+----------------------------------------+
| Number is an expression of computation |
+----------------------------------------+
Numbers are basically computational expressions (natural number plus operation,
e.g. positional number, rational, continued fraction,...). Removing operators
from the expression, what left are natural numbers. Yet, natural numbers are
basically successor functions(operation). Therefore, number is an expression of
computation. A normal number, or 'constant', can be treated as as a name,
representing a 'standardized' operation. Common computation of an expression
stops when a standardized operation is derived, because most major properties
are well known.

Number::= An expression/procedure (or a name) of computation (operation).

Positional Number::= Usual number notation, like decimal, an abbreviation of
'restricted' polynomial.

ℙosNum(B)::= Σ(r=a,b) di*B^r, di,B∈ℕ, i,r∈ℤ, B>1, ∀i, 0<=di<B

Lemma: Rational number and positional number are mutually convertible, I.e.
∀q,∃B, q∈ℚ <=> q∈ℙosNum(B)
Proof: Omitted (easy to see)

[Note] Decimal number means a positional number, of which at least one digit
represents number <1.
[Note] This lemma also shows that repeating decimals are irrational because
repeating decimals ∉ ℙosNum(B).

+-------------+
| Real number |
+-------------+
Real number is the representation of the distance between two arbitrary points
in space. Real numbers are the result of measurement of real entities, e.g.
time, distance, mass.
General recognition is that decimals are real number, including infinitely long
decimals (classified as irrational number).

Real number ℝ ::= Decimals are real number, including infinitely long decimals
which are normally expressed by infinite series.

[Note] Real number posseses properties such as trichotomy, density,..., but
this file treats it as implied properties from the above definition.
Another case to consider is closure property. Real number 'must' remain
in ℝ by 'legal' operations, but 'legal operation' is difficult to
define.
[Note] Real number could be basically this simple. Those continuous,
differentiable, convergent,...,etc. properties in real analysis/
calculus/advanced math. are considered invention of that domain,
irrelevant to this general recognition.

Theorem: No onto function exists from {0,1}^* to {0,1}^∞
Proof: Omitted

This theorem means that most real numbers are not expressible by finite length
of symbols.

Theorem: Let A(0)=0, A(n) is between A(n-1) and L by a factor s/r (0<s/r<1).
A(n)= A(n-1) + (L-A(n-1))*(s/r)
= (r*A(n-1) + (s)*L - s*A(n-1))/r
= ( s*L + (r-s)*A(n-1) )/r
Then, A(n)<L holds.

Proof: 1. The procedure can go on infinitely, n can be an infinity.
2. A(n)>=L causes contradiction from the given and A(n-1)>=A(n).

Lemma: Let L=1, r>=2, s=r-1, all form of positional number "0.999..." is less than 1.
Lemma: Let the greatest different digit of positional number A,B be a,b (same
position), then a>b => A>B
Lemma: All the digits of positional number A,B are equal iff A=B.
Note: A disputable point might be for 'infinite long ℙosNum(B)'.
Simpe conclusion: the 'algorithm' of order relationship for infinite-long
version would be very different in essence,.... The meaning (dependency)
of order relationship alters.
---------

This view of number and real number is quite simple (basically).
The view from Analysis/Calculus/Advanced math., as said, is their theoretic
problem in their domain.
I'd like to address and re-assure the number view from the point of (digital)
computers and discrete math. is reasonable and acceptabale.

wij <wyniijj2@gmail.com> writes:

> +----------------------------------------+
> | Number is an expression of computation |
> +----------------------------------------+
> Numbers are basically computational expressions (natural number plus operation,
> e.g. positional number, rational, continued fraction,...). Removing operators
> from the expression, what left are natural numbers. Yet, natural numbers are
> basically successor functions(operation). Therefore, number is an expression of
> computation. A normal number, or 'constant', can be treated as as a name,
> representing a 'standardized' operation. Common computation of an expression
> stops when a standardized operation is derived, because most major properties
> are well known.

That's one option.

> Number::= An expression/procedure (or a name) of computation
> (operation).

You would have to say exactly what operations are permitted. You reject
limits and taking least upper bounds I presume.

> Positional Number::= Usual number notation, like decimal, an abbreviation of
> 'restricted' polynomial.
>
> ℙosNum(B)::= Σ(r=a,b) di*B^r, di,B∈ℕ, i,r∈ℤ, B>1, ∀i, 0<=di<B
>
> Lemma: Rational number and positional number are mutually convertible, I.e.
> ∀q,∃B, q∈ℚ <=> q∈ℙosNum(B)
> Proof: Omitted (easy to see)
>
> [Note] Decimal number means a positional number, of which at least one digit
> represents number <1.
> [Note] This lemma also shows that repeating decimals are irrational because
> repeating decimals ∉ ℙosNum(B).

This second note does not follow. All the PosNum(B) are rational and
all rationals can be written as some PosNum(B), but that does not
exclude other representation of rationals such as repeating decimals.

> +-------------+
> | Real number |
> +-------------+
> Real number is the representation of the distance between two arbitrary points
> in space. Real numbers are the result of measurement of real entities, e.g.
> time, distance, mass.

You should pick a different name. What's the point of causing confusion
by using an existing name for something else?

> General recognition is that decimals are real number, including infinitely long
> decimals (classified as irrational number).
>
> Real number ℝ ::= Decimals are real number, including infinitely long decimals
> which are normally expressed by infinite series.

But which don't work like the infinite series everyone else uses. You
need to say how these series work. You implicitly reject find the sum,
for example.

> [Note] Real number posseses properties such as trichotomy, density,..., but
> this file treats it as implied properties from the above definition.
> Another case to consider is closure property. Real number 'must' remain
> in ℝ by 'legal' operations, but 'legal operation' is difficult to
> define.

OK, but the other definition of the reals don't have this problem. They
are already properly defined. And until you complete the definition,
there will be little reason for anyone to care and Wij-numbers.

> [Note] Real number could be basically this simple. Those continuous,
> differentiable, convergent,...,etc. properties in real analysis/
> calculus/advanced math. are considered invention of that domain,
> irrelevant to this general recognition.

So simple yet you can't define in what way the set if closed.

> Theorem: No onto function exists from {0,1}^* to {0,1}^∞
> Proof: Omitted
>
> This theorem means that most real numbers are not expressible by
> finite length of symbols.

It only means that there is at least one un-mapped sequence. To prove
that /most/ Wij-reals are not so expressible would require a much
stronger theorem.

> Theorem: Let A(0)=0, A(n) is between A(n-1) and L by a factor s/r (0<s/r<1).
> A(n)= A(n-1) + (L-A(n-1))*(s/r)
> = (r*A(n-1) + (s)*L - s*A(n-1))/r
> = ( s*L + (r-s)*A(n-1) )/r
> Then, A(n)<L holds.

You should explain how +, *, / and < work.

> Proof: 1. The procedure can go on infinitely, n can be an infinity.
> 2. A(n)>=L causes contradiction from the given and A(n-1)>=A(n).
>
> Lemma: Let L=1, r>=2, s=r-1, all form of positional number "0.999..." is less than 1.
> Lemma: Let the greatest different digit of positional number A,B be a,b (same
> position), then a>b => A>B
> Lemma: All the digits of positional number A,B are equal iff A=B.
> Note: A disputable point might be for 'infinite long ℙosNum(B)'.
> Simpe conclusion: the 'algorithm' of order relationship for infinite-long
> version would be very different in essence,.... The meaning (dependency)
> of order relationship alters.
> ---------
>
> This view of number and real number is quite simple (basically).

Yet you can't define how it is closed.

What good are Wij-numbers for calculus, engineering and physics if I
can't calculate limits?

> The view from Analysis/Calculus/Advanced math., as said, is their
> theoretic problem in their domain. I'd like to address and re-assure
> the number view from the point of (digital) computers and discrete
> math. is reasonable and acceptabale.

Have you read anyone else's proposals about that? Why should anyone
consider yours?

--
Ben.

wij <wyniijj2@gmail.com> writes:
[...]
> +-------------+
> | Real number |
> +-------------+
> Real number is the representation of the distance between two arbitrary points
> in space. Real numbers are the result of measurement of real entities, e.g.
> time, distance, mass.
> General recognition is that decimals are real number, including infinitely long
> decimals (classified as irrational number).
[...]

Is 1/3 a rational number? If the words you use have their usual
meanings, then by your definition it is not. 1/3 can be represented
by an infinitely long (unbounded) decimal: 0.333..., where the
"..." denotes an unbounded sequence. The represented value is
rational because the sequence *repeats*.

"Decimals" are one way of *representing* real numbers. I'm not
convinced that they're the best way of *defining* real numbers,
though I suppose you could.

--
Keith Thompson (The_Other_Keith) Keith.S.Thompson+u@gmail.com
Working, but not speaking, for Philips
void Void(void) { Void(); } /* The recursive call of the void */

Op 07.sep..2022 om 20:30 schreef wij:

> ... snip ...
> +-------------+
> | Real number |
> +-------------+
> Real number is the representation of the distance between two arbitrary points
> in space. Real numbers are the result of measurement of real entities, e.g.
> time, distance, mass.
> ... snip ...

Is this a definition? Or are there two (contradicting) definitions? I
assume that the first sentence is the definition and the second one a
clarification.

I wonder what you mean with 'space'. It looks as if you are basing a
number system on a physical measurable quantity. This has several
difficulties:

_Sign_
Distances are always positive. So, your 'real numbers' lack negative
quantities.

_Relatively_
Einstein's relativity theory shows that the distance between two points
in space is a relative quantity. It may be measured different for
different observers. So your 'real numbers' may be different for
different people.

_Precision_
We know that there is a fundamental limitation when measuring distances:
the Planck length. So your 'real numbers' have a finite precision, the
Planck length, which is very different from the normal real numbers,
which have unlimited precision. This means, e.g., that your 'real
numbers' do not work for values below the Planck length (including zero).

_Maximum_
We cannot measure distances larger than the diameter of the visible
Universe. This puts a limit on the maximum 'real number' you can use.

_Dimensions_
In physics different quantities have different dimensions. We cannot
compare the value of a temperature with the value of an electrical
charge. Force, power, mass, etc., they all have different dimensions. It
is not clear how you want to express these quantities in a quantity with
dimension length.

Based on these difficulties, I assume that you mean something different
with the word 'space'. Maybe an abstract mathematical space? There are
many different mathematical spaces. Some of them have negative
distances, like the curved spaces used in Einstein's general relativity
theory.
There are also discrete spaces (e.g. used in the space of electron
states of the Hydrogen atom), complex spaces (e.g., the Hilbert space
used in quantum mechanics) and many more.
A problem is that most spaces are based on a number system, so you would
get a circular dependent definition.
Maybe you can clarify what you mean with 'space' and 'measurable'.

On Wednesday, 7 September 2022 at 20:26:21 UTC+1, Ben Bacarisse wrote:
> wij <wyni...@gmail.com> writes:
>
> > +-------------+
> > | Real number |
> > +-------------+
> > Real number is the representation of the distance between two arbitrary points
> > in space. Real numbers are the result of measurement of real entities, e.g.
> > time, distance, mass.
> You should pick a different name. What's the point of causing confusion
> by using an existing name for something else?
>
Mathematicians should pick a different name.
"Real" implies "having a physical meaning", which isn't what mathematicians mean by
"real numbers".

Malcolm McLean <malcolm.arthur.mclean@gmail.com> writes:

> On Wednesday, 7 September 2022 at 20:26:21 UTC+1, Ben Bacarisse wrote:
>> wij <wyni...@gmail.com> writes:
>>
>> > +-------------+
>> > | Real number |
>> > +-------------+
>> > Real number is the representation of the distance between two arbitrary points
>> > in space. Real numbers are the result of measurement of real entities, e.g.
>> > time, distance, mass.
>> You should pick a different name. What's the point of causing confusion
>> by using an existing name for something else?
>>
> Mathematicians should pick a different name.

How do you see that happening?

And, since we are in fantasy land now, do you think that Wij's use of
the term will be any less confusing once he's actually sorted out how to
do the mathematics that engineers and scientists want? His "real
numbers" are very likely to be either useless or identical to one of the
usual extension to R.

> "Real" implies "having a physical meaning", which isn't what
> mathematicians mean by "real numbers".

Mathematicians are generally not confused by metaphor: group, function,
ring, ideal, closed, open, disc, manifold...

The problems (such as they are) occur at the interface between
mathematics and other disciplines.

--
Ben.

On Thursday, 8 September 2022 at 11:00:22 UTC+1, Ben Bacarisse wrote:
> Malcolm McLean <malcolm.ar...@gmail.com> writes:
>
> > On Wednesday, 7 September 2022 at 20:26:21 UTC+1, Ben Bacarisse wrote:
> >> wij <wyni...@gmail.com> writes:
> >>
> >> > +-------------+
> >> > | Real number |
> >> > +-------------+
> >> > Real number is the representation of the distance between two arbitrary points
> >> > in space. Real numbers are the result of measurement of real entities, e.g.
> >> > time, distance, mass.
> >> You should pick a different name. What's the point of causing confusion
> >> by using an existing name for something else?
> >>
> > Mathematicians should pick a different name.
> How do you see that happening?
>
> And, since we are in fantasy land now, do you think that Wij's use of
> the term will be any less confusing once he's actually sorted out how to
> do the mathematics that engineers and scientists want? His "real
> numbers" are very likely to be either useless or identical to one of the
> usual extension to R.
From Wij:
Real number is the representation of the distance between two arbitrary points
in space. Real numbers are the result of measurement of real entities, e.g.
time, distance, mass.

The obvious thing to say to this is that a Wij-real number should have a confidence
interval to represent the uncertainty in the measurement. That leads to quite a few
complications. It has been done before, however.

On 9/8/22 4:35 AM, Malcolm McLean wrote:
> On Wednesday, 7 September 2022 at 20:26:21 UTC+1, Ben Bacarisse wrote:
>> wij <wyni...@gmail.com> writes:
>>
>>> +-------------+
>>> | Real number |
>>> +-------------+
>>> Real number is the representation of the distance between two arbitrary points
>>> in space. Real numbers are the result of measurement of real entities, e.g.
>>> time, distance, mass.
>> You should pick a different name. What's the point of causing confusion
>> by using an existing name for something else?
>>
> Mathematicians should pick a different name.
> "Real" implies "having a physical meaning", which isn't what mathematicians mean by
> "real numbers".

My understanding is that the "original" source for these numbers was
similar to wij's definition, but rather than a "distance" as a position
on line (plane, space, ...) particularly in Geometry.

As such, it does have a correspondence to "Reality", and thus the name
"Real". Not that it is a measurement per-se, but it represents something
that could be measured.

We can talk about a line segment having a length of exactly sqrt(2), as
the diagonal of a square with sides exactly 1. Yes, we can not
physically make such a thing, but it can be a mathematical model for
reality.

The more modern models, based in part on limit theory sort of also play
to that idea, at a given level of precision, we know where we are.

On Thursday, 8 September 2022 at 16:35:35 UTC+8, Fred. Zwarts wrote:
> Op 07.sep..2022 om 20:30 schreef wij:
>
> > ... snip ...
> > +-------------+
> > | Real number |
> > +-------------+
> > Real number is the representation of the distance between two arbitrary points
> > in space. Real numbers are the result of measurement of real entities, e.g.
> > time, distance, mass.
> > ... snip ...
>
>
> Is this a definition? Or are there two (contradicting) definitions? I
> assume that the first sentence is the definition and the second one a
> clarification.

I try to explain what the real number is (in practice). Real number is not
very definable (as indicated, most of them are non-mention-able).
I am not sure whether time/distance/mass can be reduced to only distance, or not.
What is sure is that we should not project our understanding to reality.

> I wonder what you mean with 'space'. It looks as if you are basing a
> number system on a physical measurable quantity. This has several
> difficulties:

'space' above means the physical space. And, yes, I base the real number on
physical measurable quantity AT THE BOTTOM LEVEL. Like geometry in construction,
in manufacture of mechanical elements, like complex number in all kind of
Alternate-Current analysis I know. The i/o of real numbers must match the real
thing. The inside of real number is a theoretical/logical model.
Your points seem to be the 'physical limits' of current know-how and do not
allow theoretical 'prediction'.

> _Sign_
> Distances are always positive. So, your 'real numbers' lack negative
> quantities.

Yes, I missed the mention of sign.

> _Relatively_
> Einstein's relativity theory shows that the distance between two points
> in space is a relative quantity. It may be measured different for
> different observers. So your 'real numbers' may be different for
> different people.

There are many kind of _Relative_ 'space', but they must base on something
(like Cartesian coordinates) to say 'relative', otherwise losing meaning.

> _Precision_
> We know that there is a fundamental limitation when measuring distances:
> the Planck length. So your 'real numbers' have a finite precision, the
> Planck length, which is very different from the normal real numbers,
> which have unlimited precision. This means, e.g., that your 'real
> numbers' do not work for values below the Planck length (including zero).
>
> _Maximum_
> We cannot measure distances larger than the diameter of the visible
> Universe. This puts a limit on the maximum 'real number' you can use.

| Real number ℝ ::= Decimals are real number, including infinitely long decimals
| which are normally expressed by infinite series.

The _Precision_ of 'my real number' is infinite. There is no _Maximum_.
Planck length= 1.61625502 × 10-35 meters, not very small in math.
Note that 10, 10^999... and 10^-999... share the same property of non-zero
numbers in math. 'size' (big/small) is no issue in math.

> _Dimensions_
> In physics different quantities have different dimensions. We cannot
> compare the value of a temperature with the value of an electrical
> charge. Force, power, mass, etc., they all have different dimensions. It
> is not clear how you want to express these quantities in a quantity with
> dimension length.

_Dimensions_ normally means 2D,3D,... or Cartesian product. If you mean the
'unit' of measurement, different thing.

> Based on these difficulties, I assume that you mean something different
> with the word 'space'. Maybe an abstract mathematical space? There are
> many different mathematical spaces. Some of them have negative
> distances, like the curved spaces used in Einstein's general relativity
> theory.
> There are also discrete spaces (e.g. used in the space of electron
> states of the Hydrogen atom), complex spaces (e.g., the Hilbert space
> used in quantum mechanics) and many more.
> A problem is that most spaces are based on a number system, so you would
> get a circular dependent definition.
> Maybe you can clarify what you mean with 'space' and 'measurable'.

It seems you mixed 'space' and 'measurable' with lots of different things.

On Thursday, 8 September 2022 at 04:32:47 UTC+8, Keith Thompson wrote:
> wij <wyni...@gmail.com> writes:
> [...]
> > +-------------+
> > | Real number |
> > +-------------+
> > Real number is the representation of the distance between two arbitrary points
> > in space. Real numbers are the result of measurement of real entities, e.g.
> > time, distance, mass.
> > General recognition is that decimals are real number, including infinitely long
> > decimals (classified as irrational number).
> [...]
>
> Is 1/3 a rational number? If the words you use have their usual
> meanings, then by your definition it is not. 1/3 can be represented
> by an infinitely long (unbounded) decimal: 0.333..., where the
> "..." denotes an unbounded sequence. The represented value is
> rational because the sequence *repeats*.
>
> "Decimals" are one way of *representing* real numbers. I'm not
> convinced that they're the best way of *defining* real numbers,
> though I suppose you could.
>
> --
> Keith Thompson (The_Other_Keith) Keith.S.T...@gmail.com
> Working, but not speaking, for Philips
> void Void(void) { Void(); } /* The recursive call of the void */

Lots evidences have been discussed:
https://groups.google.com/g/comp.theory/c/KiajP6jsovM
https://groups.google.com/g/comp.theory/c/XJ08g-VjV1M
https://groups.google.com/g/comp.theory/c/ImnwhKe_YFE
https://groups.google.com/g/comp.theory/c/wlaV7nnV0xo

Last 'key' question are from https://groups.google.com/g/comp.theory/c/KiajP6jsovM

1. A valid logic to remove 'lim' from a limit expression to form a conclusion.
E.g. "lim 0.999...=1" (make it more formal the way you like) can yield the conclusion "0.999...=1".

2. Skep Dick already showed that you cannot even decide whether x-1 or x+1 is
closer to infinity or not in expression "lim(x->∞) 1-1/10^x = 0.999...".
Provide more specific explanations what you mean by the occurrence of infinity
in text-books.

3. there is no arithmetic rule applicable in limit calculation. If you are so
confident. Please provide arithmetic/math rules in limit calculation that people
can follow without ambiguity.

This question can be make more specific:

f(x)= x^2
f'(x)= lim(h->0) ((x+h)^2-x^2)/h // definition of derivative, h cannot be 0
= lim(h->0) (x^2+2xh+h^2-x^2)/h // h≠0 (h in ℝ?, Is h a number?)
= lim(h->0) (2xh + h^2)/h
= lim(h->0) 2x +h
= 2x // why 'lim' is removed to conclude f'(x)=2x ? because h=0 now?
// You won't have a logical answer for the absurdity

Don't bother digging this old absurdity. In general, the rule "a+0=a" (or more)
does not apply in limit calculation. We all know well, inside limit are
calculations for 'limit', not exactly equal. And, no need to be exact, because
there exist big-o or other methods.
The evidence accumulates if we continue to discuss the "0.999...=1" issue..
Solve these key questions first (and those in provided links).
I only got "convention", "belief" answers so far.

------
By now, another case emerges:

ℙosNum(B)::= Σ(r=a,b) di*B^r, di,B∈ℕ, i,r∈ℤ, B>1, ∀i, 0<=di<B

Decimal::= Positional number (plus +-), of which some digits represent numbers <1.

Lemma: Rational number ℚ and positional number ℙosNum(B) are mutually convertible.

Proof: For simplicity, only positive numbers are shown.
1. Let x="abc.ijk", x∈ℙosNum(B) by definition,
then x="abc.ijk"= "abcijk" / B^3. Therefore, x∈ℚ.
2. ∀x∈ℚ, x>0, ∃p,q∈ℤ, q>=2, such that x=p/q
x= p/q= r+s/q (r=[p/q], s=p-r*q)
Therefore, p/q can be represented by ℙosNum(q) as "r.s"..

[Note] This lemma also shows that repeating decimals are irrational because
repeating decimals ∉ ℙosNum(B).

So, 1. Provide your proof that 1/3=0.333... 2.Refute the lemma above.

Don't bother. Such problems are many, evidence can accumulate as discussing more.

wij <wyniijj2@gmail.com> writes:

> 1. A valid logic to remove 'lim' from a limit expression to form a
> conclusion. E.g. "lim 0.999...=1" (make it more formal the way you
> like) can yield the conclusion "0.999...=1".

Are you just going to repeat this stuff again and again, ignoring all
the answers that are readily available? (For example the appendix in
Tao's book that goes into detail how 0.99... = 1, referencing the
definitions and theorems elsewhere in the book.)

How does this end? You eventually read a book on real analysis and,
though you prefer to develop your own number system, you finally accept
that your "questions" were answered about a century ago?

Or do you pick up a book on non-standard analysis and number systems are
realise that what you are trying do has been done before?

Or does it end when you get bored of posting?

--
Ben.

Op 08.sep..2022 om 14:42 schreef wij:
> On Thursday, 8 September 2022 at 16:35:35 UTC+8, Fred. Zwarts wrote:
>> Op 07.sep..2022 om 20:30 schreef wij:
>>
>>> ... snip ...
>>> +-------------+
>>> | Real number |
>>> +-------------+
>>> Real number is the representation of the distance between two arbitrary points
>>> in space. Real numbers are the result of measurement of real entities, e.g.
>>> time, distance, mass.
>>> ... snip ...
>>
>>
>> Is this a definition? Or are there two (contradicting) definitions? I
>> assume that the first sentence is the definition and the second one a
>> clarification.
>
> I try to explain what the real number is (in practice). Real number is not
> very definable (as indicated, most of them are non-mention-able).
> I am not sure whether time/distance/mass can be reduced to only distance, or not.
> What is sure is that we should not project our understanding to reality.
>
>> I wonder what you mean with 'space'. It looks as if you are basing a
>> number system on a physical measurable quantity. This has several
>> difficulties:
>
> 'space' above means the physical space. And, yes, I base the real number on
> physical measurable quantity AT THE BOTTOM LEVEL. Like geometry in construction,
> in manufacture of mechanical elements, like complex number in all kind of
> Alternate-Current analysis I know. The i/o of real numbers must match the real
> thing. The inside of real number is a theoretical/logical model.
> Your points seem to be the 'physical limits' of current know-how and do not
> allow theoretical 'prediction'.
>
>> _Sign_
>> Distances are always positive. So, your 'real numbers' lack negative
>> quantities.
>
> Yes, I missed the mention of sign.
>
>> _Relatively_
>> Einstein's relativity theory shows that the distance between two points
>> in space is a relative quantity. It may be measured different for
>> different observers. So your 'real numbers' may be different for
>> different people.
>
> There are many kind of _Relative_ 'space', but they must base on something
> (like Cartesian coordinates) to say 'relative', otherwise losing meaning.
>
>> _Precision_
>> We know that there is a fundamental limitation when measuring distances:
>> the Planck length. So your 'real numbers' have a finite precision, the
>> Planck length, which is very different from the normal real numbers,
>> which have unlimited precision. This means, e.g., that your 'real
>> numbers' do not work for values below the Planck length (including zero).
>>
>> _Maximum_
>> We cannot measure distances larger than the diameter of the visible
>> Universe. This puts a limit on the maximum 'real number' you can use.
>
> | Real number ℝ ::= Decimals are real number, including infinitely long decimals
> | which are normally expressed by infinite series.
>
> The _Precision_ of 'my real number' is infinite. There is no _Maximum_.

So, your 'real numbers' are not based on real measurable values, but on
an abstraction of them. I still don't see why you would use a different
abstraction than physicists use normally. In physics we use mathematical
models as an approximation of reality. The normal real numbers give us a
very good approximation in most cases. What is the advantage of your
system? In what way would it give a better approximation?
I have the feeling that you mix reality with your own models.

> Planck length= 1.61625502 × 10-35 meters, not very small in math.
> Note that 10, 10^999... and 10^-999... share the same property of non-zero
> numbers in math. 'size' (big/small) is no issue in math.
>
>> _Dimensions_
>> In physics different quantities have different dimensions. We cannot
>> compare the value of a temperature with the value of an electrical
>> charge. Force, power, mass, etc., they all have different dimensions. It
>> is not clear how you want to express these quantities in a quantity with
>> dimension length.
>
> _Dimensions_ normally means 2D,3D,... or Cartesian product. If you mean the
> 'unit' of measurement, different thing.

In physics we call it dimensions. Units are something different. A
dimension, e.g. length, can have different units, e.g. meters, miles,
inches, feet, etc. A different dimension, e.g. mass, can have units of
grammes, pounds, ounces, etc.
>
>> Based on these difficulties, I assume that you mean something different
>> with the word 'space'. Maybe an abstract mathematical space? There are
>> many different mathematical spaces. Some of them have negative
>> distances, like the curved spaces used in Einstein's general relativity
>> theory.
>> There are also discrete spaces (e.g. used in the space of electron
>> states of the Hydrogen atom), complex spaces (e.g., the Hilbert space
>> used in quantum mechanics) and many more.
>> A problem is that most spaces are based on a number system, so you would
>> get a circular dependent definition.
>> Maybe you can clarify what you mean with 'space' and 'measurable'.
>
> It seems you mixed 'space' and 'measurable' with lots of different things.

wij <wyniijj2@gmail.com> writes:
> On Thursday, 8 September 2022 at 04:32:47 UTC+8, Keith Thompson wrote:
>> wij <wyni...@gmail.com> writes:
>> [...]
>> > +-------------+
>> > | Real number |
>> > +-------------+
>> > Real number is the representation of the distance between two arbitrary points
>> > in space. Real numbers are the result of measurement of real entities, e.g.
>> > time, distance, mass.
>> > General recognition is that decimals are real number, including infinitely long
>> > decimals (classified as irrational number).
>> [...]
>>
>> Is 1/3 a rational number? If the words you use have their usual
>> meanings, then by your definition it is not. 1/3 can be represented
>> by an infinitely long (unbounded) decimal: 0.333..., where the
>> "..." denotes an unbounded sequence. The represented value is
>> rational because the sequence *repeats*.
>>
>> "Decimals" are one way of *representing* real numbers. I'm not
>> convinced that they're the best way of *defining* real numbers,
>> though I suppose you could.

I note that you never even attempt to answer my question.

My question was "Is 1/3 a rational number?". Either "Yes" or "No" would
have been an answer.

> Lots evidences have been discussed:
> https://groups.google.com/g/comp.theory/c/KiajP6jsovM
> https://groups.google.com/g/comp.theory/c/XJ08g-VjV1M
> https://groups.google.com/g/comp.theory/c/ImnwhKe_YFE
> https://groups.google.com/g/comp.theory/c/wlaV7nnV0xo

Great. Based on those "evidences", is 1/3 a rational number or not?

> Last 'key' question are from https://groups.google.com/g/comp.theory/c/KiajP6jsovM
>
> 1. A valid logic to remove 'lim' from a limit expression to form a conclusion.
> E.g. "lim 0.999...=1" (make it more formal the way you like) can yield the conclusion "0.999...=1".
>
> 2. Skep Dick already showed that you cannot even decide whether x-1 or x+1 is
> closer to infinity or not in expression "lim(x->∞) 1-1/10^x = 0.999...".
> Provide more specific explanations what you mean by the occurrence of infinity
> in text-books.
>
> 3. there is no arithmetic rule applicable in limit calculation. If you are so
> confident. Please provide arithmetic/math rules in limit calculation that people
> can follow without ambiguity.
>
> This question can be make more specific:
>
> f(x)= x^2
> f'(x)= lim(h->0) ((x+h)^2-x^2)/h // definition of derivative, h cannot be 0
> = lim(h->0) (x^2+2xh+h^2-x^2)/h // h≠0 (h in ℝ?, Is h a number?)
> = lim(h->0) (2xh + h^2)/h
> = lim(h->0) 2x +h
> = 2x // why 'lim' is removed to conclude f'(x)=2x ? because h=0 now?
> // You won't have a logical answer for the absurdity
>
> Don't bother digging this old absurdity. In general, the rule "a+0=a" (or more)
> does not apply in limit calculation. We all know well, inside limit are
> calculations for 'limit', not exactly equal. And, no need to be exact, because
> there exist big-o or other methods.
>
> The evidence accumulates if we continue to discuss the "0.999...=1" issue.
> Solve these key questions first (and those in provided links).
> I only got "convention", "belief" answers so far.

Apparently you dislike the standard definition of "limit". I've seen no
evidence that you even understand it.

> ------
> By now, another case emerges:
>
> ℙosNum(B)::= Σ(r=a,b) di*B^r, di,B∈ℕ, i,r∈ℤ, B>1, ∀i, 0<=di<B
>
> Decimal::= Positional number (plus +-), of which some digits represent numbers <1.
>
> Lemma: Rational number ℚ and positional number ℙosNum(B) are mutually convertible.
>
> Proof: For simplicity, only positive numbers are shown.
> 1. Let x="abc.ijk", x∈ℙosNum(B) by definition,
> then x="abc.ijk"= "abcijk" / B^3. Therefore, x∈ℚ.
> 2. ∀x∈ℚ, x>0, ∃p,q∈ℤ, q>=2, such that x=p/q
> x= p/q= r+s/q (r=[p/q], s=p-r*q)
> Therefore, p/q can be represented by ℙosNum(q) as "r.s".
>
> [Note] This lemma also shows that repeating decimals are irrational because
> repeating decimals ∉ ℙosNum(B).
>
> So, 1. Provide your proof that 1/3=0.333... 2.Refute the lemma above.
>
> Don't bother. Such problems are many, evidence can accumulate as discussing more.

OK, I certainly won't bother proving that 1/3=0.333... Others have
already done so, and there's no reason to think you would accept any
such proof -- but you wouldn't be able to point out any actual flaws in
it.

I intended to ask a simple question. I see I was overly optimistic in
thinking that you'd be able to answer it.

(If you post a followup to this, please don't quote my signature.)

--
Keith Thompson (The_Other_Keith) Keith.S.Thompson+u@gmail.com
Working, but not speaking, for Philips
void Void(void) { Void(); } /* The recursive call of the void */

On 9/8/2022 12:43 PM, Keith Thompson wrote:
> wij <wyniijj2@gmail.com> writes:

<SNIP> not relevant to this off-topic posting.

> (If you post a followup to this, please don't quote my signature.)

It seems that many posters are reformatting posting when they reply
without knowing signature conventions or are using news reader software
that has the same ignorance built in.

The convention is part of a RFC (Request For Comment) whose number I do
not recall. RFCs are used to document all networking conventions or at
least did so in the past.

For those curious about the relevant convention, here if is:

In an email or USENET-like message, a line that exactly and only
contains "-- " that is two dashes and a space should be deleted in any
reply AND so should everything following it.

This convention gave an intelligent way to sign messages with signature
files and not accumulating all that BS in threads. Of course this
convention was born in a top-posting as default world. This is still
true in USENET - the default is top posting except for a few jerks that
want to troll by making threads unreadable. However, most ordinary email
is set up to top post because of tiny screens on phones, et al.
--
Jeff Barnett

Jeff Barnett <jbb@notatt.com> writes:
[...]
> This convention gave an intelligent way to sign messages with
> signature files and not accumulating all that BS in threads. Of course
> this convention was born in a top-posting as default world. This is
> still true in USENET - the default is top posting except for a few
> jerks that want to troll by making threads unreadable. However, most
> ordinary email is set up to top post because of tiny screens on
> phones, et al.

Did you accidentally flip a bit there? The default in Usenet is bottom
posting, i.e., new material following quoted material -- which is what
makes it important to trim irrelevant parts of the parent article.
Top-posting is common in email.

--
Keith Thompson (The_Other_Keith) Keith.S.Thompson+u@gmail.com
Working, but not speaking, for Philips
void Void(void) { Void(); } /* The recursive call of the void */

On 9/8/2022 1:27 PM, Keith Thompson wrote:
> Jeff Barnett <jbb@notatt.com> writes:
> [...]
>> This convention gave an intelligent way to sign messages with
>> signature files and not accumulating all that BS in threads. Of course
>> this convention was born in a top-posting as default world. This is
>> still true in USENET - the default is top posting except for a few
>> jerks that want to troll by making threads unreadable. However, most
>> ordinary email is set up to top post because of tiny screens on
>> phones, et al.
>
> Did you accidentally flip a bit there? The default in Usenet is bottom
> posting, i.e., new material following quoted material -- which is what
> makes it important to trim irrelevant parts of the parent article.
> Top-posting is common in email.
Of course you are right. Let me post this corrected:

It seems that many posters are reformatting posting when they reply
without knowing signature conventions or are using news reader software
that has the same ignorance built in.

The convention is part of a RFC (Request For Comment) whose number I do
not recall. RFCs are used to document all networking conventions or at
least did so in the past.

For those curious about the relevant convention, here if is:

In an email or USENET-like message, a line that exactly and only
contains "-- " that is two dashes and a space should be deleted in any
reply AND so should everything following it.

This convention gave an intelligent way to sign messages with signature
files and not accumulating all that BS in threads. Of course this
convention was born in a bottom-posting as default world. This is still
true in USENET - the default is bottom posting except for a few jerks
that want to troll by making threads unreadable. However, most ordinary
email is set up to top post because of tiny screens on phones, et al. so
one needs to deal with that reality.
--
Jeff Barnett

On Thursday, 8 September 2022 at 02:30:32 UTC+8, wij wrote:
> ...[cut]
> This view of number and real number is quite simple (basically).
> The view from Analysis/Calculus/Advanced math., as said, is their theoretic
> problem in their domain.
> I'd like to address and re-assure the number view from the point of (digital)
> computers and discrete math. is reasonable and acceptabale.

--Revision changes--
1. Definition of positional number changed to allow +/- sign
2. Notation 'ℙosNum(B)' changed to 'ℙosNum<B>'
3. Proof is added for the lemma that ℚ and ℙosNum<B> are mutually convertible.

.........................................................................................................................................
+----------------------------------------+
| Number is an expression of computation |
+----------------------------------------+
Numbers are basically computational expressions (natural number plus operation,
e.g. positional number, rational, continued fraction,...). Removing operators
from the expression, what left are natural numbers. Yet, natural numbers are
basically successor functions(operation). Therefore, number is an expression of
computation. A normal number, or 'constant', can be treated as as a name,
representing a 'standardized' operation. Common computation of an expression
stops when a standardized operation is derived, because most major properties
are well known.

Number::= An expression/procedure (or a name) of computation (operation).

Positional Number::= Usual number notation, like decimal, an abbreviation of
'restricted' polynomial. Positional number may be prefixed with +/- sign.

ℙosNum<B>::= ±Σ(r=a,b) di*B^r, di,B∈ℕ, i,r∈ℤ, B>1, ∀i, 0<=di<B

Decimal::= Positional number, of which some digits may represent numbers <1.

Lemma: Rational number ℚ and positional number ℙosNum<B> are mutually convertible.

Proof: For simplicity, only positive numbers are shown.
1. Let x="abc.ijk", x∈ℙosNum<B> by definition,
then x="abc.ijk"= "abcijk" / B^3. Therefore, x∈ℚ.
2. ∀x∈ℚ, x>0, ∃p,q∈ℤ, q>=2, such that x=p/q
x= p/q= r+s/q (r=[p/q], s=p-r*q)
Therefore, p/q can be represented by ℙosNum<q> as "r.s"..

[Note] This lemma also shows that repeating decimals are irrational because
repeating decimals ∉ ℙosNum<B>.

+-------------+
| Real number |
+-------------+
Real number is the representation of the distance between two arbitrary points
in space. Real numbers are the result of measurement of real entities, e.g.
time, distance, mass.
General recognition is that decimals are real number, including infinitely long
decimals (classified as irrational number).

Real number ℝ ::= Decimals are real number, including infinitely long decimals
which are normally expressed by infinite series.

[Note] Real number possesses properties such as trichotomy, density,..., but
this file treats it as implied properties from the above definition.
Another case to consider is closure property. Real number 'must' remain
in ℝ by 'legal' operations, but 'legal operation' is difficult to
define.
[Note] Real number could be basically this simple. Those continuous,
differentiable, convergent,...,etc. properties in real analysis/
calculus/advanced math. are considered invention of that domain,
irrelevant to this general recognition.

Theorem: No onto function exists from {0,1}^* to {0,1}^∞
Proof: Omitted

This theorem means that most real numbers are not expressible by finite length
of symbols.

Theorem: Let A(0)=0, A(n) is between A(n-1) and L by a factor s/r (L>0, 0<s/r<1).
A(n)= A(n-1) + (L-A(n-1))*(s/r)
= (r*A(n-1) + (s)*L - s*A(n-1))/r
= ( s*L + (r-s)*A(n-1) )/r
Then, A(n)<L holds.

Proof: 1. The procedure can go on infinitely, n can be an infinity.
2. A(n)>=L causes contradiction from the given and A(n-1)>=A(n).

Lemma: Let L=1, r>=2, s=r-1, all form of positional number "0.999..." is less
than 1.
Lemma: Let the greatest different digit (same position) of positional number
A,B (A>0,B>0) be a,b, then a>b => A>B
Lemma: All the digits of positional number A,B are equal iff A=B.

Op 09.sep..2022 om 12:56 schreef wij:
> On Thursday, 8 September 2022 at 02:30:32 UTC+8, wij wrote:
>> ...[cut]
>> This view of number and real number is quite simple (basically).
>> The view from Analysis/Calculus/Advanced math., as said, is their theoretic
>> problem in their domain.
>> I'd like to address and re-assure the number view from the point of (digital)
>> computers and discrete math. is reasonable and acceptabale.
>
> --Revision changes--
> 1. Definition of positional number changed to allow +/- sign
> 2. Notation 'ℙosNum(B)' changed to 'ℙosNum<B>'
> 3. Proof is added for the lemma that ℚ and ℙosNum<B> are mutually convertible.
>
<Snip>
>
> +-------------+
> | Real number |
> +-------------+
> Real number is the representation of the distance between two arbitrary points
> in space. Real numbers are the result of measurement of real entities, e.g.
> time, distance, mass.
> General recognition is that decimals are real number, including infinitely long
> decimals (classified as irrational number).
<Snip>

I see no change for Real numbers. You still say that real numbers are
the result of measurement. How much time would such a measurement take
for infinitely long decimals?

On Thursday, 8 September 2022 at 02:30:32 UTC+8, wij wrote:

Just finished a stage summary: Simply put, decimals are real number.
https://sourceforge.net/projects/cscall/files/MisFiles/NumberView-en.txt/download
....[snippet]
+-------------+
| Real number |
+-------------+
Real number is used to represent the distance between two arbitrary points in
space, the measuring result of real entities, e.g. time,mass,...,etc.
General recognition is that decimals are real number, including infinitely long
decimals (classified as irrational number).

Real number ℝ ::= Decimals are real number, including infinitely long decimals
which are normally expressed by infinite series.

[Note] Real number possesses properties such as trichotomy, density,..., but
this file treats it as implied properties from the above definition.
Another case to consider is closure property. Real number 'must' remain
in ℝ by 'legal' operations, but 'legal operation' is difficult to
define.
[Note] Real number could be basically this simple. Those continuous,
differentiable, convergent,...,etc. properties in real analysis/
calculus/advanced math. are considered invention of that domain,
irrelevant to this general recognition.
[Note] ∞∈ℝ (∞ is a real number).
Let f(0)=0, f(n)=(f(n-1)+L)/2, then f(∞) approaches L. (many irrational
numbers are derived from similar exhaustion method). Because f(∞)∈ℝ,
L-f(∞) (infinitesimal) and 1/(L-f(∞)) (infinity) must also be real
numbers from the closure issue.

Theorem: No onto function exists maping from set {0,1}^* to {0,1}^∞
Proof: Omitted

This theorem means that most real numbers are not expressible by finite length
of symbols.

Theorem: Let A(0)=0, A(n) is between A(n-1) and L by a factor s/r (L>0, 0<s/r<1).
A(n)= A(n-1) + (L-A(n-1))*(s/r)
= (r*A(n-1) + (s)*L - s*A(n-1))/r
= ( s*L + (r-s)*A(n-1) )/r
Then, A(n)<L holds.

Proof: 1. The procedure can go on infinitely, n can be an infinity.
2. A(n)>=L causes contradiction from the given and A(n-1)>=A(n).

Lemma: Let L=1, r>=2, s=r-1, all form of positional number "0.999..." is less
than 1.
Lemma: Let the greatest different digit (same position) of positional number
A,B (A>0,B>0) be a,b, then a>b => A>B
Lemma: All the digits of positional number A,B are equal iff A=B.

--- End of snippet

Some related topics might be interested:
.Repeating decimals are irrational (therefore 0.0.999...≠1. "Repeating decimal"
cannot denote a specific number)
.Definition of infinity '∞'
.Definition of single-point-slope
.Interpretation of infinite series