real part.
>
> 0.0[1,∞] specifies the entire set of geometric points with a value
> greater than zero on the number line which self-evidently corresponds
> to the entire set of positive reals.
>
> Notice that this entire set is countable using the set of positive
> integers.

If they are countable you can always list them in the counted order.
With my above example I can always construct numbers that are not in
your counted set. This disproves that you have counted them all. It
disproves that they are countable at all.

[James Kuyper]

On Wednesday, September 11, 2019 at 1:04:36 PM UTC-4, peteolcott wrote:
...

> I specified a whole Infinitesimal number system:

I see no such specification in this thread.

The term "whole infinitesimal number system" is made up of words that I
understand very well, but in combination, they don't mean anything to
me. They don't mean anything to Wikipedia or Google, either. The only
hit I got with either search engine was your use of that term in this
very discussion. Congratulations: you used a phrase that was a
googlenope until you used it.

> Real_Part[Infinitesimal_Part] where the Infinitesimal_Part is the number
> of geometric points offset from the real part.

That notation is unfamiliar to me as well, and that definition seems
meaningless as well. There's an infinite number of geometric points on
the number line between any two distinct real numbers. But I presume
that you could define an alternative mathematics where that isn't true,
and perhaps that notation is meant to be interpreted within the context
of some such alternative system?

> 0.0[1,∞] specifies the entire set of geometric points with a value
> greater than zero on the number line which self-evidently corresponds
> to the entire set of positive reals.

It's not particularly self-evident to me. That the set you're referring
to can be described in that fashion seems self-evidently to imply that
the set you're referring to is NOT the entire set of positive reals.
However, it's entirely possible that I've misunderstood something, so
I'll ask you a few questions about your notation.

1. Does 0.0[n] denote a real number for all positive integral values of
n?
2. What is the precise numerical value of 0.0[1]? Please use some more
conventional notation, such as a decimal expansion or an algebraic
expression to answer that question.
3. Just in case the answer to 2 is too esoteric for me to understand,
is it true that 0.0 < 0.0[1]?
Assuming that the answers to 1 and 3 are "yes", is there any
non-negative integral value of n such that 0.0[n] = (0.0 + 0.0[1])/2?
In particular, is true for either n=0 or n=1?

If 0.0[1] denotes a real number distinct from 0.0, then (0.0 + 0.0[1])/2
must denote another real number, distinct from both 0 and 0.0[1]. I
can't imagine how you could disagree with that statement, but I also
don't understand how you could make the claims you've made in that
message unless you did disagree with it. If you do disagree, please
explain.

> Notice that this entire set is countable using the set of positive integers.

Regardless of what your notation actually means, by construction the set
you describe is countable. My only objection is that it doesn't seem to
possess an important characteristic of the set of real numbers, one that is
shared by the set of rational numbers, which is countable: between any pair
of distinct points in the set, there's an infinite number of other distinct
points that are also in the set.

[peteolcott]

You can construct a number that is not on any point of the number line?
In other words you can construct a number THAT IS NOT A NUMBER.
That would be a (categorically impossible) neat trick.

> This disproves that you have counted them all. It disproves that they are countable at all.
>

[Keith Thompson]

Reinhardt Behm <rb...@hushmail.com> writes:
> On 9/12/19 1:04 AM, peteolcott wrote:

[...]

>> Notice that this entire set is countable using the set of positive
>> integers.
>
> If they are countable you can always list them in the counted order.

[...]

Reinhardt, let me urge you to take a look at comp.theory before
deciding whether it's worth your time to discuss anything with
Pete Olcott.

Either way, please drop comp.lang.c++ from the newsgroups list,
and consider whether the readers of comp.ai.philosophy want to read
about this.

[James Kuyper]

On Thursday, September 12, 2019 at 12:51:50 AM UTC-4, peteolcott wrote:
> On 9/11/2019 9:20 PM, Reinhardt Behm wrote:
> > On 9/12/19 1:04 AM, peteolcott wrote:
> >> On 9/11/2019 2:31 AM, Reinhardt Behm wrote:
> >>> On 9/10/19 11:38 PM, peteolcott wrote:
> >>>> The next real number after 3.0 is the geometric point on the number
> >>>> line that is immediately adjacent to 3.0 on its right side. This geometric
> >>>> point is an infinitesimally larger than 3.0: encoded as the first point in
> >>>> the interval: (3,4]
> >>>
> >>> And between 3.0 and your "next" number (xnext) to the right is another real number: (3.0+xnext) / 2.
> >>>
> >>
> >>
> >> I specified a whole Infinitesimal number system:
> >> Real_Part[Infinitesimal_Part] where the Infinitesimal_Part is the number
> >> of geometric points offset from the real part.
> >>
> >> 0.0[1,∞] specifies the entire set of geometric points with a value
> >> greater than zero on the number line which self-evidently corresponds
> >> to the entire set of positive reals.
> >>
> >> Notice that this entire set is countable using the set of positive integers.
> >
> > If they are countable you can always list them in the counted order. With my above example I can always construct numbers that are not in your counted set.
>
> You can construct a number that is not on any point of the number line?

No, he can construct a real number that is on the number line, but is not a member of your set, proving that your set does not contain all of the real numbers on the number line. He only refers to the countable property of your set, which suggests that he might be thinking of Cantor's diagonal argument, but your description of this set implies that the set is ordered: if n < m, then x[n] < x[m]. Is that correct? If so, then constructing such a number is much simpler. Such a set cannot even contain all of the rationals, which is a countable set.

If 0.0[1] is a positive real number distinct from 0.0, than (0.0 + 0.0[1])/2 is also a positive real number distinct from both 0.0 and 0.0[1]. If your set includes all positive real numbers, there must be a positive integral value of n such that 0.0[n] == (0.0 + 0.0[1])/2. What is that value?

[Juha Nieminen]

In comp.lang.c++ peteolcott <Here@home> wrote:
> The next real number after 3.0 is the geometric point on the number
> line that is immediately adjacent to 3.0 on its right side. This geometric
> point is an infinitesimally larger than 3.0: encoded as the first point in
> the interval: (3,4]

That would be "the smallest real number larger than 3". Which does not
exist.

Infinitesimals (no matter how you want to define them) are not going
to help you. That number still does not exist, even if you add
infinitesimals to your number system. It doesn't matter how you try
to twist it, it's not going to work. You are not going to make that
number exist. Not with infinitesimals, not with anything.

The set of real numbers is genuinely larger than the set of natural
numbers because there is no bijection between the two sets. (This
is in contrast with, for example, the set of rational numbers, or
the set of algebraic numbers, which are equally large as the set of
natural numbers because there is a bijection between them.)

[peteolcott]

On 9/12/2019 5:07 PM, Juha Nieminen wrote:
> In comp.lang.c++ peteolcott <Here@home> wrote:
>> The next real number after 3.0 is the geometric point on the number
>> line that is immediately adjacent to 3.0 on its right side. This geometric
>> point is an infinitesimally larger than 3.0: encoded as the first point in
>> the interval: (3,4]
>
> That would be "the smallest real number larger than 3". Which does not
> exist.
>
> Infinitesimals (no matter how you want to define them) are not going
> to help you. That number still does not exist, even if you add
> infinitesimals to your number system. It doesn't matter how you try
> to twist it, it's not going to work. You are not going to make that
> number exist. Not with infinitesimals, not with anything.
>

There is a contiguous set of points between (3.0, 4.0].
There is a first point in that interval, let's call it X.

Every point on the number line represents some number therefore the
first point X in the above interval represents a number, it might not
be a real number, or any other conventional named type of number.
X is a number on the number line.

> The set of real numbers is genuinely larger than the set of natural
> numbers because there is no bijection between the two sets. (This
> is in contrast with, for example, the set of rational numbers, or
> the set of algebraic numbers, which are equally large as the set of
> natural numbers because there is a bijection between them.)
>

[Keith Thompson]

peteolcott <Here@Home> writes:
[SNIP]

Pete, will you please consider *not* posting this to comp.lang.c++
and other newsgroups where it's off-topic? You've already taken
over comp.theory, and anyone who's interested can follow you there.
(I've cross-posted and redirected followups to comp.theory.)

Note to other readers: This is my first *and last* attempt to ask
Pete not to post to off-topic newsgroups.

[James Kuyper]

On Thursday, September 12, 2019 at 7:54:09 PM UTC-4, peteolcott wrote:
...

> There is a contiguous set of points between (3.0, 4.0].
> There is a first point in that interval, let's call it X.

The concept that you can identify a first point in that interval
inherently leads to a contradiction. If X is in that interval, then
3.0 < X. Therefore,(3.0+X)/2.0 has a value that is manifestly greater
than 3.0, so it should be in that interval, but is manifestly less than
X, so it should occur earlier in that interval than X.
Unless and until you say something to address that argument, I'll pay
no further attention to you - I've already paid more attention than I
should.

[Juha Nieminen]

In comp.lang.c++ peteolcott <Here@home> wrote:
> There is a contiguous set of points between (3.0, 4.0].
> There is a first point in that interval, let's call it X.

No, there isn't. Such a point doesn't exist.

Learn some number theory, will you?

[peteolcott]

APPARENTLY I AM RIGHT AND YOU ARE WRONG!!!

https://www.encyclopediaofmath.org/index.php/Interval_and_segment

Interval and segment
An interval (open interval) is a set of points on a line lying
between two fixed points and , where and themselves are considered
not to belong to the interval.

THERE IS A FIRST POINT OF THE ABOVE INTERVAL.
THERE MAY NOT BE ANY CONVENTIONALLY NAMED NUMBER TYPE ASSOCIATED WITH THIS POINT.

The fact that [0.0, 1.0) is exactly one geometric point shorter
than [0.0, 1.0] proves that infinitesimal numbers do exist.

[Mr Flibble]

On 13/09/2019 15:35, peteolcott wrote:
> The fact that [0.0, 1.0) is exactly one geometric point shorter
> than [0.0, 1.0] proves that infinitesimal numbers do exist.

Nonsense, there is always a number smaller than an "infinitesimal
number" ergo there is no "smallest" number ergo "infinitesimal number"
is a nonsense concept.

[peteolcott]

On 9/13/2019 9:55 AM, Mr Flibble wrote:
> On 13/09/2019 15:35, peteolcott wrote:
>> The fact that [0.0, 1.0) is exactly one geometric point shorter
>> than [0.0, 1.0] proves that infinitesimal numbers do exist.
>
> Nonsense, there is always a number smaller than an "infinitesimal number" ergo there is no "smallest" number ergo "infinitesimal number" is a nonsense concept.
>
> /Flibble
>

This is merely naysaying without any actual rebuttal.

If there is is always a number smaller than an "infinitesimal number"
then specify a number that is half the size of the difference in length
of the above two sequences of points.

[Mr Flibble]

On 14/09/2019 19:35, peteolcott wrote:
> On 9/13/2019 9:55 AM, Mr Flibble wrote:
>> On 13/09/2019 15:35, peteolcott wrote:
>>> The fact that [0.0, 1.0) is exactly one geometric point shorter
>>> than [0.0, 1.0] proves that infinitesimal numbers do exist.
>>
>> Nonsense, there is always a number smaller than an "infinitesimal
>> number" ergo there is no "smallest" number ergo "infinitesimal number"
>> is a nonsense concept.
>>
>> /Flibble
>>
>
> This is merely naysaying without any actual rebuttal.
>
> If there is is always a number smaller than an "infinitesimal number"
> then specify a number that is half the size of the difference in length
> of the above two sequences of points.

There is always a number which is half the size of the previous number so
my assertion stands: "infinitesimal number" is a nonsense concept.

/Flibble


--
"Snakes didn't evolve, instead talking snakes with legs changed into
snakes." - Rick C. Hodgin

[peteolcott]

On 9/14/2019 4:52 PM, Mr Flibble wrote:
> On 14/09/2019 19:35, peteolcott wrote:
>> On 9/13/2019 9:55 AM, Mr Flibble wrote:
>>> On 13/09/2019 15:35, peteolcott wrote:
>>>> The fact that [0.0, 1.0) is exactly one geometric point shorter
>>>> than [0.0, 1.0] proves that infinitesimal numbers do exist.
>>>
>>> Nonsense, there is always a number smaller than an "infinitesimal number" ergo there is no "smallest" number ergo "infinitesimal number" is a nonsense concept.
>>>
>>> /Flibble
>>>
>>
>> This is merely naysaying without any actual rebuttal.
>>
>> If there is is always a number smaller than an "infinitesimal number"
>> then specify a number that is half the size of the difference in length
>> of the above two sequences of points.
>
> There is always a number which is half the size of the previous number so my assertion stands: "infinitesimal number" is a nonsense concept.
>
> /Flibble
>
>

You reasoning goes like this:
There has never been an X, therefore there never will be an X.

[Mr Flibble]

On 14/09/2019 23:18, peteolcott wrote:
> On 9/14/2019 4:52 PM, Mr Flibble wrote:
>> On 14/09/2019 19:35, peteolcott wrote:
>>> On 9/13/2019 9:55 AM, Mr Flibble wrote:
>>>> On 13/09/2019 15:35, peteolcott wrote:
>>>>> The fact that [0.0, 1.0) is exactly one geometric point shorter
>>>>> than [0.0, 1.0] proves that infinitesimal numbers do exist.
>>>>
>>>> Nonsense, there is always a number smaller than an "infinitesimal
>>>> number" ergo there is no "smallest" number ergo "infinitesimal number"
>>>> is a nonsense concept.
>>>>
>>>> /Flibble
>>>>
>>>
>>> This is merely naysaying without any actual rebuttal.
>>>
>>> If there is is always a number smaller than an "infinitesimal number"
>>> then specify a number that is half the size of the difference in length
>>> of the above two sequences of points.
>>
>> There is always a number which is half the size of the previous number
>> so my assertion stands: "infinitesimal number" is a nonsense concept.
>>
>> /Flibble
>>
>>
> You reasoning goes like this:
> There has never been an X, therefore there never will be an X.

Eh? I think you need to take your meds m8.

[peteolcott]

On 9/14/2019 5:30 PM, Mr Flibble wrote:
> On 14/09/2019 23:18, peteolcott wrote:
>> On 9/14/2019 4:52 PM, Mr Flibble wrote:
>>> On 14/09/2019 19:35, peteolcott wrote:
>>>> On 9/13/2019 9:55 AM, Mr Flibble wrote:
>>>>> On 13/09/2019 15:35, peteolcott wrote:
>>>>>> The fact that [0.0, 1.0) is exactly one geometric point shorter
>>>>>> than [0.0, 1.0] proves that infinitesimal numbers do exist.
>>>>>
>>>>> Nonsense, there is always a number smaller than an "infinitesimal number" ergo there is no "smallest" number ergo "infinitesimal number" is a nonsense concept.
>>>>>
>>>>> /Flibble
>>>>>
>>>>
>>>> This is merely naysaying without any actual rebuttal.
>>>>
>>>> If there is is always a number smaller than an "infinitesimal number"
>>>> then specify a number that is half the size of the difference in length
>>>> of the above two sequences of points.
>>>
>>> There is always a number which is half the size of the previous number so my assertion stands: "infinitesimal number" is a nonsense concept.
>>>
>>> /Flibble
>>>
>>>
>> You reasoning goes like this:
>> There has never been an X, therefore there never will be an X.
>
> Eh? I think you need to take your meds m8.
>
> /Flibble
>

Resorting to ad hominem as you did is evidence that I am correct
because people resort to ad hominem when then run out of reasoning.

[Jeff Barnett]

Me thinks that a conjecture about you being off your meds is justified
by the complete, total non sequitur of your response. Try lithium and,
as suggested elsewhere, get your hand out of your pants when responding.
The response you criticized was not ad hominem, it was a concern for
your well being since you seem more and more nuts. Just say thank you.
--
Jeff Barnett

[Juha Nieminen]

In comp.lang.c++ peteolcott <Here@home> wrote:
> On 9/13/2019 3:14 AM, Juha Nieminen wrote:
>> In comp.lang.c++ peteolcott <Here@home> wrote:
>>> There is a contiguous set of points between (3.0, 4.0].
>>> There is a first point in that interval, let's call it X.
>>
>> No, there isn't. Such a point doesn't exist.
>>
>> Learn some number theory, will you?
>>
>
> APPARENTLY I AM RIGHT AND YOU ARE WRONG!!!
>
> https://www.encyclopediaofmath.org/index.php/Interval_and_segment

You are right and I'm wrong, even though that website does not
say what you are saying?

It doesn't matter how many arguments you present or how much you
shout, it will not make "the smallest real number larger than 3"
to start to exist.

In fact, even if you limit yourself to rational numbers, it still
doesn't exist. There is no "smallest rational number larger than 3".

Infinitesimals, no matter how you define them, do not help here.
Such a number still doesn't exist, even if you include infinitesimals
into your number system.

Removing one number from an interval of real (or even rational) numbers
doesn't make that number exist. You can remove an infinite amount of
numbers from that interval, and it still doesn't make such a number
to exist.

[peteolcott]

On 9/15/2019 3:09 AM, Juha Nieminen wrote:
> In comp.lang.c++ peteolcott <Here@home> wrote:
>> On 9/13/2019 3:14 AM, Juha Nieminen wrote:
>>> In comp.lang.c++ peteolcott <Here@home> wrote:
>>>> There is a contiguous set of points between (3.0, 4.0].
>>>> There is a first point in that interval, let's call it X.
>>>
>>> No, there isn't. Such a point doesn't exist.
>>>
>>> Learn some number theory, will you?
>>>
>>
>> APPARENTLY I AM RIGHT AND YOU ARE WRONG!!!
>>
>> https://www.encyclopediaofmath.org/index.php/Interval_and_segment
>
> You are right and I'm wrong, even though that website does not
> say what you are saying?
>
> It doesn't matter how many arguments you present or how much you
> shout, it will not make "the smallest real number larger than 3"
> to start to exist.
>

The length of this interval is exactly 3.0
(0.0, 3.0]
The smallest real number larger than 0.0 is its first point.

The length of this interval is exactly one geometric point longer than 3.0
[0.0, 3.0]

If we limit the specified points in the intervals to integers then:
(0.0, 3.0] specifies: {1,2,3}
[0.0, 3.0] specifies: {0,1,2,3}
Proving that the above intervals are not the same length.

> In fact, even if you limit yourself to rational numbers, it still
> doesn't exist. There is no "smallest rational number larger than 3".
>
> Infinitesimals, no matter how you define them, do not help here.
> Such a number still doesn't exist, even if you include infinitesimals
> into your number system.
>
> Removing one number from an interval of real (or even rational) numbers
> doesn't make that number exist. You can remove an infinite amount of
> numbers from that interval, and it still doesn't make such a number
> to exist.
>

[Mr Flibble]

On 16/09/2019 15:50, peteolcott wrote:
> On 9/15/2019 3:09 AM, Juha Nieminen wrote:
>> In comp.lang.c++ peteolcott <Here@home> wrote:
>>> On 9/13/2019 3:14 AM, Juha Nieminen wrote:
>>>> In comp.lang.c++ peteolcott <Here@home> wrote:
>>>>> There is a contiguous set of points between (3.0, 4.0].
>>>>> There is a first point in that interval, let's call it X.
>>>>
>>>> No, there isn't. Such a point doesn't exist.
>>>>
>>>> Learn some number theory, will you?
>>>>
>>>
>>> APPARENTLY I AM RIGHT AND YOU ARE WRONG!!!
>>>
>>> https://www.encyclopediaofmath.org/index.php/Interval_and_segment
>>
>> You are right and I'm wrong, even though that website does not
>> say what you are saying?
>>
>> It doesn't matter how many arguments you present or how much you
>> shout, it will not make "the smallest real number larger than 3"
>> to start to exist.
>>
>
> The length of this interval is exactly 3.0
> (0.0, 3.0]
> The smallest real number larger than 0.0 is its first point.

There isn't a smallest real number larger than 0.0; it is impossible for

such a number to exist.

>

> The length of this interval is exactly one geometric point longer than 3.0
> [0.0, 3.0]

Nonsense. 3.0 - 0.0 = 3.0

>
> If we limit the specified points in the intervals to integers then:
> (0.0, 3.0] specifies: {1,2,3}
> [0.0, 3.0] specifies: {0,1,2,3}
> Proving that the above intervals are not the same length.

More nonsense, length and set cardinality are two different things.

It really would be a good idea for you to take your meds, m8.

[peteolcott]

On 9/16/2019 11:13 AM, Mr Flibble wrote:
> On 16/09/2019 15:50, peteolcott wrote:
>> On 9/15/2019 3:09 AM, Juha Nieminen wrote:
>>> In comp.lang.c++ peteolcott <Here@home> wrote:
>>>> On 9/13/2019 3:14 AM, Juha Nieminen wrote:
>>>>> In comp.lang.c++ peteolcott <Here@home> wrote:
>>>>>> There is a contiguous set of points between (3.0, 4.0].
>>>>>> There is a first point in that interval, let's call it X.
>>>>>
>>>>> No, there isn't. Such a point doesn't exist.
>>>>>
>>>>> Learn some number theory, will you?
>>>>>
>>>>
>>>> APPARENTLY I AM RIGHT AND YOU ARE WRONG!!!
>>>>
>>>> https://www.encyclopediaofmath.org/index.php/Interval_and_segment
>>>
>>> You are right and I'm wrong, even though that website does not
>>> say what you are saying?
>>>
>>> It doesn't matter how many arguments you present or how much you
>>> shout, it will not make "the smallest real number larger than 3"
>>> to start to exist.
>>>
>>
>> The length of this interval is exactly 3.0
>> (0.0, 3.0]
>> The smallest real number larger than 0.0 is its first point.
>
> There isn't a smallest real number larger than 0.0; it is impossible for such a number to exist.
>

https://www.encyclopediaofmath.org/index.php/Interval_and_segment

An interval (open interval) is a set of points on a line lying between two fixed

points A and B, where A and B themselves are considered not to belong to the interval.

Since [0, 100) clearly specifies every point on the number line between
0 and 100 except 100 it is specifying the point immediately before 100
as the last point of this interval with no points in-between this point
and 100. THAT IS WHAT IT SAYS.

This contradicts the definition of real numbers that specifies there
is always a real number between every pair of real numbers.

>>
>> The length of this interval is exactly one geometric point longer than 3.0
>> [0.0, 3.0]
>
> Nonsense. 3.0 - 0.0 = 3.0
>
>>
>> If we limit the specified points in the intervals to integers then:
>> (0.0, 3.0] specifies: {1,2,3}
>> [0.0, 3.0] specifies: {0,1,2,3}
>> Proving that the above intervals are not the same length.
>
> More nonsense, length and set cardinality are two different things.
>
> It really would be a good idea for you to take your meds, m8.
>
> /Flibble
>


--

[Mr Flibble]

Points have a "width" of zero, dear. Please take your meds.

[peteolcott]

The relation between real numbers and points on a number line is defined inconsistently.

>> An interval (open interval) is a set of points on a line lying between two fixed
>> points A and B, where A and B themselves are considered not to belong to the interval.

The above stipulates all of the points between A and B besides A and B thus
the first point AFTER A (with no points in-between) and the first point
BEFORE B (with no points in-between) ARE SPECIFIED in this interval.

[Mr Flibble]

That definition is entirely consistent with points having a "width" of
zero, dear. Please take your meds so you can spare us any more demented posts.

[peteolcott]

(A) An open interval specifies its first point as immediately after a
specified point with no points in-between.

(B) The definition of real numbers says there are always points in-between.

(A) contradicts (B).

[Mr Flibble]

(A) is erroneous, you made "immediately after" and "no points in-between"
up based on nothing but your demented thoughts and reasoning. Take your meds.

[peteolcott]

You aren't bright enough to actually follow the reasoning.
You can only spout off what you learned by rote.

A number line has an infinite set of contiguous ascending points.
When you remove the very first point of an interval [0, 1] you are
left with an interval beginning with the very next point (0, 1].

Even if every other aspect of mathematics disagrees, none-the-less the
beginning of this interval: (0, 1] stipulates the point immediately after 0.
If every other aspect of mathematics says no such point exists,
none-the-less it is defined to exist by the definition of open interval.

Learned-by-rote people are not bright enough to discern new innovations.
To them any new idea is always incorrect because it is unconventional.
They simply are not bright enough to process reasoning instead they
merely look up the "facts" of conventional wisdom.

[Mr Flibble]

This is the most perfect example of projection that I have ever seen.

[peteolcott]

My position is diametrically opposed to the learned-by-rote position
thus proving that I did not learn it by rote.

[peteolcott]

You would have to double your wits to become a halfwit.

[James Kuyper]

On Monday, September 16, 2019 at 3:48:38 PM UTC-4, peteolcott wrote:
...

> Even if every other aspect of mathematics disagrees, none-the-less the
> beginning of this interval: (0, 1] stipulates the point immediately after 0.
> If every other aspect of mathematics says no such point exists,
> none-the-less it is defined to exist by the definition of open interval.

No, the definition of an open interval is explicitly based upon the
absence of such a point. See
<https://en.wikipedia.org/wiki/Interval_(mathematics)#Terminology>:

123456789012345678901234567890123456789012345678901234567890123456789012
"An interval is said to be left-open if and only if it contains no
minimum (an element that is smaller than all other elements); right-open
if it contains no maximum; and open if it has both properties. The
interval [0,1) = {x | 0 ≤ x < 1}, for example, is left-closed and right-
open. The empty set and the set of all reals are open intervals, while
the set of non-negative reals, for example, is a right-open but not
left-open interval."

[guinne...@gmail.com]

If you omitted those last three words you would be correct.

[Juha Nieminen]

In comp.lang.c++ peteolcott <Here@home> wrote:
> The length of this interval is exactly 3.0
> (0.0, 3.0]
> The smallest real number larger than 0.0 is its first point.

You can repeat that as many times as you want, but that will not make

such a number to exist.

There is no "smallest real number larger than 0". Learn some number
theory, will you?

[Juha Nieminen]

In comp.lang.c++ peteolcott <Here@home> wrote:
> Even if every other aspect of mathematics disagrees, none-the-less the
> beginning of this interval: (0, 1] stipulates the point immediately after 0.

There is no point immediately after 0. You might be incapable of
comprehending such a fact of mathematics, but that doesn't mean
it's incorrect.

An open interval does not mean that there must exist a particular
number at the end of that interval. There is no such number. You cannot
give its value as it's conceptually and mathematically non-existent.
It doesn't exist even in some kind of abstract meta-sense. It doesn't
exist even if you introduce the concept of "infinitesimal" into your
number system (no matter how you define "infinitesimal").

Positing that there exists such a number leads immediately to a
contradiction, and it's extraordinarily simple to prove that.
Since the assumption was that the number exists, but that assumption
leads to a contradiction, that means that the assumption was incorrect.
"Proof by contradiction" is not just some wishy-washy nebulous term.

[James Kuyper]

On 9/16/19 8:54 PM, James Kuyper wrote:
> On Monday, September 16, 2019 at 3:48:38 PM UTC-4, peteolcott wrote:
> ...
>> Even if every other aspect of mathematics disagrees, none-the-less the
>> beginning of this interval: (0, 1] stipulates the point immediately after 0.
>> If every other aspect of mathematics says no such point exists,
>> none-the-less it is defined to exist by the definition of open interval.
>
> No, the definition of an open interval is explicitly based upon the
> absence of such a point. See
> <https://en.wikipedia.org/wiki/Interval_(mathematics)#Terminology>:
>

> "An interval is said to be left-open if and only if it contains no
> minimum (an element that is smaller than all other elements); right-open

> if it contains no maximum; and open if it has both properties. ...
I realized a few additional points after posting that message.

If, as you claim, the interval (0, 1] contains a minimum value (let's
call it iota, for convenience), then it is, by definition (see above)
left-closed, not left-open. Therefore, that interval can be written as
[iota, 1].

Now, since 0 < iota and iota < 1, (0, 1] is the union of
(0, iota) and [iota, 1]. Your claim is equivalent, therefore, to the
claim that (0, iota) is the empty set.

But the Lesbesgue measure of (0, iota) is iota - 0, or simply iota.
Since iota > 0, the interval (0, iota) can't empty. Furthermore, the
Lesbesgue measure of (0, 1] is 1 - 0 = 1, whereas the Lesbesgue measure
of [iota, 1] is 1 - iota. Since
0 < iota, 1 - iota < 1. Since (0, 1] and [iota, 1] have different
measures, they can't be the same set.


For any point x in (iota, 1), since 0 < iota < x < 1, the value y =
iota*(x - iota)/(1 - iota) is real, greater than 0, and less than iota.
In other words, it is an element of (0, iota). Therefore, the claim that
(0, 1] contains a minimum value leads to a contradiction, because every
value of y is in the interval (0, 1], and is less than iota.

Note that in the above paragraph I was referring to the open interval
(iota, 1), not the closed interval [iota,1] - don't get confused about
the difference between the two intervals. I specified that open interval
because the claims I made about y would not be true for x=iota or x=1.

[peteolcott]

Keith Thompson proves that he fully understands the Infinitesimal
number system:

On 9/16/2019 11:38 PM, Keith Thompson wrote:
> I can imagine that *this* set has a least element. For example
> we can define "iota" as an infinitesimal, something that is not a
> real number but that has a location on the number line, directly
> adjacent to the real number 0. It is greater than 0 and less than
> any positive real number.

He elaborates this understanding much more completely in his
9/16/2019 11:38 PM, reply.

[peteolcott]

On 9/16/2019 11:38 PM, Keith Thompson wrote:

> Arbitrarily inventing a new notation, let _(0,1]_ be a set of, um,
> something or other. It includes all the real numbers in (0,1],
> and it may also include some other things, each of which is greater
> than 0, less than or equal to 1, and somewhere on the "number line"
> between 0 and 1. Some of the non-real elements might be points,
> or infinitesimals. I'll assert that for any two members of *this*
> set, exactly one of a<b, a=b, a>b is true.

>
> I can imagine that *this* set has a least element. For example
> we can define "iota" as an infinitesimal, something that is not a
> real number but that has a location on the number line, directly
> adjacent to the real number 0. It is greater than 0 and less than
> any positive real number.

[Mr Flibble]

On 17/09/2019 17:45, peteolcott wrote:
> On 9/17/2019 3:05 AM, Juha Nieminen wrote:
>> In comp.lang.c++ peteolcott <Here@home> wrote:
>>> The length of this interval is exactly 3.0
>>> (0.0, 3.0]
>>> The smallest real number larger than 0.0 is its first point.
>>
>> You can repeat that as many times as you want, but that will not make
>> such a number to exist.
>>
>> There is no "smallest real number larger than 0". Learn some number
>> theory, will you?
>>
>
> Keith Thompson proves that he fully understands the Infinitesimal
> number system:
>
> On 9/16/2019 11:38 PM, Keith Thompson wrote:
> > I can imagine that *this* set has a least element.  For example
> > we can define "iota" as an infinitesimal, something that is not a
> > real number but that has a location on the number line, directly
> > adjacent to the real number 0.  It is greater than 0 and less than
> > any positive real number.
>
> He elaborates this understanding much more completely in his
> 9/16/2019 11:38 PM, reply.

The fact that this Keith Thompson bloke is also talking bollocks has no
bearing on the fact that you are still hopelessly wrong. Do what someone
else suggested and learn some number theory (and take your meds).

[peteolcott]

On 9/17/2019 12:14 PM, Mr Flibble wrote:
> On 17/09/2019 17:45, peteolcott wrote:
>> On 9/17/2019 3:05 AM, Juha Nieminen wrote:
>>> In comp.lang.c++ peteolcott <Here@home> wrote:
>>>> The length of this interval is exactly 3.0
>>>> (0.0, 3.0]
>>>> The smallest real number larger than 0.0 is its first point.
>>>
>>> You can repeat that as many times as you want, but that will not make
>>> such a number to exist.
>>>
>>> There is no "smallest real number larger than 0". Learn some number
>>> theory, will you?
>>>
>>
>> Keith Thompson proves that he fully understands the Infinitesimal
>> number system:
>>
>> On 9/16/2019 11:38 PM, Keith Thompson wrote:
>>  > I can imagine that *this* set has a least element.  For example
>>  > we can define "iota" as an infinitesimal, something that is not a
>>  > real number but that has a location on the number line, directly
>>  > adjacent to the real number 0.  It is greater than 0 and less than
>>  > any positive real number.
>>
>> He elaborates this understanding much more completely in his
>> 9/16/2019 11:38 PM, reply.
>
> The fact that this Keith Thompson bloke is also talking bollocks has no bearing on the fact that you are still hopelessly wrong. Do what someone else suggested and learn some number theory (and take your meds).
>
> /Flibble
>

Mike Terry also agrees with a key element.
It is not that any of us are wrong. We would be wrong from the limited
perspective of the Real number system, yet are not confined inside the
box of this number system.

When thinking outside the box occurs different number systems that are
apparently more dense than the Real number system can be imagined.

To people that are enclosed in the tight little boxes of conventional
wisdom every new idea that contradicts their learned-by-rote seems to
be an error. Recent research shows that inflexible thinking is due to
abnormalities in the brain that show up on brain scans.

[Mike Terry]

Please do not assume that because PO says somebody or other "agrees"
with him on something, that they actually do agree. And if they do
agree ON SOME SPECIFIC POINT, within SOME QUALIFIED CONTEXT, that does
not mean they "support" PO generally, or that PO has honestly
represented their views with his selective quoting.

For all you know, PO is quoting sentences you've written and presenting
them elsewhere as "/Flibble over in xxxx agrees with me that xxxx", and
people there are saying "Hey, that /Flibble guy doesn't half talk a load
of bollocks!"

Mike.

[Keith Thompson]

peteolcott <Here@Home> writes:
> On 9/17/2019 3:05 AM, Juha Nieminen wrote:
>> In comp.lang.c++ peteolcott <Here@home> wrote:
>>> The length of this interval is exactly 3.0
>>> (0.0, 3.0]
>>> The smallest real number larger than 0.0 is its first point.
>>
>> You can repeat that as many times as you want, but that will not make
>> such a number to exist.
>>
>> There is no "smallest real number larger than 0".

[...]

Agreed.

> Keith Thompson proves that he fully understands the Infinitesimal
> number system:

I do not fully understand infinitesimals, and I have not claimed
that I do.

[...]

--
Keith Thompson (The_Other_Keith) ks...@mib.org <http://www.ghoti.net/~kst>
Will write code for food.
void Void(void) { Void(); } /* The recursive call of the void */

[Mike Terry]

LOL, he's doing it to me now! (I don't think he knows I sometimes
follow these newsgroups)

For the record, I didn't say that PO wasn't wrong. In fact I can't see
ANYTHING in the above paragraphs that matches up with ANYTHING I've said
elsewhere. It's purely PO's words.

PO believes he is an unacknowledged genius [ok, he said recently that
/maybe/ he was only borderline genius - I don't want to give the
impression PO is anything but modest!] but the truth is he's not a
genius - he's actually a "DELUDED DUMBO".

The DUMBO label comes from his inability to follow arguments and
comprehend basic concepts in the fields he spouts forth his nonsense on.
I'm sure he has previously tried to study at least a little CS and
Logic, but he had to give up because it was all just beyond him. His
knowledge of [everything] is just based on reading Wikipedia articles,
without understanding them, and coming away with just a couple of
keywords he can (mis)use. He admits to spending 15 years studying just
two pages from a book (author Linz) covering the proof of the Halting
Problem result. (And of course, he still doesn't get it...)

The DELUDED label probably I don't need to explain further! Just in
case it's not well known here:

- he believes he is an unacknowledged (borderline) genius with powers
beyond those of normal people

- he is going to gain credibility as a logician/computer scientist/??
by refuting a string of key mathematical results that he does not even
understand (Godel Incompleteness theorem, Halting Problem, Rice's
theorem, Tarski's "undefineability of Truth", and lately "refuting
Cardinality")

- Then when he goes back to Doug Lenat (CYC project), Doug will have no
choice but to put him in charge of the architecture for the Cyc upper
ontology layer.

- Then he will somehow become the first person to have created a "human
mind inside a computer".

- Umm, let's not get started on the "God beliefs" (He is God, the
"actual creator of the universe", and he can make other individuals
follow his will, just by wishing for them to do what he wants, and as
God, he has access to a mode of infallible reasoning not available to
others etc.. Hope I've got all that right!)

Seriously, the God stuff is right out there - he even used this as a
defense in a legal case in which he was involved: (Google "peter olcott
god"). Normally I wouldn't bring that to anyone's attention, but hey,
nobody likes cranks misusing their name for their own gain...

Anyway, my suggestion is that if you want PO to keep on posting his OT
rubbish in these newsgroups, keep on arguing with him to prove him
wrong! He loves all that stuff. You certainly won't make him stop by
nicely asking him not to post OT stuff...


Mike.

[peteolcott]

On 9/17/2019 1:10 PM, Keith Thompson wrote:
> peteolcott <Here@Home> writes:
>> On 9/17/2019 3:05 AM, Juha Nieminen wrote:
>>> In comp.lang.c++ peteolcott <Here@home> wrote:
>>>> The length of this interval is exactly 3.0
>>>> (0.0, 3.0]
>>>> The smallest real number larger than 0.0 is its first point.
>>>
>>> You can repeat that as many times as you want, but that will not make
>>> such a number to exist.
>>>
>>> There is no "smallest real number larger than 0".
> [...]
>
> Agreed.
>
>> Keith Thompson proves that he fully understands the Infinitesimal
>> number system:
>
> I do not fully understand infinitesimals, and I have not claimed
> that I do.

This is the key essence of it:

On 9/16/2019 11:38 PM, Keith Thompson wrote:
> we can define "iota" as an infinitesimal, something that is not a
> real number but that has a location on the number line, directly
> adjacent to the real number 0. It is greater than 0 and less than
> any positive real number.

[A, B] - {A} = (A, B] encoded as [A[1], B[0]]
[A[1], B[0]] - {A[1]} = (A[1], B[0]] encoded as [A[2], B[0]]

[Keith Thompson]

Keith Thompson <ks...@mib.org> writes:
> peteolcott <Here@Home> writes:
[SNIP]

My apologies, I didn't notice which newsgroups this was posted to.

[Mr Flibble]

On 17/09/2019 18:36, peteolcott wrote:

You seriously need to take your meds, m8, you are quite mad.

[David Brown]

You can always define extra elements and add them to sets like this.
But you can't do so in a way that is consistent in other ways. So you
can make the ordered set of "real in (0, 1] plus iota", but you lose
other properties of the reals. "reals plus iota" is not a field, nor is
it complete.

And it does not make sense to talk about "having a location on the
number line" without defining "the number line" - normally that term is
used precisely to mean the real numbers.

[peteolcott]

That you have totally run out of all reasoning as any rebuttal is
sufficient evidence of the plausibility that my assertion is correct.

You are very highly motivated to prove that I am wrong yet cannot only
because I AM NOT WRONG !!!

[peteolcott]

This is not Keith and Mike's ideas it is their agreement with my ideas:
These are the two key essences of agreement with Infinitesimal Numbers:

On 9/16/2019 11:38 PM, Keith Thompson wrote:
> we can define "iota" as an infinitesimal, something that is not a
> real number but that has a location on the number line, directly
> adjacent to the real number 0. It is greater than 0 and less than
> any positive real number.

On 9/16/2019 8:33 PM, Mike Terry wrote:
> OK, so (A,B] = [A,B] - {A}, and that can be counted as an "operation
in which we get (A,B] from [A,B].

Here are some examples of the next level of elaboration based on the
above two key essences of agreement:
Mike Terry agreed with this verbatim: [A, B] - {A} = (A, B]

Infinitesimal Numbers would encode (A, B] as [A[1], B]
We merely extrapolate the very next elaboration
[A[1], B] - {A[1]} = (A[1], B] encoded as [A[2], B]

[Juha Nieminen]

In comp.lang.c++ peteolcott <Here@home> wrote:
>> There is no "smallest real number larger than 0". Learn some number
>> theory, will you?
>>
>
> Keith Thompson proves that he fully understands the Infinitesimal
> number system:

Are you capable of understanding the "proof by contradiction" method in
mathematics?

Incidentally, the fact that there's no "smallest real number larger than 0"
is one of the *classical* simple examples of applying proof by
contradiction.

"Infinitesimals", no matter how you define them, do not help here. You
repeating your claims a million times does not change that fact.

[Öö Tiib]

Replying to Peter Olcott seems not helping him to understand anything.
BTW, is that about him?: https://www.youtube.com/watch?v=wfPPJBYc2B0

[Mike Terry]

Yes.

[peteolcott]

YES OF COURSE NOTHING HELPS WHEN YOU DON'T BOTHER TO READ WHAT IS SAID

THERE IS A POINT ON THE NUMBER LINE IMMEDIATELY ADJACENT TO 0

On 9/16/2019 11:38 PM, Keith Thompson wrote:
> we can define "iota" as an infinitesimal, something that is not a
> real number but that has a location on the number line, directly
> adjacent to the real number 0. It is greater than 0 and less than
> any positive real number.

[Mr Flibble]

On 18/09/2019 16:02, peteolcott wrote:
> On 9/18/2019 2:35 AM, Juha Nieminen wrote:
>> In comp.lang.c++ peteolcott <Here@home> wrote:
>>>> There is no "smallest real number larger than 0". Learn some number
>>>> theory, will you?
>>>>
>>>
>>> Keith Thompson proves that he fully understands the Infinitesimal
>>> number system:
>>
>> Are you capable of understanding the "proof by contradiction" method in
>> mathematics?
>>
>> Incidentally, the fact that there's no "smallest real number larger than 0"
>> is one of the *classical* simple examples of applying proof by
>> contradiction.
>>
>> "Infinitesimals", no matter how you define them, do not help here. You
>> repeating your claims a million times does not change that fact.
>>
>
> YES OF COURSE NOTHING HELPS WHEN YOU DON'T BOTHER TO READ WHAT IS SAID
>
> THERE IS A POINT ON THE NUMBER LINE IMMEDIATELY ADJACENT TO 0

For the field of real numbers, no, there isn't. Take your meds.

[Juha Nieminen]

In comp.lang.c++ peteolcott <Here@home> wrote:
> THERE IS A POINT ON THE NUMBER LINE IMMEDIATELY ADJACENT TO 0

Ok, what is that number?

[peteolcott]

On 9/18/2019 4:25 PM, Mr Flibble wrote:
> On 18/09/2019 16:02, peteolcott wrote:
>> On 9/18/2019 2:35 AM, Juha Nieminen wrote:
>>> In comp.lang.c++ peteolcott <Here@home> wrote:
>>>>> There is no "smallest real number larger than 0". Learn some number
>>>>> theory, will you?
>>>>>
>>>>
>>>> Keith Thompson proves that he fully understands the Infinitesimal
>>>> number system:
>>>
>>> Are you capable of understanding the "proof by contradiction" method in
>>> mathematics?
>>>
>>> Incidentally, the fact that there's no "smallest real number larger than 0"
>>> is one of the *classical* simple examples of applying proof by
>>> contradiction.
>>>
>>> "Infinitesimals", no matter how you define them, do not help here. You
>>> repeating your claims a million times does not change that fact.
>>>
>>
>> YES OF COURSE NOTHING HELPS WHEN YOU DON'T BOTHER TO READ WHAT IS SAID
>>
>> THERE IS A POINT ON THE NUMBER LINE IMMEDIATELY ADJACENT TO 0
>
> For the field of real numbers, no, there isn't.  Take your meds.
>
> /Flibble
>

Did I say that it was a Real number?
Quit being a jackass.

[peteolcott]

It is the Infinitesimal Number 0.0[1] one point offset from 0.0.

[Mr Flibble]

On 18/09/2019 22:41, peteolcott wrote:
> On 9/18/2019 4:25 PM, Mr Flibble wrote:
>> On 18/09/2019 16:02, peteolcott wrote:
>>> On 9/18/2019 2:35 AM, Juha Nieminen wrote:
>>>> In comp.lang.c++ peteolcott <Here@home> wrote:
>>>>>> There is no "smallest real number larger than 0". Learn some number
>>>>>> theory, will you?
>>>>>>
>>>>>
>>>>> Keith Thompson proves that he fully understands the Infinitesimal
>>>>> number system:
>>>>
>>>> Are you capable of understanding the "proof by contradiction" method in
>>>> mathematics?
>>>>
>>>> Incidentally, the fact that there's no "smallest real number larger
>>>> than 0"
>>>> is one of the *classical* simple examples of applying proof by
>>>> contradiction.
>>>>
>>>> "Infinitesimals", no matter how you define them, do not help here. You
>>>> repeating your claims a million times does not change that fact.
>>>>
>>>
>>> YES OF COURSE NOTHING HELPS WHEN YOU DON'T BOTHER TO READ WHAT IS SAID
>>>
>>> THERE IS A POINT ON THE NUMBER LINE IMMEDIATELY ADJACENT TO 0
>>
>> For the field of real numbers, no, there isn't.  Take your meds.
>>
>> /Flibble
>>
>
> Did I say that it was a Real number?
> Quit being a jackass.

YOU CANNOT MIX NUMBERS FROM DIFFERENT FIELDS ON THE SAME NUMBER LINE YOU
DEMENTED FUCKTARD. TAKE YOUR FUCKING MEDICATION.

[James Kuyper]

On Wednesday, September 18, 2019 at 5:41:28 PM UTC-4, peteolcott wrote:
> On 9/18/2019 4:25 PM, Mr Flibble wrote:
> > On 18/09/2019 16:02, peteolcott wrote:
> >> On 9/18/2019 2:35 AM, Juha Nieminen wrote:
> >>> In comp.lang.c++ peteolcott <Here@home> wrote:

...

> >>>> Keith Thompson proves that he fully understands the Infinitesimal
> >>>> number system:
> >>>
> >>> Are you capable of understanding the "proof by contradiction" method in
> >>> mathematics?
> >>>
> >>> Incidentally, the fact that there's no "smallest real number larger than 0"
> >>> is one of the *classical* simple examples of applying proof by
> >>> contradiction.
> >>>
> >>> "Infinitesimals", no matter how you define them, do not help here. You
> >>> repeating your claims a million times does not change that fact.
> >>>
> >>
> >> YES OF COURSE NOTHING HELPS WHEN YOU DON'T BOTHER TO READ WHAT IS SAID
> >>
> >> THERE IS A POINT ON THE NUMBER LINE IMMEDIATELY ADJACENT TO 0
> >
> > For the field of real numbers, no, there isn't.  Take your meds.
> >
> > /Flibble
> >
>
> Did I say that it was a Real number?

See <https://en.wikipedia.org/wiki/Number_line>
"In basic mathematics, a number line is a picture of a graduated
straight line that serves as abstraction for real numbers, denoted by R.
Every point of a number line is assumed to correspond to a real number,
and every real number to a point."
...
"In advanced mathematics, the expressions real number line, or real line
are typically used to indicate the above-mentioned concept that every
point on a straight line corresponds to a single real number, and vice
versa."

When you say that you're talking about the "number line", you're being,
at best, confusing, and at worst, deliberately misleading, since if it
contains things that aren't real numbers, then it isn't the "real number
line" that most people think of when you say "number line". You should
give your concept a special name to clearly distinguish it from the real
number line. For the purposes of the following discussion, I'll call it
the Olcott Number Line, ONL for short.

Keith was not agreeing with you - he was pointing out an inherent
limitation to your results due to the fact that you're talking about the
ONL, rather than the conventional number line.

You have not described your infinitesimals in any detail on this
newsgroup, nor have you described how the ONL differs from the real
number line. Your description of your a[i] notation doesn't make any
sense when applied to the real number line, and what you haven't
explained about the ONL means we can't be sure what it might mean when
applied to the ONL. I don't have enough interest in your ideas to
justify bothering to track down whether you've ever addressed those
issues anywhere else. Depending upon those details that you haven't
provided, it might be the case that you can prove that the ONL is
countable. That in turn implies that the real numbers which are on the
ONL are also countable; but that would not imply that the real numbers
which aren't on the ONL are countable - that's the fundamental problem
that you face.

If the things you've said about the ONL are true (namely that 0.0[i] for
i an integer from 1 to infinity generates every point on the ONL to the
right of 0.0), then there must be real numbers which are not in the ONL
- a lot of them - in fact, the infinitely overwhelming majority of real
numbers cannot be on the ONL. Your description of your a[i] notation
implies that that the real numbers in the ONL are ordered (I have no
idea whether the infinitesimals are ordered, since you've provided no
description of their properties). In other words, if a[i] and a[j] are
both real, then a[i] < a[j] iff i < j. Is that true? If so, it's not
only feasible, but trivial, to prove that most real numbers aren't on
the ONL.

[Juha Nieminen]

In comp.lang.c++ peteolcott <Here@home> wrote:
> On 9/18/2019 4:34 PM, Juha Nieminen wrote:
>> In comp.lang.c++ peteolcott <Here@home> wrote:
>>> THERE IS A POINT ON THE NUMBER LINE IMMEDIATELY ADJACENT TO 0
>>
>> Ok, what is that number?
>>
>
> It is the Infinitesimal Number 0.0[1] one point offset from 0.0.

What are the properties of that number that make it different from 0?

[peteolcott]

On 9/19/2019 12:54 PM, Juha Nieminen wrote:
> In comp.lang.c++ peteolcott <Here@home> wrote:
>> On 9/18/2019 4:34 PM, Juha Nieminen wrote:
>>> In comp.lang.c++ peteolcott <Here@home> wrote:
>>>> THERE IS A POINT ON THE NUMBER LINE IMMEDIATELY ADJACENT TO 0
>>>
>>> Ok, what is that number?
>>>
>>
>> It is the Infinitesimal Number 0.0[1] one point offset from 0.0.
>
> What are the properties of that number that make it different from 0?
>

[0, 1] - (0, 1] = {0} the first interval has exactly one point that the
second interval does not have, making the first interval Infinitesimally
longer than the second one.

The first interval begins at 0.0[0] and
The second interval begins at 0.0[1].

[Siri Cruise]

In article
<581781046.589713241.7...@news.eternal-september.org>,
Stuart Redmann <DerT...@web.de> wrote:

> Goedel showed that this is not always the case (see the continuum
> hypothesis). This implies that there are things in mathematics that cannot
> be proven. This culminated in Goedels incompleteness theorem, which is
> considered by many mathematicians as a severe throwback in maths.

Goedel's incompleteness theorem is not that a specific proposition is unprovable
but that any system complex enough to prove number theory theorems will have
either true propositions which cannot be proven (incompleteness) or false
propositions which can be proven (w-inconsistency).

Picture the boundary between truth and falseness as if a fractal curve, truth
and falseness become so entangled that it takes a nondenumerably infinite proof
to distinguish.

We can force the issue on a finite subset of propositions asserted or denied as
postulates without introducing inconsistency. Experience has shown the continuum
hypothesis like the parallel postulates does not lead to inconsistency.

--
:-<> Siri Seal of Disavowal #000-001. Disavowed. Denied. Deleted. @
'I desire mercy, not sacrifice.' /|\
The first law of discordiamism: The more energy This post / \
to make order is nore energy made into entropy. insults Islam. Mohammed

[Siri Cruise]

In article <T9mdnX__WZbR9OvA...@giganews.com>,
peteolcott <Here@Home> wrote:

> The number of real numbers is the number of adjacent geometric
> points on a number line, countable using integers.

You're misreading the axiom of choice. While the axiom can be assumed for
nondenumerable sets, no has come up with an orderring of, say, reals that an
open set has a distinguished minimum.

[Siri Cruise]

In article <pMqdnQFBSMfCXurA...@giganews.com>,
peteolcott <Here@Home> wrote:

> The next real number after 3.0 is the geometric point on the number
> line that is immediately adjacent to 3.0 on its right side. This geometric
> point is an infinitesimally larger than 3.0: encoded as the first point in
> the interval: (3,4]

The axiom of choice says the set has a distinguished minimum with respect to
some orderring, but not what that orderring is. The usual < order does not work;
nobody has yet discoverred an orderring that does. Assuming choice produces
unsettling results like this, but not inconsistencies, so it is allowed.

Real numbers do not have infinitesimals. That was discoverred a couple of
hundred years of ago. This is why most treatments of calculus use limits instead
of the infinitesimals of Newton and Liebniz.

[Siri Cruise]

In article <qlefjm$300c$1...@adenine.netfront.net>,
Juha Nieminen <nos...@thanks.invalid> wrote:

> In comp.lang.c++ peteolcott <Here@home> wrote:
> > The next real number after 3.0 is the geometric point on the number
> > line that is immediately adjacent to 3.0 on its right side. This geometric
> > point is an infinitesimally larger than 3.0: encoded as the first point in
> > the interval: (3,4]
>

> That would be "the smallest real number larger than 3". Which does not
> exist.

Not for the usual < order. Someday someone might come up with an order that can
do this. According the axiom of choice, it's out there. Somewhere.

[X-Files theme music] Do you believe?

[peteolcott]

On 9/19/2019 2:34 PM, Siri Cruise wrote:
> In article <T9mdnX__WZbR9OvA...@giganews.com>,
> peteolcott <Here@Home> wrote:
>
>> The number of real numbers is the number of adjacent geometric
>> points on a number line, countable using integers.
>
> You're misreading the axiom of choice. While the axiom can be assumed for
> nondenumerable sets, no has come up with an orderring of, say, reals that an
> open set has a distinguished minimum.
>

I have updated this on the basis of feedback:

(1) There cannot be a real number associated with the point
immediately adjacent to 0.0 because real numbers must be divisible
by 2 and single points are indivisible.

(2) If there is no real number immediately adjacent to 0.0 then
there is always a measurable gap after 0.0. By dividing the
size of the gap by 2 it gets smaller and smaller yet never closes,
there are always points in the gap that are unaccounted for.

This is where Infinitesimal numbers come in:
[0,1] - (0,1] = {0} a single point on the number line.
Thus proving a difference (in length) of one point between the first
interval and the second interval.

In all of the years where the width of a single point has been defined
to be 0 it was defined incorrectly, the actual width is Infinitesimal.

Infinitesimals sometimes act as if they had zero width and other times
act as if they minimum width. This is similar to the dichotomy of the
particle versus wave theories of light.

When we adapt (0,1] to the Infinitesimal number notational conventions it becomes
[0[1], 1] indicating an interval beginning with a single point offset from 0.

This notational convention allows us to reference subsequent immediately
adjacent points on the number line: [0[1], 1] - {0[1]} = [0[2], 1].

[peteolcott]

On 9/19/2019 2:28 PM, Siri Cruise wrote:
> In article
> <581781046.589713241.7...@news.eternal-september.org>,
> Stuart Redmann <DerT...@web.de> wrote:
>
>> Goedel showed that this is not always the case (see the continuum
>> hypothesis). This implies that there are things in mathematics that cannot
>> be proven. This culminated in Goedels incompleteness theorem, which is
>> considered by many mathematicians as a severe throwback in maths.
>
> Goedel's incompleteness theorem is not that a specific proposition is unprovable
> but that any system complex enough to prove number theory theorems will have
> either true propositions which cannot be proven (incompleteness) or false
> propositions which can be proven (w-inconsistency).
>
> Picture the boundary between truth and falseness as if a fractal curve, truth
> and falseness become so entangled that it takes a nondenumerably infinite proof
> to distinguish.
>
> We can force the issue on a finite subset of propositions asserted or denied as
> postulates without introducing inconsistency. Experience has shown the continuum
> hypothesis like the parallel postulates does not lead to inconsistency.
>

Bottom line (A great simplification of Wittgenstein's notorious paragraph)
http://liarparadox.org/Wittgenstein.pdf

For every formal system the only way that we know that a sentence
is true is that this sentence satisfies a sequence of finite strings
that concurrently make this same sentence provable.

[peteolcott]

On 9/19/2019 2:54 PM, Siri Cruise wrote:
> In article <qlefjm$300c$1...@adenine.netfront.net>,
> Juha Nieminen <nos...@thanks.invalid> wrote:
>
>> In comp.lang.c++ peteolcott <Here@home> wrote:
>>> The next real number after 3.0 is the geometric point on the number
>>> line that is immediately adjacent to 3.0 on its right side. This geometric
>>> point is an infinitesimally larger than 3.0: encoded as the first point in
>>> the interval: (3,4]
>>
>> That would be "the smallest real number larger than 3". Which does not
>> exist.
>
> Not for the usual < order. Someday someone might come up with an order that can
> do this. According the axiom of choice, it's out there. Somewhere.
>
> [X-Files theme music] Do you believe?
>

As explained in this reply

On 9/19/2019 2:58 PM, peteolcott wrote:
Provably unprovable eliminates incompleteness [ COMPLETE-2 ]

There is no such real number, because all Reals are divisible by 2
there is such an Infinitesimal number: a single indivisible point
on the number line.

[Siri Cruise]

In article <qlkrir$1iva$1...@adenine.netfront.net>,
Juha Nieminen <nos...@thanks.invalid> wrote:

> In fact, even if you limit yourself to rational numbers, it still
> doesn't exist. There is no "smallest rational number larger than 3".

You can enumerate rationals, E:rationals to naturals. Then define the order O to
be xOy = E(x)<E(y). The smallest rational larger than 3 would be 4 or
thereabouts. This might not be the most useful order but does show the axiom of
choice works on rationals.

Note that you define a total order on R^n: each real in the orderred n-tuples
has a decimal representation. You can interleave the digits of the n reals into
a single real and define the total order of R^n to be the usual < on this R^1
set of interleaved reals. Again not a particularly useful order, but it does
prove you can total order R^n for any natural n. Once you understand tricks like
this are allowed, you realise maybe there is some order on reals that satisfies
the axiom of choice.

[Siri Cruise]

In article <O9udnZdFPogAkBzA...@giganews.com>,

peteolcott <Here@Home> wrote:

> Keith Thompson proves that he fully understands the Infinitesimal
> number system:
>

> On 9/16/2019 11:38 PM, Keith Thompson wrote:

> > I can imagine that *this* set has a least element. For example

> > we can define "iota" as an infinitesimal, something that is not a
> > real number but that has a location on the number line, directly
> > adjacent to the real number 0. It is greater than 0 and less than
> > any positive real number.

iota > iota^2 > iota^3 > ...

> He elaborates this understanding much more completely in his
> 9/16/2019 11:38 PM, reply.

Infinitesimals cannot be reals. A couple of centuries ago it was shown this
leads to inconsistency. Recently nonstandard analysis has reintroduced
infinitesimals and their reciprocals. Positive infinitesimals are an infinite
set each greater than zero and less than any positive real. Their reciprocals
are greater than any real number and defined since infinitesimals are nonzero.

Note that for an infinitesimal dx, dx > dx^2 > dx^3 > ... . So with the usual <
infinitesimals also lack a minimum element. Choice assumes an order exists, but
we haven't found it.

[Siri Cruise]

In article <PYKdnZj3lqVg2x_A...@giganews.com>,

peteolcott <Here@Home> wrote:

> On 9/18/2019 2:35 AM, Juha Nieminen wrote:
> > In comp.lang.c++ peteolcott <Here@home> wrote:
> >>> There is no "smallest real number larger than 0". Learn some number
> >>> theory, will you?
> >>>
> >>
> >> Keith Thompson proves that he fully understands the Infinitesimal
> >> number system:
> >
> > Are you capable of understanding the "proof by contradiction" method in
> > mathematics?
> >
> > Incidentally, the fact that there's no "smallest real number larger than 0"
> > is one of the *classical* simple examples of applying proof by
> > contradiction.
> >
> > "Infinitesimals", no matter how you define them, do not help here. You
> > repeating your claims a million times does not change that fact.
> >
>
> YES OF COURSE NOTHING HELPS WHEN YOU DON'T BOTHER TO READ WHAT IS SAID
>
> THERE IS A POINT ON THE NUMBER LINE IMMEDIATELY ADJACENT TO 0
> On 9/16/2019 11:38 PM, Keith Thompson wrote:
> > we can define "iota" as an infinitesimal, something that is not a
> > real number but that has a location on the number line, directly
> > adjacent to the real number 0. It is greater than 0 and less than
> > any positive real number.

Unless you can prove the set of numbers satisfying this definition is finite,
all you're doing is adding another infinite set to the issue.

Hint: it's an infinite set.

[peteolcott]

On 9/19/2019 3:33 PM, Siri Cruise wrote:
> In article <O9udnZdFPogAkBzA...@giganews.com>,
> peteolcott <Here@Home> wrote:
>
>> Keith Thompson proves that he fully understands the Infinitesimal
>> number system:
>>
>> On 9/16/2019 11:38 PM, Keith Thompson wrote:
>> > I can imagine that *this* set has a least element. For example
>> > we can define "iota" as an infinitesimal, something that is not a
>> > real number but that has a location on the number line, directly
>> > adjacent to the real number 0. It is greater than 0 and less than
>> > any positive real number.
>
> iota > iota^2 > iota^3 > ...
>
>> He elaborates this understanding much more completely in his
>> 9/16/2019 11:38 PM, reply.
>
> Infinitesimals cannot be reals. A couple of centuries ago it was shown this
> leads to inconsistency. Recently nonstandard analysis has reintroduced
> infinitesimals and their reciprocals. Positive infinitesimals are an infinite
> set each greater than zero and less than any positive real. Their reciprocals
> are greater than any real number and defined since infinitesimals are nonzero.
>
> Note that for an infinitesimal dx, dx > dx^2 > dx^3 > ... . So with the usual <
> infinitesimals also lack a minimum element. Choice assumes an order exists, but
> we haven't found it.
>

This is the Pete Olcott copyright 2018, 2019 notion of
infinitesimals that have a bijection to every point on
the number line indexed by integers.

Reals have no point immediately adjacent to 0.0, infinitesimals
name this point 0.0[1].

[peteolcott]

If all of the points on the number line can be counted using integers
then all infinite sets have the same cardinality.

[Siri Cruise]

In article <m6ydnRGwy51ucB7A...@giganews.com>,

peteolcott <Here@Home> wrote:

> If all of the points on the number line can be counted using integers
> then all infinite sets have the same cardinality.

They can't be.

[peteolcott]

On 9/19/2019 5:04 PM, Siri Cruise wrote:
> In article <m6ydnRGwy51ucB7A...@giganews.com>,
> peteolcott <Here@Home> wrote:
>
>> If all of the points on the number line can be counted using integers
>> then all infinite sets have the same cardinality.
>
> They can't be.
>

I already proved this in other replies to you.

[Siri Cruise]

In article <m6ydnRawy53gcB7A...@giganews.com>,

peteolcott <Here@Home> wrote:

> Reals have no point immediately adjacent to 0.0, infinitesimals
> name this point 0.0[1].

For positive reals exponents can be any reals, so that you can have a
nondenumerable semi-open set of exponents A = [1, oo) and thus a nondenumerable
semiopen set for real x X(x) = {x^a: 0<x<1, a in A} which has no minimum under <
and a maximium of x. This is a nondenumerable version of an infinite strictly
decreasing bounded sequence.

To maintain definitions of exponentiation and less than for surreal
infinitesimals, X(dx) for infinitesimal dx is also a nondenumerable subset of
(0, dx]; thus a nondenumerable set separates 0.0 and dx. There is no point
immediately adjacent to 0.0 even for surreals.

You can, of course, define a set of infinitesimals which are not surreal, but it
becomes your responsibility to prove your element 0.0[1] uniquely exists and can
be meaningfully orderred among reals.

I can define the set of all four corner triangles. However it is empty: it has
no (unique) elements.

I can also define < on naturals and < on triangles but that alone does not make
4 < triangle ABC a meaningful statement.

[Siri Cruise]

In article <UuadneUXFb9XQB7A...@giganews.com>,

peteolcott <Here@Home> wrote:

> This is where Infinitesimal numbers come in:
> [0,1] - (0,1] = {0} a single point on the number line.
> Thus proving a difference (in length) of one point between the first
> interval and the second interval.

You have to provide a length function and prove its properties. The best I can
think of is d(S) = d(inf S, sup S). Since inf[0, 1] = inf(0, 1], under my
definition d[0, 1] = d(0, 1].

I switch min to inf because you end up with lim d(x, 1], x/=0, x->0, and that's
just a cladded definition of inf.

[peteolcott]

On 9/19/2019 5:48 PM, Siri Cruise wrote:
> In article <m6ydnRawy53gcB7A...@giganews.com>,
> peteolcott <Here@Home> wrote:
>
>> Reals have no point immediately adjacent to 0.0, infinitesimals
>> name this point 0.0[1].
>
> For positive reals exponents can be any reals, so that you can have a
> nondenumerable semi-open set of exponents A = [1, oo) and thus a nondenumerable
> semiopen set for real x X(x) = {x^a: 0<x<1, a in A} which has no minimum under <
> and a maximium of x. This is a nondenumerable version of an infinite strictly
> decreasing bounded sequence.
>
> To maintain definitions of exponentiation and less than for surreal
> infinitesimals, X(dx) for infinitesimal dx is also a nondenumerable subset of
> (0, dx]; thus a nondenumerable set separates 0.0 and dx. There is no point
> immediately adjacent to 0.0 even for surreals.
>
> You can, of course, define a set of infinitesimals which are not surreal, but it
> becomes your responsibility to prove your element 0.0[1] uniquely exists and can
> be meaningfully orderred among reals.
>
> I can define the set of all four corner triangles. However it is empty: it has
> no (unique) elements.
>
> I can also define < on naturals and < on triangles but that alone does not make
> 4 < triangle ABC a meaningful statement.
>

I don't pay any attention to any of these things. I simply derived a
notational convention that allows all of the adjacent points on a number
line to be directly referenced.

I also proved why this is required in that Real numbers leave gaps
in the number line with points that are unaccounted for.

[peteolcott]

On 9/19/2019 5:57 PM, Siri Cruise wrote:
> In article <UuadneUXFb9XQB7A...@giganews.com>,
> peteolcott <Here@Home> wrote:
>
>> This is where Infinitesimal numbers come in:
>> [0,1] - (0,1] = {0} a single point on the number line.
>> Thus proving a difference (in length) of one point between the first
>> interval and the second interval.
>
> You have to provide a length function and prove its properties. The best I can
> think of is d(S) = d(inf S, sup S). Since inf[0, 1] = inf(0, 1], under my
> definition d[0, 1] = d(0, 1].
>
> I switch min to inf because you end up with lim d(x, 1], x/=0, x->0, and that's
> just a cladded definition of inf.
>

I don't understand any of that stuff.
I do understand that (0, 1] is one point longer than (0, 1).

Which entails that the definition of zero width points is incorrect.
The first interval clearly has one more point than the second.

The first interval also has a length of exactly 1.0 and the
second interval has a length that is one point less than 1.0
because the missing point clearly has a numerical value that
is greater than its predecessor.

[David Brown]

On 20/09/2019 01:00, peteolcott wrote:

> I don't pay any attention to any of these things. I simply derived a
> notational convention that allows all of the adjacent points on a number
> line to be directly referenced.

You don't seem to pay attention to anything much. Making up a notation
is not the same as forming a mathematical theory.

>
> I also proved why this is required in that Real numbers leave gaps
> in the number line with points that are unaccounted for.
>

You haven't proven anything. "Proof by repeated assertion" is not valid
in mathematics.

Perhaps the real people could give up this thread? I had hoped that
there might be some interesting mathematics here, or even that PO could
learn something. But while we have seen that there are people in
c.l.c++ who are competent and interested in mathematics, I don't think
there is anyone here with a hammer big enough to knock some sense
through PO's thick skull. And his continued wilful ignorance makes it
difficult to have an intelligent discussion on the topic (or anything else).

[Siri Cruise]

In article <qm1svn$qe1$1...@dont-email.me>, David Brown <david...@hesbynett.no>
wrote:

> > I also proved why this is required in that Real numbers leave gaps
> > in the number line with points that are unaccounted for.
> >
>
> You haven't proven anything. "Proof by repeated assertion" is not valid
> in mathematics.

Infinity is counterintuitive. Nondenumerable infinite sets are doubleplus
counterinitutive. God made integers; all else is the work of man.

[Juha Nieminen]

In comp.lang.c++ peteolcott <Here@home> wrote:
>> What are the properties of that number that make it different from 0?
>>
>
> [0, 1] - (0, 1] = {0} the first interval has exactly one point that the
> second interval does not have, making the first interval Infinitesimally
> longer than the second one.
>
> The first interval begins at 0.0[0] and
> The second interval begins at 0.0[1].

You didn't give a single property of that number. You only made claims
about an interval.

[Juha Nieminen]

In comp.lang.c++ peteolcott <Here@home> wrote:

> On 9/19/2019 5:04 PM, Siri Cruise wrote:
>> In article <m6ydnRGwy51ucB7A...@giganews.com>,
>> peteolcott <Here@Home> wrote:
>>
>>> If all of the points on the number line can be counted using integers
>>> then all infinite sets have the same cardinality.
>>
>> They can't be.
>>
>
> I already proved this in other replies to you.

No, you didn't. Making a claim is not a mathematical proof.

[David Brown]

On 20/09/2019 09:10, Siri Cruise wrote:
> In article <qm1svn$qe1$1...@dont-email.me>, David Brown <david...@hesbynett.no>
> wrote:
>
>>> I also proved why this is required in that Real numbers leave gaps
>>> in the number line with points that are unaccounted for.
>>>
>>
>> You haven't proven anything. "Proof by repeated assertion" is not valid
>> in mathematics.
>
> Infinity is counterintuitive. Nondenumerable infinite sets are doubleplus
> counterinitutive.

And the difference between infinite ordinals and infinite cardinals
makes it even worse!

> God made integers; all else is the work of man.
>

I remember making Peano integers in a Haskell-like functional
programming language. First year maths students made the integers - all
else is the work of second or third year students!

[Juha Nieminen]

In comp.lang.c++ peteolcott <Here@home> wrote:

> This is the Pete Olcott copyright 2018, 2019 notion of
> infinitesimals that have a bijection to every point on
> the number line indexed by integers.
>
> Reals have no point immediately adjacent to 0.0, infinitesimals
> name this point 0.0[1].

So you take an uncountable set, the set of reals, you add numbers
to that set, and that somehow magically makes the set countable.

And your proof of that is a simple assertion.

You can't even provide a bijection between this expanded set of
reals of yours, and the natural numbers. But surely it must be
countable because you say it is. That's all the proof you need.

Also, you can't give a single property of that "0.0[1]" number
which distinguishes it from zero.

[Mr Flibble]

On 20/09/2019 00:10, peteolcott wrote:
> On 9/19/2019 5:57 PM, Siri Cruise wrote:
>> In article <UuadneUXFb9XQB7A...@giganews.com>,
>>   peteolcott <Here@Home> wrote:
>>
>>> This is where Infinitesimal numbers come in:
>>> [0,1] - (0,1] = {0} a single point on the number line.
>>> Thus proving a difference (in length) of one point between the first
>>> interval and the second interval.
>>
>> You have to provide a length function and prove its properties. The best
>> I can
>> think of is d(S) = d(inf S, sup S). Since inf[0, 1] = inf(0, 1], under my
>> definition d[0, 1] = d(0, 1].
>>
>> I switch min to inf because you end up with lim d(x, 1], x/=0, x->0, and
>> that's
>> just a cladded definition of inf.
>>
>
> I don't understand any of that stuff.
> I do understand that (0, 1] is one point longer than (0, 1).
>
> Which entails that the definition of zero width points is incorrect.
> The first interval clearly has one more point than the second.

Your assertion in the form of an erroneous premise followed by an equally
erroneous conclusion is, obviously, also erroneous. You don't get to
define what is meant by a "geometric point"; we already have a well
established and accepted definition for that concept.

>
> The first interval also has a length of exactly 1.0 and the
> second interval has a length that is one point less than 1.0
> because the missing point clearly has a numerical value that
> is greater than its predecessor.

More erroneous nonsense.

You are either:

1) a troll;
2) a fucktard;
or
3) not taking your medication.

Which is it?

/Flibble

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