Snippet from https://sourceforge.net/projects/cscall/files/MisFiles/NumberView-en.txt/download

Number::= Symbol of an expression of computation (calculation algorithm).

Positional Number::= Usual number notation, like decimal, an abbreviation of
restricted polynomial.
ℙosNum(B)::= dn*B^n +d(n-1)*B^(n-1) +...+d0 + f1*B^(-1)+ f2*B^(-2)+...+fm*B^(-m)
where B,m,n∈ℕ, B>1, ∀m,n, dn<B, fm<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] This lemma also agrees that repeating fractionals are irrational since
repeating fractionals ∉ ℙosNum(B).
------
This may be another way to see that 0.999...≠1.

On Tuesday, March 22, 2022 at 2:21:13 AM UTC-4, wij wrote:
> Snippet from https://sourceforge.net/projects/cscall/files/MisFiles/NumberView-en.txt/download
>
> Number::= Symbol of an expression of computation (calculation algorithm).
>
> Positional Number::= Usual number notation, like decimal, an abbreviation of
> restricted polynomial.
> ℙosNum(B)::= dn*B^n +d(n-1)*B^(n-1) +...+d0 + f1*B^(-1)+ f2*B^(-2)+...+fm*B^(-m)
> where B,m,n∈ℕ, B>1, ∀m,n, dn<B, fm<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] This lemma also agrees that repeating fractionals are irrational since
> repeating fractionals ∉ ℙosNum(B).
> ------
> This may be another way to see that 0.999...≠1.

Choose B = 12. B is a positional number. Since the definition is self referential there is a problem.

The first idea as well is bad. Number is what we do calculations with. Operators and values are distinct.
Without the number the 'calculation algorithm' will not compute.
To me number and value are one and the same. Not being tricky here.
Thereby the rational number is not fundamental as is accepted by modern and pre-modern mathematicians.
3/5
is two values and an operator; two numbers and an operator.
Your positional value has greater integrity in that the prior use of the radix to develop the natural number representation using radix 10 numbers is in use. This reuse of the radix within the decimal notation is not even an accurate statement. Pull the decimal point and the value literally is a natural value. Ignore sign, please. The usage of forever numbers (0.333...) can be shelved when epsilon/delta is absorbed. Close enough is good enough. Claims of infinite precision on any value that is real; as in physical; is dubious. In hindsight even our selection of unity on the continuum is arbitrary.
That the ordinary decimal value such as 2.01 carries this gray sense by engineers, but not by mathematicians: this problem deserves further scrutiny. I don't see that your 'positional number' is doing anything to alleviate these ambiguities. To me this bottoms out as an interpretation of the continuum as gray rather than so keenly defined as the mathematician claims.

wij formulated on Tuesday :
> Snippet from
> https://sourceforge.net/projects/cscall/files/MisFiles/NumberView-en.txt/download
>
> Number::= Symbol of an expression of computation (calculation algorithm).
>
> Positional Number::= Usual number notation, like decimal, an abbreviation of
> restricted polynomial.
> ℙosNum(B)::= dn*B^n +d(n-1)*B^(n-1) +...+d0 + f1*B^(-1)+
> f2*B^(-2)+...+fm*B^(-m) where B,m,n∈ℕ, B>1, ∀m,n, dn<B, fm<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] This lemma also agrees that repeating fractionals are irrational
> since repeating fractionals ∉ ℙosNum(B).
> ------
> This may be another way to see that 0.999...≠1.

IIUC your restricted polynomials are only a countable proper subset of
the reals anyway.

On Tuesday, 22 March 2022 at 22:58:11 UTC+8, timba...@gmail.com wrote:
> On Tuesday, March 22, 2022 at 2:21:13 AM UTC-4, wij wrote:
> > Snippet from https://sourceforge.net/projects/cscall/files/MisFiles/NumberView-en.txt/download
> >
> > Number::= Symbol of an expression of computation (calculation algorithm).
> >
> > Positional Number::= Usual number notation, like decimal, an abbreviation of
> > restricted polynomial.
> > ℙosNum(B)::= dn*B^n +d(n-1)*B^(n-1) +...+d0 + f1*B^(-1)+ f2*B^(-2)+...+fm*B^(-m)
> > where B,m,n∈ℕ, B>1, ∀m,n, dn<B, fm<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] This lemma also agrees that repeating fractionals are irrational since
> > repeating fractionals ∉ ℙosNum(B).
> > ------
> > This may be another way to see that 0.999...≠1.
> Choose B = 12. B is a positional number. Since the definition is self referential there is a problem.
>
> The first idea as well is bad. Number is what we do calculations with. Operators and values are distinct.
> Without the number the 'calculation algorithm' will not compute.

12= 11+1 = 10+1+1= 9+1+1+1= ...
= 1+1+1+1+1+1+1+1+1+1+1+1= S(S(S(S(S(S(S(S(S(S(S(S(0))))))))))))
= inc()+...+inc()

It's OK if you don't see what number really is. Anyway, there may be many
different theories at the fundamental level.

> To me number and value are one and the same. Not being tricky here.
> Thereby the rational number is not fundamental as is accepted by modern and pre-modern mathematicians.
> 3/5
> is two values and an operator; two numbers and an operator.
> Your positional value has greater integrity in that the prior use of the radix to develop the natural number representation using radix 10 numbers is in use. This reuse of the radix within the decimal notation is not even an accurate statement. Pull the decimal point and the value literally is a natural value. Ignore sign, please. The usage of forever numbers (0.333...) can be shelved when epsilon/delta is absorbed. Close enough is good enough. Claims of infinite precision on any value that is real; as in physical; is dubious. In hindsight even our selection of unity on the continuum is arbitrary.
> That the ordinary decimal value such as 2.01 carries this gray sense by engineers, but not by mathematicians: this problem deserves further scrutiny.. I don't see that your 'positional number' is doing anything to alleviate these ambiguities. To me this bottoms out as an interpretation of the continuum as gray rather than so keenly defined as the mathematician claims.

ℙosNum==ℚ ? (may be difficult for modern Pythagoreans)

On Tuesday, 22 March 2022 at 23:54:55 UTC+8, FromTheRafters wrote:
> wij formulated on Tuesday :
> > Snippet from
> > https://sourceforge.net/projects/cscall/files/MisFiles/NumberView-en.txt/download
> >
> > Number::= Symbol of an expression of computation (calculation algorithm).
> >
> > Positional Number::= Usual number notation, like decimal, an abbreviation of
> > restricted polynomial.
> > ℙosNum(B)::= dn*B^n +d(n-1)*B^(n-1) +...+d0 + f1*B^(-1)+
> > f2*B^(-2)+...+fm*B^(-m) where B,m,n∈ℕ, B>1, ∀m,n, dn<B, fm<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] This lemma also agrees that repeating fractionals are irrational
> > since repeating fractionals ∉ ℙosNum(B).
> > ------
> > This may be another way to see that 0.999...≠1.
> IIUC your restricted polynomials are only a countable proper subset of
> the reals anyway.

What is not 'countable'? You think card(Dedekind cuts of [0,1]) is not?
How many Cauchy sequences can be defined in [0,1]?
Note the main point is ℙosNum==ℚ.