This updated file should solve most doubts I encountered. Hope, useful to you
(of course, not in official exam if that is your 'real')
https://sourceforge.net/projects/cscall/files/MisFiles/RealNumber-en.txt/download

+-------------+
| Real Number | ('computational' may be added to modify terms used in this file
+-------------+ if needed)

n-ary Fixed-Point Number::= Number represented by a string of digits, the
string may contain a minus sign or a point:

<fixed_point_number>::= [-] <dstr1> [ . <dstr2> ]
<dstr1>::= 0 | <nzd> { 0, <nzd> }
<dstr2>::= { 0, <nzd> } <nzd>
<nzd> ::= (1, 2, 3, 4, 5, 6, 7, 8, 9) // 'digit' varys depending on n-ary

Two n-ary fixed-point number (same n-ary) x,y are equal iff their
<fixed_point_number> representation are identical.

Real Nunmber(ℝ)::= {x| x is finitely represented by n-ary <fixed_point_number>
and those that cannot be finitely represented }

Note: Numbers that is not finitely representable cannot all be explicitly
defined, this is the property of real number based on discrete symbols
(like quantum?). E.g.

A= lim(n->∞) 1-3/10^n = 0.999...
B= lim(n->∞) 1-2/2^n = 0.999...
C= lim(n->∞) 1-1/n = 0.999...
...

IOW, by repeatedly multiplying 0.999... with 10, you can only see 9,
the structure of the rear end of 0.999... is not seen.

Since <fixed_point_number> is very definitely real and infinity is
involved, theories that composed of finite words cannot be too
exclusive about such a ℝ. 'Completeness' is impossible.

Note: This definition implies that repeating decimals are irrational number.
Let's list a common magic proof in the way as a brief explanation:
(1) x= 0.999...
(2) 10x= 9+x // 10x= 9.999...
(3) 9x=9
(4) x=1
Ans: There is no axiom or theorem to prove (1) => (2).

Note: To determine whether a repeating decimal x is rational or not, we can
repeatedly subtract the repeating pattern p(i) from x.
If x-p(1)-p(2)-...=0 can be verified in finite steps, then x is
rational. Otherwise, x is irrational, because, if x is rational, the
last remaining piece r(i)= x-p(1)-p(2)-... must exactly be the
repeating pattern p(i). But, by definition of 'repeating', r(i) cannot
be pattern p(i). Therefore, repeating decimal is irrational.

Real number is just this simple. The limit theory is a methodology for finding
derivative, nothing to do with what the real number is (otherwise, a definition
like the above must be defined in advance to avoid circular-reasoning).

+-------+
| Limit |
+-------+
Limit::= lim(x->a) f(x)=L
http://www.math.ntu.edu.tw/~mathcal/download/precal/PPT/Chapter%2002_04..pdf
http://www.math.ncu.edu.tw/~yu/ecocal98/boards/lec6_ec_98.pdf
https://en.wikipedia.org/wiki/Limit_(mathematics)
https://en.wikipedia.org/wiki/Limit_of_a_function

L is defined as the limit (a number) while x approaches a (f(a) may not
be defined, although, while f is continuous, L=f(a)). L is a defined value,
not "if something infinitely close ... then equal" (no such logic).

Ex1: A= lim(n->∞) 1-1/n= lim(n->0⁺) 1-n= lim 0.999...=1
B= lim(n->∞) 1+1/n= lim(n->0⁺) 1+n= lim 1.000..?=1

Ex2: A=lim(x->ℵ₀) f(x), B=lim(x->ℵ₁) f(x) // ℵ₀,ℵ₁ being proper or not is
// another issue here. But problematic
// for "finally equal" interpretation.

Limit defines A=B, does not mean the contents of the limit are equal. If the
"x approaches..., then equal" notion is adopted, lots of logical issues arise.

Note: The equation of limit may be questionable
lim(x->c) (f(x)*g(x))= (lim(x->c) f(x))*(lim(x->c) g(x)):

Let A=lim(n->∞) (1-1/n)= 1
A*A*..*A= ... = lim(n->∞) (1-1/n)^n // 1=1/e ?

+--------------------------------------+
| Restoring Interpretation of Calculus |
+--------------------------------------+
http://www.math.ntu.edu.tw/~mathcal/download/precal/PPT/Chapter%2002_08.pdf
https://en.wikipedia.org/wiki/Derivative

Assume calculus is basically the area problem of a function: Let F compute the
the area of f. From the meaning of area, we can have:

(F(x+h)-F(x)) ≒ (f(x+h)+f(x))*(h/2) // h is a sufficiently small (test)offset
<=> (F(x+h)-F(x))/h ≒ (f(x+h)+f(x))/2 // the limit(h->0) of rhs is f(x)

Expected property of F: (1)Error |lhs-rhs| strictly decreases with the tiny
(test) offset h (2)When h=0, lhs=rhs.
Because the h in the lhs cannot be 0, the basic problem of calculus is
finding such a F (or f) that satisfies the expected porperty above...Thus,

D(f(x))= lim(h->0) (F(x+h)-F(x))/h = f(x)

Note: Hope that this interpretation can avoid the interpretation of infinity
/infinitesimal, and provide more correct foundation for some theories
, e.g. Zeno paradoxes, repeating decimal,...,and more (exponiential,
Cantor set, infinite series...).

-------------------------------------------------------------------------------

Op 28.mrt.2024 om 13:29 schreef wij:
> This updated file should solve most doubts I encountered. Hope, useful to you
> (of course, not in official exam if that is your 'real')
> https://sourceforge.net/projects/cscall/files/MisFiles/RealNumber-en.txt/download
>
> +-------------+
> | Real Number | ('computational' may be added to modify terms used in this file
> +-------------+ if needed)
>
> n-ary Fixed-Point Number::= Number represented by a string of digits, the
> string may contain a minus sign or a point:
>
> <fixed_point_number>::= [-] <dstr1> [ . <dstr2> ]
> <dstr1>::= 0 | <nzd> { 0, <nzd> }
> <dstr2>::= { 0, <nzd> } <nzd>
> <nzd> ::= (1, 2, 3, 4, 5, 6, 7, 8, 9) // 'digit' varys depending on n-ary
>
> Two n-ary fixed-point number (same n-ary) x,y are equal iff their
> <fixed_point_number> representation are identical.
>
> Real Nunmber(ℝ)::= {x| x is finitely represented by n-ary <fixed_point_number>
> and those that cannot be finitely represented }
>
> Note: Numbers that is not finitely representable cannot all be explicitly
> defined, this is the property of real number based on discrete symbols
> (like quantum?). E.g.
>
> A= lim(n->∞) 1-3/10^n = 0.999...
> B= lim(n->∞) 1-2/2^n = 0.999...
> C= lim(n->∞) 1-1/n = 0.999...
> ...
>
> IOW, by repeatedly multiplying 0.999... with 10, you can only see 9,
> the structure of the rear end of 0.999... is not seen.
>
> Since <fixed_point_number> is very definitely real and infinity is
> involved, theories that composed of finite words cannot be too
> exclusive about such a ℝ. 'Completeness' is impossible.
>
> Note: This definition implies that repeating decimals are irrational number.
> Let's list a common magic proof in the way as a brief explanation:
> (1) x= 0.999...
> (2) 10x= 9+x // 10x= 9.999...
> (3) 9x=9
> (4) x=1
> Ans: There is no axiom or theorem to prove (1) => (2).
>
> Note: To determine whether a repeating decimal x is rational or not, we can
> repeatedly subtract the repeating pattern p(i) from x.
> If x-p(1)-p(2)-...=0 can be verified in finite steps, then x is
> rational. Otherwise, x is irrational, because, if x is rational, the
> last remaining piece r(i)= x-p(1)-p(2)-... must exactly be the
> repeating pattern p(i). But, by definition of 'repeating', r(i) cannot
> be pattern p(i). Therefore, repeating decimal is irrational.
>
> Real number is just this simple. The limit theory is a methodology for finding
> derivative, nothing to do with what the real number is (otherwise, a definition
> like the above must be defined in advance to avoid circular-reasoning).
>
> +-------+
> | Limit |
> +-------+
> Limit::= lim(x->a) f(x)=L
> http://www.math.ntu.edu.tw/~mathcal/download/precal/PPT/Chapter%2002_04.pdf
> http://www.math.ncu.edu.tw/~yu/ecocal98/boards/lec6_ec_98.pdf
> https://en.wikipedia.org/wiki/Limit_(mathematics)
> https://en.wikipedia.org/wiki/Limit_of_a_function
>
> L is defined as the limit (a number) while x approaches a (f(a) may not
> be defined, although, while f is continuous, L=f(a)). L is a defined value,
> not "if something infinitely close ... then equal" (no such logic).
>
> Ex1: A= lim(n->∞) 1-1/n= lim(n->0⁺) 1-n= lim 0.999...=1
> B= lim(n->∞) 1+1/n= lim(n->0⁺) 1+n= lim 1.000..?=1
>
> Ex2: A=lim(x->ℵ₀) f(x), B=lim(x->ℵ₁) f(x) // ℵ₀,ℵ₁ being proper or not is
> // another issue here. But problematic
> // for "finally equal" interpretation.
>
> Limit defines A=B, does not mean the contents of the limit are equal. If the
> "x approaches..., then equal" notion is adopted, lots of logical issues arise.
>
> Note: The equation of limit may be questionable
> lim(x->c) (f(x)*g(x))= (lim(x->c) f(x))*(lim(x->c) g(x)):
>
> Let A=lim(n->∞) (1-1/n)= 1
> A*A*..*A= ... = lim(n->∞) (1-1/n)^n // 1=1/e ?
>
> +--------------------------------------+
> | Restoring Interpretation of Calculus |
> +--------------------------------------+
> http://www.math.ntu.edu.tw/~mathcal/download/precal/PPT/Chapter%2002_08.pdf
> https://en.wikipedia.org/wiki/Derivative
>
> Assume calculus is basically the area problem of a function: Let F compute the
> the area of f. From the meaning of area, we can have:
>
> (F(x+h)-F(x)) ≒ (f(x+h)+f(x))*(h/2) // h is a sufficiently small (test)offset
> <=> (F(x+h)-F(x))/h ≒ (f(x+h)+f(x))/2 // the limit(h->0) of rhs is f(x)
>
> Expected property of F: (1)Error |lhs-rhs| strictly decreases with the tiny
> (test) offset h (2)When h=0, lhs=rhs.
> Because the h in the lhs cannot be 0, the basic problem of calculus is
> finding such a F (or f) that satisfies the expected porperty above...Thus,
>
> D(f(x))= lim(h->0) (F(x+h)-F(x))/h = f(x)
>
> Note: Hope that this interpretation can avoid the interpretation of infinity
> /infinitesimal, and provide more correct foundation for some theories
> , e.g. Zeno paradoxes, repeating decimal,...,and more (exponiential,
> Cantor set, infinite series...).
>
> -------------------------------------------------------------------------------
>
>

It seems that wij wants to define a number type that is different than
the real numbers, but wij uses the same name Real. Very confusing.

Further, it seems he only defines how these number are written down.
There is no explanation of how to interpret these writings. No order is
defined. (Is 0.333333 less, equal, or greater than 0.9?) No operations
are defined, such as + - / * etc. It is not explained whether the same
number can have different representations, such as normal real numbers
have (such as 3E10 and 3.0E10 and 30E9 all for the same number). How can
we see whether different writings are about different numbers?

Talking about this proposal is possible only when not only the
representation of the numbers is defined, but also what they mean and
how to work with them.

Then suddenly the term 'lim' is used without definition. What does that
mean in this context?

On Thu, 2024-03-28 at 14:16 +0100, Fred. Zwarts wrote:
> Op 28.mrt.2024 om 13:29 schreef wij:
> > This updated file should solve most doubts I encountered. Hope, useful to you
> > (of course, not in official exam if that is your 'real')
> > https://sourceforge.net/projects/cscall/files/MisFiles/RealNumber-en.txt/download
> >
> > +-------------+
> > > Real Number | ('computational' may be added to modify terms used in this file
> > +-------------+   if needed)
> >
> > n-ary Fixed-Point Number::= Number represented by a string of digits, the
> >     string may contain a minus sign or a point:
> >
> >       <fixed_point_number>::= [-] <dstr1> [ .. <dstr2> ]
> >       <dstr1>::= 0 | <nzd> { 0, <nzd> }
> >       <dstr2>::= { 0, <nzd> } <nzd>
> >       <nzd> ::= (1, 2, 3, 4, 5, 6, 7, 8, 9)  // 'digit' varys depending on n-ary
> >
> >     Two n-ary fixed-point number (same n-ary) x,y are equal iff their
> >     <fixed_point_number> representation are identical.
> >
> > Real Nunmber(ℝ)::= {x| x is finitely represented by n-ary <fixed_point_number>
> >     and those that cannot be finitely represented }
> >
> >     Note: Numbers that is not finitely representable cannot all be explicitly
> >           defined, this is the property of real number based on discrete symbols
> >           (like quantum?). E.g.
> >
> >           A= lim(n->∞) 1-3/10^n = 0.999...
> >           B= lim(n->∞) 1-2/2^n  = 0.999...
> >           C= lim(n->∞) 1-1/n    = 0.999...
> >           ...
> >
> >           IOW, by repeatedly multiplying 0.999... with 10, you can only see 9,
> >           the structure of the rear end of 0.999... is not seen.
> >
> >           Since <fixed_point_number> is very definitely real and infinity is
> >           involved, theories that composed of finite words cannot be too
> >           exclusive about such a ℝ. 'Completeness' is impossible.
> >
> >     Note: This definition implies that repeating decimals are irrational number.
> >           Let's list a common magic proof in the way as a brief explanation:
> >             (1) x= 0.999...
> >             (2) 10x= 9+x  // 10x= 9.999...
> >             (3) 9x=9
> >             (4) x=1
> >           Ans: There is no axiom or theorem to prove (1) => (2).
> >
> >     Note: To determine whether a repeating decimal x is rational or not, we can
> >           repeatedly subtract the repeating pattern p(i) from x.
> >           If x-p(1)-p(2)-....=0 can be verified in finite steps, then x is
> >           rational. Otherwise, x is irrational, because, if x is rational, the
> >           last remaining piece r(i)= x-p(1)-p(2)-... must exactly be the
> >           repeating pattern p(i). But, by definition of 'repeating', r(i) cannot
> >           be pattern p(i). Therefore, repeating decimal is irrational.
> >
> > Real number is just this simple. The limit theory is a methodology for finding
> > derivative, nothing to do with what the real number is (otherwise, a definition
> > like the above must be defined in advance to avoid circular-reasoning).
> >
> > +-------+
> > > Limit |
> > +-------+
> > Limit::=  lim(x->a) f(x)=L
> >      http://www.math.ntu.edu.tw/~mathcal/download/precal/PPT/Chapter%2002_04.pdf
> >      http://www.math.ncu.edu.tw/~yu/ecocal98/boards/lec6_ec_98.pdf
> >      https://en.wikipedia.org/wiki/Limit_(mathematics)
> >      https://en.wikipedia.org/wiki/Limit_of_a_function
> >
> >      L is defined as the limit (a number) while x approaches a (f(a) may not
> >      be defined, although, while f is continuous, L=f(a)). L is a defined value,
> >      not "if something infinitely close ... then equal" (no such logic).
> >
> >      Ex1: A= lim(n->∞) 1-1/n= lim(n->0⁺) 1-n= lim 0.999...=1
> >           B= lim(n->∞) 1+1/n= lim(n->0⁺) 1+n= lim 1.000..?=1
> >
> >      Ex2: A=lim(x->ℵ₀) f(x), B=lim(x->ℵ₁) f(x)  // ℵ₀,ℵ₁ being proper or not is
> >                                           // another issue here. But problematic
> >                                           // for "finally equal" interpretation.
> >
> >      Limit defines A=B, does not mean the contents of the limit are equal. If the
> >      "x approaches..., then equal" notion is adopted, lots of logical issues arise.
> >
> >      Note: The equation of limit may be questionable
> >            lim(x->c) (f(x)*g(x))= (lim(x->c) f(x))*(lim(x->c) g(x)):
> >
> >            Let A=lim(n->∞) (1-1/n)= 1
> >            A*A*..*A= ... = lim(n->∞) (1-1/n)^n    // 1=1/e ?
> >
> > +--------------------------------------+
> > > Restoring Interpretation of Calculus |
> > +--------------------------------------+
> > http://www.math.ntu.edu.tw/~mathcal/download/precal/PPT/Chapter%2002_08..pdf
> > https://en.wikipedia.org/wiki/Derivative
> >
> > Assume calculus is basically the area problem of a function: Let F compute the
> > the area of f. From the meaning of area, we can have:
> >
> >      (F(x+h)-F(x)) ≒ (f(x+h)+f(x))*(h/2)  // h is a sufficiently small (test)offset
> > <=> (F(x+h)-F(x))/h ≒ (f(x+h)+f(x))/2    // the limit(h->0) of rhs is f(x)
> >
> > Expected property of F: (1)Error |lhs-rhs| strictly decreases with the tiny
> >      (test) offset h (2)When h=0, lhs=rhs.
> >      Because the h in the lhs cannot be 0, the basic problem of calculus is
> >      finding such a F (or f) that satisfies the expected porperty above...Thus,
> >
> >      D(f(x))= lim(h->0) (F(x+h)-F(x))/h = f(x)
> >
> >      Note: Hope that this interpretation can avoid the interpretation of infinity
> >            /infinitesimal, and provide more correct foundation for some theories
> >            , e.g. Zeno paradoxes, repeating decimal,...,and more (exponiential,
> >            Cantor set, infinite series...).
> >
> > -------------------------------------------------------------------------------
> >
> >
>
> It seems that wij wants to define a number type that is different than
> the real numbers, but wij uses the same name Real. Very confusing.
>
> Further, it seems he only defines how these number are written down.
> There is no explanation of how to interpret these writings. No order is
> defined. (Is 0.333333 less, equal, or greater than 0.9?) No operations
> are defined, such as + - / * etc. It is not explained whether the same
> number can have different representations, such as normal real numbers
> have (such as 3E10 and 3.0E10 and 30E9 all for the same number). How can
> we see whether different writings are about different numbers?
>
> Talking about this proposal is possible only when not only the
> representation of the numbers is defined, but also what they mean and
> how to work with them.
>
> Then suddenly the term 'lim' is used without definition. What does that
> mean in this context?
>

Idiot. Show me how your real is constructed.

On 28/03/2024 13:16, Fred. Zwarts wrote:
> It seems that wij wants to define a number type that is different
> than the real numbers, but wij uses the same name Real. Very
> confusing.

It seems to me to be worse than that. Wij apparently thinks he
/is/ defining the real numbers, and that the traditional definitions are
wrong in some way that he has never managed to explain. But as he uses
infinity and infinitesimals [in an unexplained way], he is breaking the
Archimedean/Eudoxian axiom, so Wij-reals are not R, and they seem also
not to be any of the other usual real-like number systems. So the whole
of mathematical physics, engineering, ... is left in limbo, with all the
standard theorems inapplicable unless/until Wij tells us much more, and
probably not even then judging by Wij's responses thus far.

> Further, it seems he only defines how these number are written down.
> There is no explanation of how to interpret these writings.

Well, quite. It seems that we're supposed to use the standard
processes of arithmetic until we get to infinity and similar. But of
course mathematics is concerned with numbers much more than with how
they are notated.

All might become clear if Wij could explain what problem he is
really trying to solve. What bridges fall down if "traditional" maths
is used but stay up with Wij-reals? What new puzzles are soluble? Are
they somehow more logical, or easier to teach? He seems to think that
"trad" maths is full of holes that he sees but that all the great minds
of the past 2500 years have overlooked. Perhaps it's all or mostly lost
in translation, but it's more likely that he is joining the PO Club.

--
Andy Walker, Nottingham.
Andy's music pages: www.cuboid.me.uk/andy/Music
Composer of the day: www.cuboid.me.uk/andy/Music/Composers/Couperin

On 3/28/2024 10:59 AM, Andy Walker wrote:
> On 28/03/2024 13:16, Fred. Zwarts wrote:
>> It seems that wij wants to define a number type that is different
>> than the real numbers, but wij uses the same name Real. Very
>> confusing.
>
>     It seems to me to be worse than that.  Wij apparently thinks he
> /is/ defining the real numbers, and that the traditional definitions are
> wrong in some way that he has never managed to explain.  But as he uses
> infinity and infinitesimals [in an unexplained way], he is breaking the
> Archimedean/Eudoxian axiom, so Wij-reals are not R, and they seem also
> not to be any of the other usual real-like number systems.  So the whole
> of mathematical physics, engineering, ... is left in limbo, with all the
> standard theorems inapplicable unless/until Wij tells us much more, and
> probably not even then judging by Wij's responses thus far.
>

Yet it seems that wij is correct that 0.999... would seem to
be infinitesimally < 1.0. One geometric point on the number line.
[0.0, 1.0) < [0.0, 1.0] by one geometric point.

>> Further, it seems he only defines how these number are written down.
>> There is no explanation of how to interpret these writings.
>
>     Well, quite.  It seems that we're supposed to use the standard
> processes of arithmetic until we get to infinity and similar.  But of
> course mathematics is concerned with numbers much more than with how
> they are notated.
>
>     All might become clear if Wij could explain what problem he is
> really trying to solve.  What bridges fall down if "traditional" maths
> is used but stay up with Wij-reals?  What new puzzles are soluble?  Are
> they somehow more logical, or easier to teach?  He seems to think that
> "trad" maths is full of holes that he sees but that all the great minds
> of the past 2500 years have overlooked.  Perhaps it's all or mostly lost
> in translation, but it's more likely that he is joining the PO Club.
>

--
Copyright 2024 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer

On 28/03/2024 16:07, olcott wrote:
> Yet it seems that wij is correct that 0.999... would seem to
> be infinitesimally < 1.0.

That /cannot/ be correct in the "real" numbers, in which there
are no infinitesimals [basic axiom of the reals]. In other systems of
numbers, it could be correct, but that will depend on what is meant by
"0.999..", and note that if you appeal to something that mentions limits
to define this, then you have to explain how infinite and infinitesimal
numbers are handled in the definition.

> One geometric point on the number line.
> [0.0, 1.0) < [0.0, 1.0] by one geometric point.

Until you describe the axioms of what you mean by "geometric
point" and "number line", this is meaningless verbiage. Give your
axioms, and it becomes possible to discuss this. Until then, we are
entitled to assume that you and Wij are talking about the "traditional"
"real" numbers [as used in engineering, etc.] in which there are no
infinitesimals, and so the only interpretation we can make of the size
of "one geometric point" is the usual "measure", which is zero.

To repeat [to both you and Wij]: *Show us your axioms, and this
may perhaps be worth discussing.* In particular, we need to know where
and why you are departing from standard axiomatisations of the reals.
[For the latter, simplest is to google for "axioms of real numbers",
which throws up dozens of articles ranging from elementary to extremely
advanced.]

--
Andy Walker, Nottingham.
Andy's music pages: www.cuboid.me.uk/andy/Music
Composer of the day: www.cuboid.me.uk/andy/Music/Composers/Couperin

On 3/28/2024 11:43 AM, Andy Walker wrote:
> On 28/03/2024 16:07, olcott wrote:
>> Yet it seems that wij is correct that 0.999... would seem to
>> be infinitesimally < 1.0.
>
>     That /cannot/ be correct in the "real" numbers, in which there
> are no infinitesimals [basic axiom of the reals].  In other systems of
> numbers, it could be correct,

Yes.

> but that will depend on what is meant by
> "0.999..",

Approaching yet never reaching 1.0.

> and note that if you appeal to something that mentions limits
> to define this, then you have to explain how infinite and infinitesimal
> numbers are handled in the definition.
>
>>                   One geometric point on the number line.
>> [0.0, 1.0) < [0.0, 1.0] by one geometric point.
>
>     Until you describe the axioms of what you mean by "geometric
> point" and "number line", this is meaningless verbiage.  Give your

Of course by geometric point I must mean a box of chocolates and by
number line I mean a pretty pink bow. No one would ever suspect that
these terms have their conventional meanings.

> axioms, and it becomes possible to discuss this.  Until then, we are
> entitled to assume that you and Wij are talking about the "traditional"
> "real" numbers [as used in engineering, etc.] in which there are no
> infinitesimals, and so the only interpretation we can make of the size
> of "one geometric point" is the usual "measure", which is zero.
>

Yet it is never actually zero because it is possible to specify a
line segment that is exactly one geometric point longer than another.
[0.0, 1.0] - [0.0, 1.0) = one geometric point.

>     To repeat [to both you and Wij]:  *Show us your axioms, and this
> may perhaps be worth discussing.*  In particular, we need to know where
> and why you are departing from standard axiomatisations of the reals.
> [For the latter, simplest is to google for "axioms of real numbers",
> which throws up dozens of articles ranging from elementary to extremely
> advanced.]
>

--
Copyright 2024 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer

On Thu, 2024-03-28 at 11:53 -0500, olcott wrote:
> On 3/28/2024 11:43 AM, Andy Walker wrote:
> > On 28/03/2024 16:07, olcott wrote:
> > > Yet it seems that wij is correct that 0.999... would seem to
> > > be infinitesimally < 1.0.
> >
> >      That /cannot/ be correct in the "real" numbers, in which there
> > are no infinitesimals [basic axiom of the reals].  In other systems of
> > numbers, it could be correct,
>
> Yes.
>
> > but that will depend on what is meant by
> > "0.999..",
>
> Approaching yet never reaching 1.0.
>
> > and note that if you appeal to something that mentions limits
> > to define this, then you have to explain how infinite and infinitesimal
> > numbers are handled in the definition.
> >
> > >                   One geometric point on the number line.
> > > [0.0, 1.0) < [0.0, 1.0] by one geometric point.
> >
> >      Until you describe the axioms of what you mean by "geometric
> > point" and "number line", this is meaningless verbiage.  Give your
>
> Of course by geometric point I must mean a box of chocolates and by
> number line I mean a pretty pink bow. No one would ever suspect that
> these terms have their conventional meanings.
>
> > axioms, and it becomes possible to discuss this.  Until then, we are
> > entitled to assume that you and Wij are talking about the "traditional"
> > "real" numbers [as used in engineering, etc.] in which there are no
> > infinitesimals, and so the only interpretation we can make of the size
> > of "one geometric point" is the usual "measure", which is zero.
> >
>
> Yet it is never actually zero because it is possible to specify a
> line segment that is exactly one geometric point longer than another.
> [0.0, 1.0] - [0.0, 1.0) = one geometric point.
>
> >      To repeat [to both you and Wij]:  *Show us your axioms, and this
> > may perhaps be worth discussing.*  In particular, we need to know where
> > and why you are departing from standard axiomatisations of the reals.
> > [For the latter, simplest is to google for "axioms of real numbers",
> > which throws up dozens of articles ranging from elementary to extremely
> > advanced.]
> >
>

Finally, olcott showed some IQ.

I saw lots of inconsistency in Andy Walker's response. I think the simple 
way to solve his doubt is for him to prove "repeating decimal is rational".
He cannot throw in 'advanced math.", because doing this will not
only indicate ignorance, but also invalid for elementary arithmetic.

Op 28.mrt.2024 om 16:09 schreef wij:
> On Thu, 2024-03-28 at 14:16 +0100, Fred. Zwarts wrote:
>> Op 28.mrt.2024 om 13:29 schreef wij:
>>> This updated file should solve most doubts I encountered. Hope, useful to you
>>> (of course, not in official exam if that is your 'real')
>>> https://sourceforge.net/projects/cscall/files/MisFiles/RealNumber-en.txt/download
>>>
>>> +-------------+
>>>> Real Number | ('computational' may be added to modify terms used in this file
>>> +-------------+   if needed)
>>>
>>> n-ary Fixed-Point Number::= Number represented by a string of digits, the
>>>     string may contain a minus sign or a point:
>>>
>>>       <fixed_point_number>::= [-] <dstr1> [ . <dstr2> ]
>>>       <dstr1>::= 0 | <nzd> { 0, <nzd> }
>>>       <dstr2>::= { 0, <nzd> } <nzd>
>>>       <nzd> ::= (1, 2, 3, 4, 5, 6, 7, 8, 9)  // 'digit' varys depending on n-ary
>>>
>>>     Two n-ary fixed-point number (same n-ary) x,y are equal iff their
>>>     <fixed_point_number> representation are identical.
>>>
>>> Real Nunmber(ℝ)::= {x| x is finitely represented by n-ary <fixed_point_number>
>>>     and those that cannot be finitely represented }
>>>
>>>     Note: Numbers that is not finitely representable cannot all be explicitly
>>>           defined, this is the property of real number based on discrete symbols
>>>           (like quantum?). E.g.
>>>
>>>           A= lim(n->∞) 1-3/10^n = 0.999...
>>>           B= lim(n->∞) 1-2/2^n  = 0.999...
>>>           C= lim(n->∞) 1-1/n    = 0.999...
>>>           ...
>>>
>>>           IOW, by repeatedly multiplying 0.999... with 10, you can only see 9,
>>>           the structure of the rear end of 0.999... is not seen.
>>>
>>>           Since <fixed_point_number> is very definitely real and infinity is
>>>           involved, theories that composed of finite words cannot be too
>>>           exclusive about such a ℝ. 'Completeness' is impossible.
>>>
>>>     Note: This definition implies that repeating decimals are irrational number.
>>>           Let's list a common magic proof in the way as a brief explanation:
>>>             (1) x= 0.999...
>>>             (2) 10x= 9+x  // 10x= 9.999...
>>>             (3) 9x=9
>>>             (4) x=1
>>>           Ans: There is no axiom or theorem to prove (1) => (2).
>>>
>>>     Note: To determine whether a repeating decimal x is rational or not, we can
>>>           repeatedly subtract the repeating pattern p(i) from x.
>>>           If x-p(1)-p(2)-...=0 can be verified in finite steps, then x is
>>>           rational. Otherwise, x is irrational, because, if x is rational, the
>>>           last remaining piece r(i)= x-p(1)-p(2)-... must exactly be the
>>>           repeating pattern p(i). But, by definition of 'repeating', r(i) cannot
>>>           be pattern p(i). Therefore, repeating decimal is irrational.
>>>
>>> Real number is just this simple. The limit theory is a methodology for finding
>>> derivative, nothing to do with what the real number is (otherwise, a definition
>>> like the above must be defined in advance to avoid circular-reasoning).
>>>
>>> +-------+
>>>> Limit |
>>> +-------+
>>> Limit::=  lim(x->a) f(x)=L
>>>      http://www.math.ntu.edu.tw/~mathcal/download/precal/PPT/Chapter%2002_04.pdf
>>>      http://www.math.ncu.edu.tw/~yu/ecocal98/boards/lec6_ec_98.pdf
>>>      https://en.wikipedia.org/wiki/Limit_(mathematics)
>>>      https://en.wikipedia.org/wiki/Limit_of_a_function
>>>
>>>      L is defined as the limit (a number) while x approaches a (f(a) may not
>>>      be defined, although, while f is continuous, L=f(a)). L is a defined value,
>>>      not "if something infinitely close ... then equal" (no such logic).
>>>
>>>      Ex1: A= lim(n->∞) 1-1/n= lim(n->0⁺) 1-n= lim 0.999...=1
>>>           B= lim(n->∞) 1+1/n= lim(n->0⁺) 1+n= lim 1.000..?=1
>>>
>>>      Ex2: A=lim(x->ℵ₀) f(x), B=lim(x->ℵ₁) f(x)  // ℵ₀,ℵ₁ being proper or not is
>>>                                           // another issue here. But problematic
>>>                                           // for "finally equal" interpretation.
>>>
>>>      Limit defines A=B, does not mean the contents of the limit are equal. If the
>>>      "x approaches..., then equal" notion is adopted, lots of logical issues arise.
>>>
>>>      Note: The equation of limit may be questionable
>>>            lim(x->c) (f(x)*g(x))= (lim(x->c) f(x))*(lim(x->c) g(x)):
>>>
>>>            Let A=lim(n->∞) (1-1/n)= 1
>>>            A*A*..*A= ... = lim(n->∞) (1-1/n)^n    // 1=1/e ?
>>>
>>> +--------------------------------------+
>>>> Restoring Interpretation of Calculus |
>>> +--------------------------------------+
>>> http://www.math.ntu.edu.tw/~mathcal/download/precal/PPT/Chapter%2002_08.pdf
>>> https://en.wikipedia.org/wiki/Derivative
>>>
>>> Assume calculus is basically the area problem of a function: Let F compute the
>>> the area of f. From the meaning of area, we can have:
>>>
>>>      (F(x+h)-F(x)) ≒ (f(x+h)+f(x))*(h/2)  // h is a sufficiently small (test)offset
>>> <=> (F(x+h)-F(x))/h ≒ (f(x+h)+f(x))/2    // the limit(h->0) of rhs is f(x)
>>>
>>> Expected property of F: (1)Error |lhs-rhs| strictly decreases with the tiny
>>>      (test) offset h (2)When h=0, lhs=rhs.
>>>      Because the h in the lhs cannot be 0, the basic problem of calculus is
>>>      finding such a F (or f) that satisfies the expected porperty above...Thus,
>>>
>>>      D(f(x))= lim(h->0) (F(x+h)-F(x))/h = f(x)
>>>
>>>      Note: Hope that this interpretation can avoid the interpretation of infinity
>>>            /infinitesimal, and provide more correct foundation for some theories
>>>            , e.g. Zeno paradoxes, repeating decimal,...,and more (exponiential,
>>>            Cantor set, infinite series...).
>>>
>>> -------------------------------------------------------------------------------
>>>
>>>
>>
>> It seems that wij wants to define a number type that is different than
>> the real numbers, but wij uses the same name Real. Very confusing.
>>
>> Further, it seems he only defines how these number are written down.
>> There is no explanation of how to interpret these writings. No order is
>> defined. (Is 0.333333 less, equal, or greater than 0.9?) No operations
>> are defined, such as + - / * etc. It is not explained whether the same
>> number can have different representations, such as normal real numbers
>> have (such as 3E10 and 3.0E10 and 30E9 all for the same number). How can
>> we see whether different writings are about different numbers?
>>
>> Talking about this proposal is possible only when not only the
>> representation of the numbers is defined, but also what they mean and
>> how to work with them.
>>
>> Then suddenly the term 'lim' is used without definition. What does that
>> mean in this context?
>>
>
> Idiot. Show me how your real is constructed.
>

Ad hominem attacks show your lack of reasoning. I will ignore it now,
but I am tempted to stop the discussion for this reason.

Concerning the construction of reals, see:
> https://en.wikipedia.org/wiki/Construction_of_the_real_numbers

In particular note, that it does not use the representation to define
real numbers. Of course that is because the representation does not tell
us what a real number is. It is only a way to write down real numbers.

On Thu, 2024-03-28 at 20:27 +0100, Fred. Zwarts wrote:
> Op 28.mrt.2024 om 16:09 schreef wij:
> > On Thu, 2024-03-28 at 14:16 +0100, Fred. Zwarts wrote:
> > > Op 28.mrt.2024 om 13:29 schreef wij:
> > > > This updated file should solve most doubts I encountered. Hope, useful to you
> > > > (of course, not in official exam if that is your 'real')
> > > > https://sourceforge.net/projects/cscall/files/MisFiles/RealNumber-en.txt/download
> > > >
> > > > +-------------+
> > > > > Real Number | ('computational' may be added to modify terms used in this file
> > > > +-------------+   if needed)
> > > >
> > > > n-ary Fixed-Point Number::= Number represented by a string of digits, the
> > > >      string may contain a minus sign or a point:
> > > >
> > > >        <fixed_point_number>::= [-] <dstr1> [ . <dstr2> ]
> > > >        <dstr1>::= 0 | <nzd> { 0, <nzd> }
> > > >        <dstr2>::= { 0, <nzd> } <nzd>
> > > >        <nzd> ::= (1, 2, 3, 4, 5, 6, 7, 8, 9)  // 'digit' varys depending on n-ary
> > > >
> > > >      Two n-ary fixed-point number (same n-ary) x,y are equal iff their
> > > >      <fixed_point_number> representation are identical.
> > > >
> > > > Real Nunmber(ℝ)::= {x| x is finitely represented by n-ary <fixed_point_number>
> > > >      and those that cannot be finitely represented }
> > > >
> > > >      Note: Numbers that is not finitely representable cannot all be explicitly
> > > >            defined, this is the property of real number based on discrete symbols
> > > >            (like quantum?). E.g.
> > > >
> > > >            A= lim(n->∞) 1-3/10^n = 0.999...
> > > >            B= lim(n->∞) 1-2/2^n  = 0.999...
> > > >            C= lim(n->∞) 1-1/n    = 0.999...
> > > >            ...
> > > >
> > > >            IOW, by repeatedly multiplying 0.999... with 10, you can only see 9,
> > > >            the structure of the rear end of 0.999... is not seen.
> > > >
> > > >            Since <fixed_point_number> is very definitely real and infinity is
> > > >            involved, theories that composed of finite words cannot be too
> > > >            exclusive about such a ℝ. 'Completeness' is impossible.
> > > >
> > > >      Note: This definition implies that repeating decimals are irrational number.
> > > >            Let's list a common magic proof in the way as a brief explanation:
> > > >              (1) x= 0.999...
> > > >              (2) 10x= 9+x  // 10x= 9.999...
> > > >              (3) 9x=9
> > > >              (4) x=1
> > > >            Ans: There is no axiom or theorem to prove (1) => (2).
> > > >
> > > >      Note: To determine whether a repeating decimal x is rational or not, we can
> > > >            repeatedly subtract the repeating pattern p(i) from x.
> > > >            If x-p(1)-p(2)-...=0 can be verified in finite steps, then x is
> > > >            rational. Otherwise, x is irrational, because, if x is rational, the
> > > >            last remaining piece r(i)= x-p(1)-p(2)-... must exactly be the
> > > >            repeating pattern p(i). But, by definition of 'repeating', r(i) cannot
> > > >            be pattern p(i). Therefore, repeating decimal is irrational.
> > > >
> > > > Real number is just this simple. The limit theory is a methodology for finding
> > > > derivative, nothing to do with what the real number is (otherwise, a definition
> > > > like the above must be defined in advance to avoid circular-reasoning).
> > > >
> > > > +-------+
> > > > > Limit |
> > > > +-------+
> > > > Limit::=  lim(x->a) f(x)=L
> > > >       http://www.math.ntu.edu.tw/~mathcal/download/precal/PPT/Chapter%2002_04.pdf
> > > >       http://www.math.ncu.edu.tw/~yu/ecocal98/boards/lec6_ec_98.pdf
> > > >       https://en.wikipedia.org/wiki/Limit_(mathematics)
> > > >       https://en.wikipedia.org/wiki/Limit_of_a_function
> > > >
> > > >       L is defined as the limit (a number) while x approaches a (f(a) may not
> > > >       be defined, although, while f is continuous, L=f(a)). L is a defined value,
> > > >       not "if something infinitely close .... then equal" (no such logic).
> > > >
> > > >       Ex1: A= lim(n->∞) 1-1/n= lim(n->0⁺) 1-n= lim 0.999...=1
> > > >            B= lim(n->∞) 1+1/n= lim(n->0⁺) 1+n= lim 1.000..?=1
> > > >
> > > >       Ex2: A=lim(x->ℵ₀) f(x), B=lim(x->ℵ₁) f(x)  // ℵ₀,ℵ₁ being proper or not is
> > > >                                            // another issue here. But problematic
> > > >                                            // for "finally equal" interpretation.
> > > >
> > > >       Limit defines A=B, does not mean the contents of the limit are equal. If the
> > > >       "x approaches..., then equal" notion is adopted, lots of logical issues arise.
> > > >
> > > >       Note: The equation of limit may be questionable
> > > >             lim(x->c) (f(x)*g(x))= (lim(x->c) f(x))*(lim(x->c) g(x)):
> > > >
> > > >             Let A=lim(n->∞) (1-1/n)= 1
> > > >             A*A*..*A= ... = lim(n->∞) (1-1/n)^n    // 1=1/e ?
> > > >
> > > > +--------------------------------------+
> > > > > Restoring Interpretation of Calculus |
> > > > +--------------------------------------+
> > > > http://www.math.ntu.edu.tw/~mathcal/download/precal/PPT/Chapter%2002_08.pdf
> > > > https://en.wikipedia.org/wiki/Derivative
> > > >
> > > > Assume calculus is basically the area problem of a function: Let F compute the
> > > > the area of f. From the meaning of area, we can have:
> > > >
> > > >       (F(x+h)-F(x)) ≒ (f(x+h)+f(x))*(h/2)  // h is a sufficiently small (test)offset
> > > > <=> (F(x+h)-F(x))/h ≒ (f(x+h)+f(x))/2    // the limit(h->0) of rhs is f(x)
> > > >
> > > > Expected property of F: (1)Error |lhs-rhs| strictly decreases with the tiny
> > > >       (test) offset h (2)When h=0, lhs=rhs.
> > > >       Because the h in the lhs cannot be 0, the basic problem of calculus is
> > > >       finding such a F (or f) that satisfies the expected porperty above...Thus,
> > > >
> > > >       D(f(x))= lim(h->0) (F(x+h)-F(x))/h = f(x)
> > > >
> > > >       Note: Hope that this interpretation can avoid the interpretation of infinity
> > > >             /infinitesimal, and provide more correct foundation for some theories
> > > >             , e.g. Zeno paradoxes, repeating decimal,...,and more (exponiential,
> > > >             Cantor set, infinite series...).
> > > >
> > > > -------------------------------------------------------------------------------
> > > >
> > > >
> > >
> > > It seems that wij wants to define a number type that is different than
> > > the real numbers, but wij uses the same name Real. Very confusing.
> > >
> > > Further, it seems he only defines how these number are written down.
> > > There is no explanation of how to interpret these writings. No order is
> > > defined. (Is 0.333333 less, equal, or greater than 0.9?) No operations
> > > are defined, such as + - / * etc. It is not explained whether the same
> > > number can have different representations, such as normal real numbers
> > > have (such as 3E10 and 3.0E10 and 30E9 all for the same number). How can
> > > we see whether different writings are about different numbers?
> > >
> > > Talking about this proposal is possible only when not only the
> > > representation of the numbers is defined, but also what they mean and
> > > how to work with them.
> > >
> > > Then suddenly the term 'lim' is used without definition. What does that
> > > mean in this context?
> > >
> >
> > Idiot. Show me how your real is constructed.
> >
>
> Ad hominem attacks show your lack of reasoning. I will ignore it now,
> but I am tempted to stop the discussion for this reason.
>
> Concerning the construction of reals, see:
> > https://en.wikipedia.org/wiki/Construction_of_the_real_numbers
>
> In particular note, that it does not use the representation to define
> real numbers. Of course that is because the representation does not tell
> us what a real number is. It is only a way to write down real numbers.

Op 28.mrt.2024 om 20:39 schreef wij:
> On Thu, 2024-03-28 at 20:27 +0100, Fred. Zwarts wrote:
>> Op 28.mrt.2024 om 16:09 schreef wij:
>>> On Thu, 2024-03-28 at 14:16 +0100, Fred. Zwarts wrote:
>>>> Op 28.mrt.2024 om 13:29 schreef wij:
>>>>> This updated file should solve most doubts I encountered. Hope, useful to you
>>>>> (of course, not in official exam if that is your 'real')
>>>>> https://sourceforge.net/projects/cscall/files/MisFiles/RealNumber-en.txt/download
>>>>>
>>>>> +-------------+
>>>>>> Real Number | ('computational' may be added to modify terms used in this file
>>>>> +-------------+   if needed)
>>>>>
>>>>> n-ary Fixed-Point Number::= Number represented by a string of digits, the
>>>>>      string may contain a minus sign or a point:
>>>>>
>>>>>        <fixed_point_number>::= [-] <dstr1> [ . <dstr2> ]
>>>>>        <dstr1>::= 0 | <nzd> { 0, <nzd> }
>>>>>        <dstr2>::= { 0, <nzd> } <nzd>
>>>>>        <nzd> ::= (1, 2, 3, 4, 5, 6, 7, 8, 9)  // 'digit' varys depending on n-ary
>>>>>
>>>>>      Two n-ary fixed-point number (same n-ary) x,y are equal iff their
>>>>>      <fixed_point_number> representation are identical.
>>>>>
>>>>> Real Nunmber(ℝ)::= {x| x is finitely represented by n-ary <fixed_point_number>
>>>>>      and those that cannot be finitely represented }
>>>>>
>>>>>      Note: Numbers that is not finitely representable cannot all be explicitly
>>>>>            defined, this is the property of real number based on discrete symbols
>>>>>            (like quantum?). E.g.
>>>>>
>>>>>            A= lim(n->∞) 1-3/10^n = 0.999...
>>>>>            B= lim(n->∞) 1-2/2^n  = 0.999...
>>>>>            C= lim(n->∞) 1-1/n    = 0.999...
>>>>>            ...
>>>>>
>>>>>            IOW, by repeatedly multiplying 0.999... with 10, you can only see 9,
>>>>>            the structure of the rear end of 0.999... is not seen.
>>>>>
>>>>>            Since <fixed_point_number> is very definitely real and infinity is
>>>>>            involved, theories that composed of finite words cannot be too
>>>>>            exclusive about such a ℝ. 'Completeness' is impossible.
>>>>>
>>>>>      Note: This definition implies that repeating decimals are irrational number.
>>>>>            Let's list a common magic proof in the way as a brief explanation:
>>>>>              (1) x= 0.999...
>>>>>              (2) 10x= 9+x  // 10x= 9.999...
>>>>>              (3) 9x=9
>>>>>              (4) x=1
>>>>>            Ans: There is no axiom or theorem to prove (1) => (2).
>>>>>
>>>>>      Note: To determine whether a repeating decimal x is rational or not, we can
>>>>>            repeatedly subtract the repeating pattern p(i) from x.
>>>>>            If x-p(1)-p(2)-...=0 can be verified in finite steps, then x is
>>>>>            rational. Otherwise, x is irrational, because, if x is rational, the
>>>>>            last remaining piece r(i)= x-p(1)-p(2)-... must exactly be the
>>>>>            repeating pattern p(i). But, by definition of 'repeating', r(i) cannot
>>>>>            be pattern p(i). Therefore, repeating decimal is irrational.
>>>>>
>>>>> Real number is just this simple. The limit theory is a methodology for finding
>>>>> derivative, nothing to do with what the real number is (otherwise, a definition
>>>>> like the above must be defined in advance to avoid circular-reasoning).
>>>>>
>>>>> +-------+
>>>>>> Limit |
>>>>> +-------+
>>>>> Limit::=  lim(x->a) f(x)=L
>>>>>       http://www.math.ntu.edu.tw/~mathcal/download/precal/PPT/Chapter%2002_04.pdf
>>>>>       http://www.math.ncu.edu.tw/~yu/ecocal98/boards/lec6_ec_98.pdf
>>>>>       https://en.wikipedia.org/wiki/Limit_(mathematics)
>>>>>       https://en.wikipedia.org/wiki/Limit_of_a_function
>>>>>
>>>>>       L is defined as the limit (a number) while x approaches a (f(a) may not
>>>>>       be defined, although, while f is continuous, L=f(a)). L is a defined value,
>>>>>       not "if something infinitely close ... then equal" (no such logic).
>>>>>
>>>>>       Ex1: A= lim(n->∞) 1-1/n= lim(n->0⁺) 1-n= lim 0.999...=1
>>>>>            B= lim(n->∞) 1+1/n= lim(n->0⁺) 1+n= lim 1.000..?=1
>>>>>
>>>>>       Ex2: A=lim(x->ℵ₀) f(x), B=lim(x->ℵ₁) f(x)  // ℵ₀,ℵ₁ being proper or not is
>>>>>                                            // another issue here. But problematic
>>>>>                                            // for "finally equal" interpretation.
>>>>>
>>>>>       Limit defines A=B, does not mean the contents of the limit are equal. If the
>>>>>       "x approaches..., then equal" notion is adopted, lots of logical issues arise.
>>>>>
>>>>>       Note: The equation of limit may be questionable
>>>>>             lim(x->c) (f(x)*g(x))= (lim(x->c) f(x))*(lim(x->c) g(x)):
>>>>>
>>>>>             Let A=lim(n->∞) (1-1/n)= 1
>>>>>             A*A*..*A= ... = lim(n->∞) (1-1/n)^n    // 1=1/e ?
>>>>>
>>>>> +--------------------------------------+
>>>>>> Restoring Interpretation of Calculus |
>>>>> +--------------------------------------+
>>>>> http://www.math.ntu.edu.tw/~mathcal/download/precal/PPT/Chapter%2002_08.pdf
>>>>> https://en.wikipedia.org/wiki/Derivative
>>>>>
>>>>> Assume calculus is basically the area problem of a function: Let F compute the
>>>>> the area of f. From the meaning of area, we can have:
>>>>>
>>>>>       (F(x+h)-F(x)) ≒ (f(x+h)+f(x))*(h/2)  // h is a sufficiently small (test)offset
>>>>> <=> (F(x+h)-F(x))/h ≒ (f(x+h)+f(x))/2    // the limit(h->0) of rhs is f(x)
>>>>>
>>>>> Expected property of F: (1)Error |lhs-rhs| strictly decreases with the tiny
>>>>>       (test) offset h (2)When h=0, lhs=rhs.
>>>>>       Because the h in the lhs cannot be 0, the basic problem of calculus is
>>>>>       finding such a F (or f) that satisfies the expected porperty above...Thus,
>>>>>
>>>>>       D(f(x))= lim(h->0) (F(x+h)-F(x))/h = f(x)
>>>>>
>>>>>       Note: Hope that this interpretation can avoid the interpretation of infinity
>>>>>             /infinitesimal, and provide more correct foundation for some theories
>>>>>             , e.g. Zeno paradoxes, repeating decimal,...,and more (exponiential,
>>>>>             Cantor set, infinite series...).
>>>>>
>>>>> -------------------------------------------------------------------------------
>>>>>
>>>>>
>>>>
>>>> It seems that wij wants to define a number type that is different than
>>>> the real numbers, but wij uses the same name Real. Very confusing.
>>>>
>>>> Further, it seems he only defines how these number are written down.
>>>> There is no explanation of how to interpret these writings. No order is
>>>> defined. (Is 0.333333 less, equal, or greater than 0.9?) No operations
>>>> are defined, such as + - / * etc. It is not explained whether the same
>>>> number can have different representations, such as normal real numbers
>>>> have (such as 3E10 and 3.0E10 and 30E9 all for the same number). How can
>>>> we see whether different writings are about different numbers?
>>>>
>>>> Talking about this proposal is possible only when not only the
>>>> representation of the numbers is defined, but also what they mean and
>>>> how to work with them.
>>>>
>>>> Then suddenly the term 'lim' is used without definition. What does that
>>>> mean in this context?
>>>>
>>>
>>> Idiot. Show me how your real is constructed.
>>>
>>
>> Ad hominem attacks show your lack of reasoning. I will ignore it now,
>> but I am tempted to stop the discussion for this reason.
>>
>> Concerning the construction of reals, see:
>>> https://en.wikipedia.org/wiki/Construction_of_the_real_numbers
>>
>> In particular note, that it does not use the representation to define
>> real numbers. Of course that is because the representation does not tell
>> us what a real number is. It is only a way to write down real numbers.
>
> If you like to trigger me intention. You are idiot you don't understand
> the same as olcott. Because you can't even prove 1+2=3 as rigorously as
> you requested from me.
>
>

On 28/03/2024 16:53, olcott wrote:
>>> Yet it seems that wij is correct that 0.999... would seem to
>>> be infinitesimally < 1.0.
>>      That /cannot/ be correct in the "real" numbers, in which there
>> are no infinitesimals [basic axiom of the reals].  In other systems of
>> numbers, it could be correct,
> Yes.
>> but that will depend on what is meant by
>> "0.999..",
> Approaching yet never reaching 1.0.

That is a property of the numbers 0.9, 0.99, 0.999 and so
on arranged as a sequence [and of many other sequences], but is not
/yet/ a value. Not until you explain what you mean. In conventional
mathematics, it is usually taken to mean the limit of that sequence
expressed as a real number, where "limit" has a precise meaning as
discussed and formalised in the 19thC. That limit is 1. Not a tiny
bit less than one, not some new sort of object, but 1, exactly. You
and Wij may find that surprising, or even nonsensical, but it is what
the mathematics tells us from the axioms of the real numbers and from
the definition of "limit". If you want the answer to be different,
then that must follow from different axioms and definitions. Until
you and/or Wij tell us what those are, there is nothing further useful
to be said.

>> and note that if you appeal to something that mentions limits
>> to define this, then you have to explain how infinite and infinitesimal
>> numbers are handled in the definition.

Again, there are no infinite or infinitesimal real numbers, so
if you want an infinitesimal in your answer, it is incumbent on you to
explain what you are using /other than/ conventional maths.

>>>                   One geometric point on the number line.
>>> [0.0, 1.0) < [0.0, 1.0] by one geometric point.
>>      Until you describe the axioms of what you mean by "geometric
>> point" and "number line", this is meaningless verbiage.  Give your
> Of course by geometric point I must mean a box of chocolates and by
> number line I mean a pretty pink bow. No one would ever suspect that
> these terms have their conventional meanings.

I didn't ask what "geometric point" and "number line" are, but
what axioms you think they have. In conventional mathematics, those two
intervals have /exactly/ the same measure even though they are not
exactly the same sets of points. If you get a different answer [and
have not simply made a mistake], it /must/ be because you are using
different axioms. What are they?

>> axioms, and it becomes possible to discuss this.  Until then, we are
>> entitled to assume that you and Wij are talking about the "traditional"
>> "real" numbers [as used in engineering, etc.] in which there are no
>> infinitesimals, and so the only interpretation we can make of the size
>> of "one geometric point" is the usual "measure", which is zero.
> Yet it is never actually zero because it is possible to specify a
> line segment that is exactly one geometric point longer than another.
> [0.0, 1.0] - [0.0, 1.0) = one geometric point.

But "one geometric point" has measure zero. Not "never actually
zero", but actually and really zero. Unless, that is, you are using some
different and as yet unexplained axioms/definitions. Which are ...?

--
Andy Walker, Nottingham.
Andy's music pages: www.cuboid.me.uk/andy/Music
Composer of the day: www.cuboid.me.uk/andy/Music/Composers/Couperin

On 3/28/2024 3:38 PM, Andy Walker wrote:
> On 28/03/2024 16:53, olcott wrote:
>>>> Yet it seems that wij is correct that 0.999... would seem to
>>>> be infinitesimally < 1.0.
>>>      That /cannot/ be correct in the "real" numbers, in which there
>>> are no infinitesimals [basic axiom of the reals].  In other systems of
>>> numbers, it could be correct,
>> Yes.
>>> but that will depend on what is meant by
>>> "0.999..",
>> Approaching yet never reaching 1.0.
>
>     That is a property of the numbers 0.9, 0.99, 0.999 and so
> on arranged as a sequence [and of many other sequences], but is not
> /yet/ a value.  Not until you explain what you mean.  In conventional
> mathematics, it is usually taken to mean the limit of that sequence
> expressed as a real number, where "limit" has a precise meaning as
> discussed and formalised in the 19thC.  That limit is 1.

I disagree yet prior to my infinitesimal number system there was no
way to say this. 0.999... is exactly one geometric point less than 1.0
the same way that this line segment [0.0,1.0] > [0.0,1.0) by exactly
one geometric on the number line.

> Not a tiny
> bit less than one, not some new sort of object, but 1, exactly.  You
> and Wij may find that surprising, or even nonsensical, but it is what
> the mathematics tells us from the axioms of the real numbers and from
> the definition of "limit".  If you want the answer to be different,
> then that must follow from different axioms and definitions.  Until
> you and/or Wij tell us what those are, there is nothing further useful
> to be said.
>

Yes we all agree that 0.999... never gets to 1.0.

>>> and note that if you appeal to something that mentions limits
>>> to define this, then you have to explain how infinite and infinitesimal
>>> numbers are handled in the definition.
>
>     Again, there are no infinite or infinitesimal real numbers, so
> if you want an infinitesimal in your answer, it is incumbent on you to
> explain what you are using /other than/ conventional maths.
>

Been there done that.

>>>>                   One geometric point on the number line.
>>>> [0.0, 1.0) < [0.0, 1.0] by one geometric point.
>>>      Until you describe the axioms of what you mean by "geometric
>>> point" and "number line", this is meaningless verbiage.  Give your
>> Of course by geometric point I must mean a box of chocolates and by
>> number line I mean a pretty pink bow. No one would ever suspect that
>> these terms have their conventional meanings.
>
>     I didn't ask what "geometric point" and "number line" are, but
> what axioms you think they have.  In conventional mathematics, those two
> intervals have /exactly/ the same measure even though they are not
> exactly the same sets of points.

That is inconsistent. They are exactly the same points up until
the last point at 1.0 is reached by one yet not the other.

> If you get a different answer [and
> have not simply made a mistake], it /must/ be because you are using
> different axioms.  What are they?
>

I am just showing EXACTLY where the conventional notions lead.

>>> axioms, and it becomes possible to discuss this.  Until then, we are
>>> entitled to assume that you and Wij are talking about the "traditional"
>>> "real" numbers [as used in engineering, etc.] in which there are no
>>> infinitesimals, and so the only interpretation we can make of the size
>>> of "one geometric point" is the usual "measure", which is zero.
>> Yet it is never actually zero because it is possible to specify a
>> line segment that is exactly one geometric point longer than another.
>> [0.0, 1.0] - [0.0, 1.0) = one geometric point.
>
>     But "one geometric point" has measure zero.  Not "never actually

I just proved otherwise. [0.0, 1.0] has all of the same points
as [0.0, 1.0) except that it has one more point.

> zero", but actually and really zero.  Unless, that is, you are using some
> different and as yet unexplained axioms/definitions.  Which are ...?
>

Conventional interval notion proves otherwise.
Most logicians abhor thinking outside-the-box.

--
Copyright 2024 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer

On 3/28/24 12:07 PM, olcott wrote:
> On 3/28/2024 10:59 AM, Andy Walker wrote:
>> On 28/03/2024 13:16, Fred. Zwarts wrote:
>>> It seems that wij wants to define a number type that is different
>>> than the real numbers, but wij uses the same name Real. Very
>>> confusing.
>>
>>      It seems to me to be worse than that.  Wij apparently thinks he
>> /is/ defining the real numbers, and that the traditional definitions are
>> wrong in some way that he has never managed to explain.  But as he uses
>> infinity and infinitesimals [in an unexplained way], he is breaking the
>> Archimedean/Eudoxian axiom, so Wij-reals are not R, and they seem also
>> not to be any of the other usual real-like number systems.  So the whole
>> of mathematical physics, engineering, ... is left in limbo, with all the
>> standard theorems inapplicable unless/until Wij tells us much more, and
>> probably not even then judging by Wij's responses thus far.
>>
>
> Yet it seems that wij is correct that 0.999... would seem to
> be infinitesimally < 1.0. One geometric point on the number line.
> [0.0, 1.0) < [0.0, 1.0] by one geometric point.

And that depends on WHAT number system you are working in.

With the classical "Reals", 0.9999.... is 1.00000

In some of the hyper real systems, there can be a hyper-finite real
number between them.

The number system that allow for such numbers also define what you can
do with these numbers (and what you can't do).

The problem with poorly defined systems is you can't actually try to do
anything with them, because you don't have any tools.

>
>>> Further, it seems he only defines how these number are written down.
>>> There is no explanation of how to interpret these writings.
>>
>>      Well, quite.  It seems that we're supposed to use the standard
>> processes of arithmetic until we get to infinity and similar.  But of
>> course mathematics is concerned with numbers much more than with how
>> they are notated.
>>
>>      All might become clear if Wij could explain what problem he is
>> really trying to solve.  What bridges fall down if "traditional" maths
>> is used but stay up with Wij-reals?  What new puzzles are soluble?  Are
>> they somehow more logical, or easier to teach?  He seems to think that
>> "trad" maths is full of holes that he sees but that all the great minds
>> of the past 2500 years have overlooked.  Perhaps it's all or mostly lost
>> in translation, but it's more likely that he is joining the PO Club.
>>
>

On 3/28/2024 7:07 PM, Richard Damon wrote:
> On 3/28/24 12:07 PM, olcott wrote:
>> On 3/28/2024 10:59 AM, Andy Walker wrote:
>>> On 28/03/2024 13:16, Fred. Zwarts wrote:
>>>> It seems that wij wants to define a number type that is different
>>>> than the real numbers, but wij uses the same name Real. Very
>>>> confusing.
>>>
>>>      It seems to me to be worse than that.  Wij apparently thinks he
>>> /is/ defining the real numbers, and that the traditional definitions are
>>> wrong in some way that he has never managed to explain.  But as he uses
>>> infinity and infinitesimals [in an unexplained way], he is breaking the
>>> Archimedean/Eudoxian axiom, so Wij-reals are not R, and they seem also
>>> not to be any of the other usual real-like number systems.  So the whole
>>> of mathematical physics, engineering, ... is left in limbo, with all the
>>> standard theorems inapplicable unless/until Wij tells us much more, and
>>> probably not even then judging by Wij's responses thus far.
>>>
>>
>> Yet it seems that wij is correct that 0.999... would seem to
>> be infinitesimally < 1.0. One geometric point on the number line.
>> [0.0, 1.0) < [0.0, 1.0] by one geometric point.
>
> And that depends on WHAT number system you are working in.
>
> With the classical "Reals", 0.9999.... is 1.00000
>

Yet that is NOT what 0.999... actually says.
It says that it gets infinitely close to 1.0 without every actually
getting there. In other words it is infinitesimally less than 1.0.

> In some of the hyper real systems, there can be a hyper-finite real
> number between them.
>
> The number system that allow for such numbers also define what you can
> do with these numbers (and what you can't do).
>
> The problem with poorly defined systems is you can't actually try to do
> anything with them, because you don't have any tools.
>
>>
>>>> Further, it seems he only defines how these number are written down.
>>>> There is no explanation of how to interpret these writings.
>>>
>>>      Well, quite.  It seems that we're supposed to use the standard
>>> processes of arithmetic until we get to infinity and similar.  But of
>>> course mathematics is concerned with numbers much more than with how
>>> they are notated.
>>>
>>>      All might become clear if Wij could explain what problem he is
>>> really trying to solve.  What bridges fall down if "traditional" maths
>>> is used but stay up with Wij-reals?  What new puzzles are soluble?  Are
>>> they somehow more logical, or easier to teach?  He seems to think that
>>> "trad" maths is full of holes that he sees but that all the great minds
>>> of the past 2500 years have overlooked.  Perhaps it's all or mostly lost
>>> in translation, but it's more likely that he is joining the PO Club.
>>>
>>
>

--
Copyright 2024 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer

On 3/28/24 9:56 PM, olcott wrote:
> On 3/28/2024 7:07 PM, Richard Damon wrote:
>> On 3/28/24 12:07 PM, olcott wrote:
>>> On 3/28/2024 10:59 AM, Andy Walker wrote:
>>>> On 28/03/2024 13:16, Fred. Zwarts wrote:
>>>>> It seems that wij wants to define a number type that is different
>>>>> than the real numbers, but wij uses the same name Real. Very
>>>>> confusing.
>>>>
>>>>      It seems to me to be worse than that.  Wij apparently thinks he
>>>> /is/ defining the real numbers, and that the traditional definitions
>>>> are
>>>> wrong in some way that he has never managed to explain.  But as he uses
>>>> infinity and infinitesimals [in an unexplained way], he is breaking the
>>>> Archimedean/Eudoxian axiom, so Wij-reals are not R, and they seem also
>>>> not to be any of the other usual real-like number systems.  So the
>>>> whole
>>>> of mathematical physics, engineering, ... is left in limbo, with all
>>>> the
>>>> standard theorems inapplicable unless/until Wij tells us much more, and
>>>> probably not even then judging by Wij's responses thus far.
>>>>
>>>
>>> Yet it seems that wij is correct that 0.999... would seem to
>>> be infinitesimally < 1.0. One geometric point on the number line.
>>> [0.0, 1.0) < [0.0, 1.0] by one geometric point.
>>
>> And that depends on WHAT number system you are working in.
>>
>> With the classical "Reals", 0.9999.... is 1.00000
>>
>
> Yet that is NOT what 0.999... actually says.
> It says that it gets infinitely close to 1.0 without every actually
> getting there. In other words it is infinitesimally less than 1.0.

But so close that no number exists between it and 1.0, so they are the
same number.

That comes out of the ACTUAL definitions of Real Numbers

>
>> In some of the hyper real systems, there can be a hyper-finite real
>> number between them.
>>
>> The number system that allow for such numbers also define what you can
>> do with these numbers (and what you can't do).
>>
>> The problem with poorly defined systems is you can't actually try to
>> do anything with them, because you don't have any tools.
>>
>>>
>>>>> Further, it seems he only defines how these number are written down.
>>>>> There is no explanation of how to interpret these writings.
>>>>
>>>>      Well, quite.  It seems that we're supposed to use the standard
>>>> processes of arithmetic until we get to infinity and similar.  But of
>>>> course mathematics is concerned with numbers much more than with how
>>>> they are notated.
>>>>
>>>>      All might become clear if Wij could explain what problem he is
>>>> really trying to solve.  What bridges fall down if "traditional" maths
>>>> is used but stay up with Wij-reals?  What new puzzles are soluble?  Are
>>>> they somehow more logical, or easier to teach?  He seems to think that
>>>> "trad" maths is full of holes that he sees but that all the great minds
>>>> of the past 2500 years have overlooked.  Perhaps it's all or mostly
>>>> lost
>>>> in translation, but it's more likely that he is joining the PO Club.
>>>>
>>>
>>
>

On 3/28/2024 9:23 PM, Richard Damon wrote:
> On 3/28/24 9:56 PM, olcott wrote:
>> On 3/28/2024 7:07 PM, Richard Damon wrote:
>>> On 3/28/24 12:07 PM, olcott wrote:
>>>> On 3/28/2024 10:59 AM, Andy Walker wrote:
>>>>> On 28/03/2024 13:16, Fred. Zwarts wrote:
>>>>>> It seems that wij wants to define a number type that is different
>>>>>> than the real numbers, but wij uses the same name Real. Very
>>>>>> confusing.
>>>>>
>>>>>      It seems to me to be worse than that.  Wij apparently thinks he
>>>>> /is/ defining the real numbers, and that the traditional
>>>>> definitions are
>>>>> wrong in some way that he has never managed to explain.  But as he
>>>>> uses
>>>>> infinity and infinitesimals [in an unexplained way], he is breaking
>>>>> the
>>>>> Archimedean/Eudoxian axiom, so Wij-reals are not R, and they seem also
>>>>> not to be any of the other usual real-like number systems.  So the
>>>>> whole
>>>>> of mathematical physics, engineering, ... is left in limbo, with
>>>>> all the
>>>>> standard theorems inapplicable unless/until Wij tells us much more,
>>>>> and
>>>>> probably not even then judging by Wij's responses thus far.
>>>>>
>>>>
>>>> Yet it seems that wij is correct that 0.999... would seem to
>>>> be infinitesimally < 1.0. One geometric point on the number line.
>>>> [0.0, 1.0) < [0.0, 1.0] by one geometric point.
>>>
>>> And that depends on WHAT number system you are working in.
>>>
>>> With the classical "Reals", 0.9999.... is 1.00000
>>>
>>
>> Yet that is NOT what 0.999... actually says.
>> It says that it gets infinitely close to 1.0 without every actually
>> getting there. In other words it is infinitesimally less than 1.0.
>
> But so close that no number exists between it and 1.0, so they are the
> same number.
>

You just admitted that they are not the same number.
It seems dead obvious that 0.999... is infinitesimally less than 1.0.

That we can say this in English yet not say this in conventional
number systems proves the need for another number system that can
say this.

--
Copyright 2024 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer

On Thu, 2024-03-28 at 22:23 -0400, Richard Damon wrote:
> On 3/28/24 9:56 PM, olcott wrote:
> > On 3/28/2024 7:07 PM, Richard Damon wrote:
> > > On 3/28/24 12:07 PM, olcott wrote:
> > > > On 3/28/2024 10:59 AM, Andy Walker wrote:
> > > > > On 28/03/2024 13:16, Fred. Zwarts wrote:
> > > > > > It seems that wij wants to define a number type that is different
> > > > > > than the real numbers, but wij uses the same name Real. Very
> > > > > > confusing.
> > > > >
> > > > >      It seems to me to be worse than that.  Wij apparently thinks he
> > > > > /is/ defining the real numbers, and that the traditional definitions
> > > > > are
> > > > > wrong in some way that he has never managed to explain.  But as he uses
> > > > > infinity and infinitesimals [in an unexplained way], he is breaking the
> > > > > Archimedean/Eudoxian axiom, so Wij-reals are not R, and they seem also
> > > > > not to be any of the other usual real-like number systems.  So the
> > > > > whole
> > > > > of mathematical physics, engineering, ... is left in limbo, with all
> > > > > the
> > > > > standard theorems inapplicable unless/until Wij tells us much more, and
> > > > > probably not even then judging by Wij's responses thus far.
> > > > >
> > > >
> > > > Yet it seems that wij is correct that 0.999... would seem to
> > > > be infinitesimally < 1.0. One geometric point on the number line.
> > > > [0.0, 1.0) < [0.0, 1.0] by one geometric point.
> > >
> > > And that depends on WHAT number system you are working in.
> > >
> > > With the classical "Reals", 0.9999.... is 1.00000
> > >
> >
> > Yet that is NOT what 0.999... actually says.
> > It says that it gets infinitely close to 1.0 without every actually
> > getting there. In other words it is infinitesimally less than 1.0.
>
> But so close that no number exists between it and 1.0, so they are the
> same number.
>
> That comes out of the ACTUAL definitions of Real Numbers
>

Richard Damon's limit.

x approach c, but cannot be c.
https://math.stackexchange.com/questions/3868253/why-do-we-need-x-neq-c-in-epsilon-delta-definition-of-limits

> >
> > > In some of the hyper real systems, there can be a hyper-finite real
> > > number between them.
> > >
> > > The number system that allow for such numbers also define what you can
> > > do with these numbers (and what you can't do).
> > >
> > > The problem with poorly defined systems is you can't actually try to
> > > do anything with them, because you don't have any tools.
> > >
> > > >
> > > > > > Further, it seems he only defines how these number are written down.
> > > > > > There is no explanation of how to interpret these writings.
> > > > >
> > > > >      Well, quite.  It seems that we're supposed to use the standard
> > > > > processes of arithmetic until we get to infinity and similar.  But of
> > > > > course mathematics is concerned with numbers much more than with how
> > > > > they are notated.
> > > > >
> > > > >      All might become clear if Wij could explain what problem he is
> > > > > really trying to solve.  What bridges fall down if "traditional" maths
> > > > > is used but stay up with Wij-reals?  What new puzzles are soluble?  Are
> > > > > they somehow more logical, or easier to teach?  He seems to think that
> > > > > "trad" maths is full of holes that he sees but that all the great minds
> > > > > of the past 2500 years have overlooked.  Perhaps it's all or mostly
> > > > > lost
> > > > > in translation, but it's more likely that he is joining the PO Club.
> > > > >
> > > >
> > >
> >
>

On 3/28/24 10:43 PM, olcott wrote:
> On 3/28/2024 9:23 PM, Richard Damon wrote:
>> On 3/28/24 9:56 PM, olcott wrote:
>>> On 3/28/2024 7:07 PM, Richard Damon wrote:
>>>> On 3/28/24 12:07 PM, olcott wrote:
>>>>> On 3/28/2024 10:59 AM, Andy Walker wrote:
>>>>>> On 28/03/2024 13:16, Fred. Zwarts wrote:
>>>>>>> It seems that wij wants to define a number type that is different
>>>>>>> than the real numbers, but wij uses the same name Real. Very
>>>>>>> confusing.
>>>>>>
>>>>>>      It seems to me to be worse than that.  Wij apparently thinks he
>>>>>> /is/ defining the real numbers, and that the traditional
>>>>>> definitions are
>>>>>> wrong in some way that he has never managed to explain.  But as he
>>>>>> uses
>>>>>> infinity and infinitesimals [in an unexplained way], he is
>>>>>> breaking the
>>>>>> Archimedean/Eudoxian axiom, so Wij-reals are not R, and they seem
>>>>>> also
>>>>>> not to be any of the other usual real-like number systems.  So the
>>>>>> whole
>>>>>> of mathematical physics, engineering, ... is left in limbo, with
>>>>>> all the
>>>>>> standard theorems inapplicable unless/until Wij tells us much
>>>>>> more, and
>>>>>> probably not even then judging by Wij's responses thus far.
>>>>>>
>>>>>
>>>>> Yet it seems that wij is correct that 0.999... would seem to
>>>>> be infinitesimally < 1.0. One geometric point on the number line.
>>>>> [0.0, 1.0) < [0.0, 1.0] by one geometric point.
>>>>
>>>> And that depends on WHAT number system you are working in.
>>>>
>>>> With the classical "Reals", 0.9999.... is 1.00000
>>>>
>>>
>>> Yet that is NOT what 0.999... actually says.
>>> It says that it gets infinitely close to 1.0 without every actually
>>> getting there. In other words it is infinitesimally less than 1.0.
>>
>> But so close that no number exists between it and 1.0, so they are the
>> same number.
>>
>
> You just admitted that they are not the same number.
> It seems dead obvious that 0.999... is infinitesimally less than 1.0.
>
> That we can say this in English yet not say this in conventional
> number systems proves the need for another number system that can
> say this.
>

Nope.

1+1 is just another name for 2, do we need another number system to
expalin that?

olcott <polcott2@gmail.com> writes:
[...]
> It seems dead obvious that 0.999... is infinitesimally less than 1.0.

Yes, it *seems* dead obvious. That doesn't make it true, and in fact it
isn't.

0.999... denotes a *limit*. In particular, it's the limit of the value
as the number of 9s increases without bound. That's what the notation
"0.999..." *means*. (There are more precise notations for the same
thing, such as "0.9̅" (that's a 9 with an overbar, or "vinculum") or
"0.(9)".

You have a sequence of numbers:

0.9
0.99
0.999
0.9999
0.99999
...

Each member of that sequence is strictly less than 1.0, but the *limit*
is exactly 1.0. The limit of a sequence doesn't have to be a member of
the sequence. The limit is, informally, the value that members of the
sequence approach arbitrarily closely.

<https://en.wikipedia.org/wiki/Limit_of_a_sequence>

> That we can say this in English yet not say this in conventional
> number systems proves the need for another number system that can
> say this.

Then I have good news for you. There are several such systems, for
example <https://en.wikipedia.org/wiki/Hyperreal_number>.

If your point is that you personally like hyperreals better than you
like reals, that's fine, as long as you're clear which number system
you're using. If you talk about things like "0.999..." without
qualification, everyone will assume you're talking about real numbers.

And if you're going to play with hyperreal numbers, or surreal numbers,
or any of a number of other extensions to the real numbers, I suggest
that understanding the real numbers is a necessary prerequisite. That
includes understanding that no real number is either infinitesimal or
infinite.

Disclaimer: I'm not a mathematician. I welcome corrections.

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

On 3/28/2024 10:36 PM, Keith Thompson wrote:
> olcott <polcott2@gmail.com> writes:
> [...]
>> It seems dead obvious that 0.999... is infinitesimally less than 1.0.
>
> Yes, it *seems* dead obvious. That doesn't make it true, and in fact it
> isn't.
>

0.999... means that is never reaches 1.0.
and math simply stipulates that it does even though it does not.

> 0.999... denotes a *limit*. In particular, it's the limit of the value
> as the number of 9s increases without bound. That's what the notation

That is how it has been misinterpreted yet it has always meant
infinitesimally less than 1.0.

> "0.999..." *means*. (There are more precise notations for the same
> thing, such as "0.9̅" (that's a 9 with an overbar, or "vinculum") or
> "0.(9)".
>

I already know all that.

> You have a sequence of numbers:
>
> 0.9
> 0.99
> 0.999
> 0.9999
> 0.99999
> ...
>
> Each member of that sequence is strictly less than 1.0, but the *limit*
> is exactly 1.0. The limit of a sequence doesn't have to be a member of
> the sequence. The limit is, informally, the value that members of the
> sequence approach arbitrarily closely.
>

Yet never reaching.

> <https://en.wikipedia.org/wiki/Limit_of_a_sequence>
>
>> That we can say this in English yet not say this in conventional
>> number systems proves the need for another number system that can
>> say this.
>
> Then I have good news for you. There are several such systems, for
> example <https://en.wikipedia.org/wiki/Hyperreal_number>.
>

Infinitesimally less than 1.0 means one single geometric point
on the number line less than 1.0.

> If your point is that you personally like hyperreals better than you
> like reals, that's fine, as long as you're clear which number system
> you're using.

The Infinitesimal number system that I created.

> If you talk about things like "0.999..." without
> qualification, everyone will assume you're talking about real numbers.
>

It is already the case that 0.999...
specifies Infinitesimally less than 1.0.

> And if you're going to play with hyperreal numbers, or surreal numbers,
> or any of a number of other extensions to the real numbers, I suggest
> that understanding the real numbers is a necessary prerequisite. That
> includes understanding that no real number is either infinitesimal or
> infinite.
>
> Disclaimer: I'm not a mathematician. I welcome corrections.
>

--
Copyright 2024 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer

olcott <polcott2@gmail.com> writes:
> On 3/28/2024 10:36 PM, Keith Thompson wrote:
[...]
>> If your point is that you personally like hyperreals better than you
>> like reals, that's fine, as long as you're clear which number system
>> you're using.
>
> The Infinitesimal number system that I created.

Ah, then you're not talking about the conventional real numbers. That's
all I needed to know.

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

On 3/28/2024 11:15 PM, Keith Thompson wrote:
> olcott <polcott2@gmail.com> writes:
>> On 3/28/2024 10:36 PM, Keith Thompson wrote:
> [...]
>>> If your point is that you personally like hyperreals better than you
>>> like reals, that's fine, as long as you're clear which number system
>>> you're using.
>>
>> The Infinitesimal number system that I created.
>
> Ah, then you're not talking about the conventional real numbers. That's
> all I needed to know.
>

x is said to be infinitesimal
if, and only if, |x| < 1/n for all integers n.
https://en.wikipedia.org/wiki/Hyperreal_number

0.999... specifies infinitesimally < 1.0
and math guys have no way to say that so they
simply round up to 1.0

--
Copyright 2024 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer

Op 29.mrt.2024 om 05:15 schreef Keith Thompson:
> olcott <polcott2@gmail.com> writes:
>> On 3/28/2024 10:36 PM, Keith Thompson wrote:
> [...]
>>> If your point is that you personally like hyperreals better than you
>>> like reals, that's fine, as long as you're clear which number system
>>> you're using.
>>
>> The Infinitesimal number system that I created.
>
> Ah, then you're not talking about the conventional real numbers. That's
> all I needed to know.
>

Exactly! It does not make sense to discuss a number system that is not
specified.
The construction of real numbers is specified very clearly. See e.g.

> https://en.wikipedia.org/wiki/Construction_of_the_real_numbers

I think the construction of reals from Cauchy sequences is the best
known method.
It should be noted that according to this method, numbers can be written
in several ways. E.g., 0 can be written as:
0, 0.0, lim N→∞ (1/N), or lim N→∞ (1/2N), etc.
From this construction it follows directly that 0.999... = 1, as the
article shows.
Anyone claiming that 0.999... ≠ 1 should first tell in which number
system he/she works and which of the axioms of reals she/he wants to
change and how to change them. If that is not specified, then there is
no basis for a discussion.
Such a claim creates confusion when the new number system is also called
real numbers. But we know that some persons here are famous for naming
different things with the same names.

On 28/03/2024 19:22, wij wrote:
> I saw lots of inconsistency in Andy Walker's response. I think the simple
> way to solve his doubt is for him to prove "repeating decimal is rational".
Yet you cannot actually describe any inconsistency in what I [and for
that matter Fred, Richard, Keith and perhaps others] say. My "doubt" is not
about what *I* know about mathematics, but about (a) your abuse of the terms
"R" and "real number" to describe mathematical objects to which you ascribe
properties which contradict the Archimedean axiom of R; and (b) the lack of
any discernible rationale for your proposals. So I ask again -- what problem
do Wij-numbers solve that use of the traditional real numbers fails to solve?

There is no difficulty in evaluating a "repeating decimal" in R, and
the answer is easily seen to be rational. If you hybridise R with some other
system which permits infinitesimals, then it's not surprising that you manage
to confuse yourself.

--
Andy Walker, Nottingham.
Andy's music pages: www.cuboid.me.uk/andy/Music
Composer of the day: www.cuboid.me.uk/andy/Music/Composers/Gottschalk