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Real Number --- Merely numbers whose digits can be infinitely long

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  Real Number --- Merely numbers whose digits can be infinitely long wij <wyniijj5@gmail.com> - 2024-04-29 04:42 +0800
    Re: Real Number --- Merely numbers whose digits can be infinitely long Ben Bacarisse <ben.usenet@bsb.me.uk> - 2024-04-29 11:57 +0100
      Re: Real Number --- Merely numbers whose digits can be infinitely long Andy Walker <anw@cuboid.co.uk> - 2024-04-29 13:35 +0100
      Re: Real Number --- Merely numbers whose digits can be infinitely long wij <wyniijj5@gmail.com> - 2024-04-30 11:03 +0800
        Re: Real Number --- Merely numbers whose digits can be infinitely long Keith Thompson <Keith.S.Thompson+u@gmail.com> - 2024-04-29 22:02 -0700
        Re: Real Number --- Merely numbers whose digits can be infinitely long Ben Bacarisse <ben.usenet@bsb.me.uk> - 2024-05-01 22:58 +0100
          Re: Real Number --- Merely numbers whose digits can be infinitely long wij <wyniijj5@gmail.com> - 2024-05-02 09:33 +0800
            Re: Real Number --- Merely numbers whose digits can be infinitely long Keith Thompson <Keith.S.Thompson+u@gmail.com> - 2024-05-01 18:38 -0700
              Re: Real Number --- Merely numbers whose digits can be infinitely long wij <wyniijj5@gmail.com> - 2024-05-02 10:03 +0800
                Re: Real Number --- Merely numbers whose digits can be infinitely long Keith Thompson <Keith.S.Thompson+u@gmail.com> - 2024-05-01 20:38 -0700
                  Re: Real Number --- Merely numbers whose digits can be infinitely long Keith Thompson <Keith.S.Thompson+u@gmail.com> - 2024-05-01 20:46 -0700
                    Re: Real Number --- Merely numbers whose digits can be infinitely long Ross Finlayson <ross.a.finlayson@gmail.com> - 2024-05-01 21:52 -0700
                      Re: Real Number --- Merely numbers whose digits can be infinitely long Ross Finlayson <ross.a.finlayson@gmail.com> - 2024-05-01 22:40 -0700
                      Re: Real Number --- Merely numbers whose digits can be infinitely long Andy Walker <anw@cuboid.co.uk> - 2024-05-02 09:36 +0100
                        Re: Real Number --- Merely numbers whose digits can be infinitely long Ross Finlayson <ross.a.finlayson@gmail.com> - 2024-05-02 13:22 -0700
                    Re: Real Number --- Merely numbers whose digits can be infinitely long wij <wyniijj5@gmail.com> - 2024-05-02 12:46 +0800
                      Re: Real Number --- Merely numbers whose digits can be infinitely long Keith Thompson <Keith.S.Thompson+u@gmail.com> - 2024-05-02 13:51 -0700
                        Re: Real Number --- Merely numbers whose digits can be infinitely long wij <wyniijj5@gmail.com> - 2024-05-03 08:43 +0800
                          Re: Real Number --- Merely numbers whose digits can be infinitely long Keith Thompson <Keith.S.Thompson+u@gmail.com> - 2024-05-02 18:02 -0700
                            Re: Real Number --- Merely numbers whose digits can be infinitely long wij <wyniijj5@gmail.com> - 2024-05-03 09:44 +0800
                              Re: Real Number --- Merely numbers whose digits can be infinitely long Keith Thompson <Keith.S.Thompson+u@gmail.com> - 2024-05-02 19:58 -0700
                                Re: Real Number --- Merely numbers whose digits can be infinitely long Ross Finlayson <ross.a.finlayson@gmail.com> - 2024-05-02 20:23 -0700
                                  Re: Real Number --- Merely numbers whose digits can be infinitely long Ross Finlayson <ross.a.finlayson@gmail.com> - 2024-05-04 10:36 -0700
            Re: Real Number --- Merely numbers whose digits can be infinitely long Ben Bacarisse <ben.usenet@bsb.me.uk> - 2024-05-02 23:03 +0100
              Re: Real Number --- Merely numbers whose digits can be infinitely long wij <wyniijj5@gmail.com> - 2024-05-03 08:41 +0800
                Re: Real Number --- Merely numbers whose digits can be infinitely long Ben Bacarisse <ben.usenet@bsb.me.uk> - 2024-05-04 00:02 +0100
                  Re: Real Number --- Merely numbers whose digits can be infinitely long wij <wyniijj5@gmail.com> - 2024-05-04 08:28 +0800
                    Re: Real Number --- Merely numbers whose digits can be infinitely long Ben Bacarisse <ben.usenet@bsb.me.uk> - 2024-05-05 10:14 +0100

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#104047 — Real Number --- Merely numbers whose digits can be infinitely long

The purpose this text is for establishing the bases for computational algorithm.
This file https://sourceforge.net/projects/cscall/files/MisFiles/RealNumber-en.txt/download
may be updated anytime.

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

     Ex: 78, -12.345, 3.1414159...(π)

   Addition/subtraction of n-ary fixed-point numbers are the same as what is 
   taught in elementary schools (or on abacus). Any two n-ary fixed-point 
   number (same n-ary) a,b are equal iff their <fixed_point_number> 
   representation are identical. Otherwise, a>b or a<b, exactly one of these
   two holds (Law of trichotomy).

A=B ::= 1. A≡A                 // A≡B means "strictly isomorphic"
        2. A≡B <=> A±x ≡ B±x   // The recursion steps must be finite
                               // The meaning of A,B,± is analogous to what in
                               // abstract algebra

   Note: 'Number' is an abstract concept. But, ultimately, the semantics of 
         'abstract' must refer to objective symbols (or model, or something
         concrete). In this view, 'numbers' are  procedures of computation of 
         0's and 1's that correspond to our operations. So, in the end, 
         'numbers' (or, even mathematics) are mostly likely involved with, and
         maybe modeled by, computer programs or algorithms.

Real Nunmber (ℝ)::= {x| x is finitely represented by n-ary <fixed_point_number>
   and those that cannot be finitely represented (ref. Appendix)}. ℝ must be 
   able to provide an algorithm to make the marks of a physical ruler with
   arbitrary precision. In this respect, ℝ is not entirely pure theoretical and
   the basic reason we can describe the universe.

   Note: Numbers that are not finitely representable cannot all be explicitly
         defined because most of those numbers are inexpressible. This is the
         property of real number based on discrete symbols (like quantum?). E.g.

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

         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. (The process of
         10*0.999.. changed the structure of the number. For infinite series,
         this is important, because the number defined by the infinite series
         may thus have been implicitly modified)

         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).
              (2) is an interpretation of (1) from infinite possibilities.

   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 not rational (.i.e.
         irrational).

Theorem: x∈ℚ,x>0 iff there exist finite number of q's,q∈ℚ, 0<q<x, such that
         x=q1+q2+...
  Proof: Let prosition A=lhs, proposition B=rhs, then we can have a truth table:
      A B
      ----+---
      T T | T  // x∈ℚ,x>0 and, finite q's,...,such that x=q1+q2+...
      T F | F  // x∈ℚ,x>0 and, non-finite q's,...,such that x=q1+q2+...
      F T | F  // x∉ℚ,x>0 and, finite q's,...,such that x=q1+q2+...
      F F | T  // x∉ℚ,x>0 and, non-finite q's,...,such that x=q1+q2+...

Theorem: The number represented by finite length of n-ary <fixed_pointr_number>
  is rational, and that by infinite length of n-ary <fixed_pointr_number> is
  irrational.
  Proof: The main condition when the number is an infinitely long fraction can
         be proved by using the above theorem.

Real number is just this simple. Limit defines derivative and provides method
for finding it, 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
    https://www.geneseo.edu/~aguilar/public/notes/Real-Analysis-HTML/ch4-limits.html
   
    The essence of limit is a provision that allowing us to specify a number (
    i.e. L) not via equations. Limit says: When x approaches a (x≠a), the limit
    of f(x) is L (ε-δ description is satisfied), not "When x approaches a, 
    finally, f(a)=L".

    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:
                                         // Are f(ℵ₀),f(ℵ₁) the same?

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

    Note: Limit is defined on existing number system, it cannot be used to
          define the number in the approaching sequence.

    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 ?

    Note: Infinitesimal is not small in that roughly every 'list-able' interval
          [x,c) in the approaching sequence of limit might still be 1-1
          corresponding with ℝ.

+--------------------------------------+
| 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 (see Note1) 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)

    Note1: What is in text book is a bit different. Expected property of F just
           needs the number (i.e. limit L) be uniquely defined, others.., don't
           know.
    Note2: Hope that this interpretation can temporarily separate the
           interpretation of infinity/infinitesimal, and provide more correct
           foundation for some other theories.

+-----------------+ 
| Infinite Series |
+-----------------+
Series::= S= Σ(n=0,k) a(n)= a(0)+ a(1)+ a(2) +... +a(k)
  a(n) is called the general term, a(0),a(1),... the addend, summand or just
  term. n is referred to as the index. Series S is the sum from the first term
  a(0) to the last term a(k). The sum of those first terms (n<k) is called the
  partial sum. "a(0)+...+a(k)" is called expanded form.

Infinite Series::= If the series S refers to infinite terms/addend (n=∞), S is
  called an infinite series. Note that there are infinite(NEVER terminate)
  addends. I.e. basically, the addition of addends cannot be completed in 
  finite steps by definition.

Convergent::= The sequence of the partial sum of infinite series has limit.

Operation Principle of Infinite Series: The last addend of the expanded form
  (the index is ∞) must be shown to indicate the general term.

  The arithmetic of the expanded form is the same as finite series:
  Ex1: Let S= Σ(n=0,∞) a^n = 1+a+a^2+...+a^∞)
    S= 1+a*(1+a+a^2+...+a^∞)- a*a^∞
    <=> S= 1+a*S-a^(∞+1)
    <=> S(1-a)=1-a^(∞+1)
    <=> S= (1-a^(∞+1))/(1-a)

  Ex2: Let S= Σ(n=1,∞) n = 1+2+3+...+n
    S= 1+2+3+...+n  // (1)
    S= n+...+3+2+1  // (2)
    2S= n*(n+1)     // (1)+(2)
    <=> S= n*(n+1)/2

  If the last addend is missing, the expanded form is prone to magic tricks,
  because the rearrangement of the expanded form may likely change the 
  definition of the series:
  Ex1: S can be any number from a rearrangement:
       S= Σ(n=1,∞) n= 1+2+3+... =1+1+1+1+...= (1+1)+(1+1+1)+...
       = Σ(n=1,∞) n+1  // S is modified
         (or S=(1+2)+(3+4)+... = Σ(n=1,∞) 4*n-1)

  Ex2:
    S=1+2+4+8+...          // The last addend is omitted (ill-formed)
    <=> S=1+2(1+2+4+8+...) // Various possibilities of interpretation
    <=> S=1+2S
    <=> S=-1

  Last addend is shown:
    S=1+2+4+8+...+2^∞
    <=> S=1+2(1+2+4+...+2^(∞-1))
    <=> S=1+2S-2^(∞+1)
    <=> S=2^(∞+1)-1   // Lots of similar "magic calculation" deriving the result
                      // S=-1 can be found in youtube (from the omission of the
                      // term containing ∞).

Theorem1: s1=s2 <=> s1-s2=0

Theorem2: Σ(n=0,∞) a(n)= a(0)+ Σ(n=1,∞) a(n)
                    = a(∞)+ Σ(n=0,∞-1) a(n)

Theorem3: Σ(n=0,∞) f(n) ± Σ(n=0,∞) g(n) = Σ(n=0,∞) f(n)±g(n)
Theorem4: Σ(n=0,∞) c*f(n)= c*(Σ(n=0,∞) f(n))
  Proof: Omitted (Can be derived from the expanded form. Trivial rules are also
         omitted)

  Basically, formula for finite series are also applicable to infinite series(
  but mathematical inducion cannot prove such formula because by definition,
  ∞ means 'the procedure never terminate' and the Peano axiom is only valid in 
  finite steps).

  Note: Many 'equations' of infinite series (esp. about π,e) can be proved 
        false by the theorems above. They are actually approximates (limits).
        Ex: Σ(n=1,∞) 1/n² ≒ π²/6
            Σ(n=0,∞) (-1^n)*(1/(2n+1)) ≒ π/4
            Σ(n=0,∞) k^n/n! ≒ e^k

+----------+
| Appendix |
+----------+
Appendix1: Just as an example, the theoretical ℝ might be made (yes, mathematics
    ,truth are made up) like bellow for proving equations:
    Eℕ ::= {n| n is the number defined by Peano axioms (n∈ ℕ<0,S>), including
           axiom "n∈ℕ => S(n)∈ℕ" may be applied infinitely to define infinitely
           many infinities. }
    Eℤ ::= {n| n∈Eℕ or -n∈Eℕ }
    ℝ  ::= {p/q| p,q∈Eℤ, q>=2 }

Appendix2: No 1-1 correspondence procedure exists between set ℝ and ℕ.
    Proof: Let X= {x| x is the number defined by Peano axioms including those
    (defined by the successor) that the procedure does not terminate }. Then,
    no 1-1 correspondence (procedure) can be established between X and ℕ 
    (simply because of termination/non-termination issues).
    And, because X and ℝ are isomophic, no 1-1 correspondence can be established
    ℝ and ℕ (This proof is sloop, but should be fine). This property indirectly
    explains that most numbers in ℝ are inexpressible by finite symbols.
    Note: Set X can be used to demonstrate that "repeating decimal 0.999..." (
          as 1-nary number) is able to define an isomorphic set of real number.
          Anyway, "repeating decimal 0.999..." actually can refer to a very
          large set of numbers.

Appendix3: I am not experienced with infinity ∞. Just provide several principles
    that are more sure about:
    1. From the definition, ℝ also contains infinity ∞, which share the same
       property with finite numbers in that they can compare and perform +-*/.
       There are infinite number of infinities. Take a common contradiction for
       instance: The concept of "1+∞ =∞" contradicts the concept of approach in
       "lim(x->∞) f(x)". Because if so, we cannot tell whether x+1 or x-1
       approach ∞ of leave ∞. The semantics of infinity must be unique.

       Note: If infinity does not in ℝ, the 'obsolete real' should not talk
             about infinite series. The 'obsolete real' cannot be both proven
             consistent and complete. E.g. A number of such number system(s) is
             defined on number (self-referential) (sorry, I am not even sure
             how many 'obsolete real' really are. Ex. Archimedean property 
             asserts a number system that has no infinity and no infinitesimal,
             while allowing infinite additions and using lim(x->∞) ...? And, the
             Dedekind cut theory, except mentioned, declares "it can construct
             every real number", which is of course false). Therefore, if it 
             were consistent, at most, such a number system was a subset of the
             Real defined in this file.
       
       In all, ℝ satisfies [1] Trichotomy [2] Dense property (infinite steps)
       [3] Close under +* (including the steps are infinite). Therefore, the
       very signature of ℝ is that it contains infinity.

    2. The semantics of infinity is basically about a procedural loop (as in
       Peano axioms). The characteristics is "never terminate" (might mean
       'undecidable','undefined' or 'controdiction' in procedural proofs).
       Another relative issue is about the dense property, which is different
       when applied to rational and real in that the procedure must terminate
       for the former and not necessarily for the latter (similar to Peano
       axioms).

    3. Infinity ∞ also contains Cantor's cardinal number, because the concept
       of 'number' is very primitive. Although cardinal number has its 
       definition, its concept is still based on counting (and, op +* in ℝ is
       closed).

    4. If the meaning of '∞' is unique, the meaning of 'e' (base of natural 
       logarithm) in the sense of 'number' of the following three are different:
       e::= lim(n->∞) (1+1/n)^n = (1+1/∞)^∞
       e^k= lim(n->∞) (1+k/n)^n = (1+k/∞)^∞
       e^k= Σ(n=0,∞) k^n/n!

    5. This example may help explain the concept of infinity ∞:
       Let A(0)= 0
           A(n)= (A(n-1)+1)/2
       Ask: A(∞)= ?
       Ans: The value 'A(∞)' does not exist in the range of the given problem.
            This answer also applies to the basic Zeno paradox and Supertask
            paradox.

Appendix4: 2D-number can express plane. In 2D-number, as long as the distance
    postulate (1. Distance between points is invariant by movement 2.The
    ratio of distance between points is invariant by scalar multiplication) are
    satisfied, Euclidean geometry system can be established. What is meant to 
    say is that: Such a 'mass-point universe' is constructed based on our
    preset property. We are ultimately exploring the semantics of our own
    knowledge. And, as long as the logic holds, the respective reality should
    be expected. Inversely, exploring 'real number' by physics is basicly valid.
    In the digital era, universe (semantics) is a natural computer.

Appendix5: Convergent (that has limit) infinite series should use "lim" or '≒' 
    (except definition) to be more precise and less prone to error, E.g:
    lim(k->∞) Σ(n=0,k) (-1^n)*(1/(2n+1)) = π/4, or
    Σ(n=0,∞) (-1^n)*(1/(2n+1))≒ π/4.
-------------------------------------------------------------------------------

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#104056

wij <wyniijj5@gmail.com> writes:

> The purpose this text is for establishing the bases for computational algorithm.
> This file https://sourceforge.net/projects/cscall/files/MisFiles/RealNumber-en.txt/download
> may be updated anytime.
>
> +-------------+
> | 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>::= [-] <wnum> [ . <frac> ]
>      <wnum>::= 0 | <nzd> { 0, <nzd> }

Some readers will be confused by this.  When {}s are used for "zero or
more repetitions" what goes inside is the same kind of thing that goes
on the RHS of ::= and there (in production rules) you use | for
alternation.  Hence

  <wnum> ::= 0 | { 0 | <nzd> }

would be clearer.  Of course, if you want "," to denote alternation, you
could write

  <wnum> ::= 0, { 0, <nzd> }

instead.  It's the mixing up of the notation that's pointless and
potentially confusing.

>      <frac>::= { 0, <nzd> } <nzd>
>      <nzd> ::= (1, 2, 3, 4, 5, 6, 7, 8, 9)  // 'digit' varys depending on n-ary

Since you already have a notation for alternatives, it would be clearer
to write

  <nzd> ::= 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9

otherwise, you need to say what that the brackets with commas mean.

>      Ex: 78, -12.345, 3.1414159...(π)

The third example does not match the grammar.  Also, I would guess you
intended to use the digits of pi.

>    Addition/subtraction of n-ary fixed-point numbers are the same as what is 
>    taught in elementary schools (or on abacus). Any two n-ary fixed-point 
>    number (same n-ary) a,b are equal iff their <fixed_point_number> 
>    representation are identical. Otherwise, a>b or a<b, exactly one of these
>    two holds (Law of trichotomy).

So which of -0>0 and -0<0 holds?  I'll take what you write about limits
with a pinch of salt until the basics are clear.

-- 
Ben.

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#104062

On 29/04/2024 11:57, Ben Bacarisse wrote [to Wij]:
> [...] I'll take what you write about limits
> with a pinch of salt until the basics are clear.
	Nothing he writes is clear.  Start with the very basic -- is
this an attempt to put the existing theory of real numbers on a sound
[in his view] basis or an attempt to produce a alternative theory of
[real-like] numbers?  He seems to reject the Archimedean axiom, but
without spelling out what axiom, if any, replaces it;  he seems to be
completely ignorant of the surreal and hyperreal numbers and unwilling
to put in the effort to learn about them, even though they solve many
of the problems he seems to have with [traditional] reals.

	It's all compounded by the difficulty he has in expressing
himself in English.  I don't criticise him for that [I would be at
least as bad in any foreign language], and would be willing to try
to help him knock what he writes into shape if I could make any sense
at all of it;  but I can't.

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

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#104093

On Mon, 2024-04-29 at 11:57 +0100, Ben Bacarisse wrote:
> wij <wyniijj5@gmail.com> writes:
> 
> > The purpose this text is for establishing the bases for computational algorithm.
> > This file https://sourceforge.net/projects/cscall/files/MisFiles/RealNumber-en.txt/download
> > may be updated anytime.
> > 
> > +-------------+
> > > 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>::= [-] <wnum> [ . <frac> ]
> >      <wnum>::= 0 | <nzd> { 0, <nzd> }
> 
> Some readers will be confused by this.  When {}s are used for "zero or
> more repetitions" what goes inside is the same kind of thing that goes
> on the RHS of ::= and there (in production rules) you use | for
> alternation.  Hence
> 
>   <wnum> ::= 0 | { 0 | <nzd> }
> 
> would be clearer.  Of course, if you want "," to denote alternation, you
> could write
> 
>   <wnum> ::= 0, { 0, <nzd> }
> 
> instead.  It's the mixing up of the notation that's pointless and
> potentially confusing.
> 
> >      <frac>::= { 0, <nzd> } <nzd>
> >      <nzd> ::= (1, 2, 3, 4, 5, 6, 7, 8, 9)  // 'digit' varys depending on n-ary
> 
> Since you already have a notation for alternatives, it would be clearer
> to write
> 
>   <nzd> ::= 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9
> 
> otherwise, you need to say what that the brackets with commas mean.
> 
> >      Ex: 78, -12.345, 3.1414159...(π)
> 
> The third example does not match the grammar.  Also, I would guess you
> intended to use the digits of pi.
> 
> >    Addition/subtraction of n-ary fixed-point numbers are the same as what is 
> >    taught in elementary schools (or on abacus). Any two n-ary fixed-point 
> >    number (same n-ary) a,b are equal iff their <fixed_point_number> 
> >    representation are identical. Otherwise, a>b or a<b, exactly one of these
> >    two holds (Law of trichotomy).
> 
> So which of -0>0 and -0<0 holds?  I'll take what you write about limits
> with a pinch of salt until the basics are clear.
> 

Got your idea. I'll try use '|' exclusively. Thanks for the suggestions:

     <fixed_point_number>::= [-] <wnum> [ . <frac> ]  // excluding "-0" case
     <wnum>::= 0
     <wnum>::= <nzd> { 0 | <nzd> }
     <frac>::= { 0 | <nzd> } <nzd>
     <nzd> ::= 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 // 'digit' varys depending on n-ary

    Ex: 78, -12.345, 3.1414159

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#104094

wij <wyniijj5@gmail.com> writes:
[...]
> Got your idea. I'll try use '|' exclusively. Thanks for the suggestions:
>
>      <fixed_point_number>::= [-] <wnum> [ . <frac> ]  // excluding "-0" case
>      <wnum>::= 0
>      <wnum>::= <nzd> { 0 | <nzd> }
>      <frac>::= { 0 | <nzd> } <nzd>
>      <nzd> ::= 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 // 'digit' varys depending on n-ary
>
>     Ex: 78, -12.345, 3.1414159

Am I correct in assuming that this is intended to match only finite
strings?  (The fact that <nzd>, non-zero digit, appears at the end of
<frac> seems to imply that there must be an end.)

If so, your <fixed_point_number> can only represent integer multiples of
powers of 10 (or of the base if you allow bases other than decimal).

Is 1/3 not a fixed-point number?

Your subject line refers to *real* numbers; is π not a real number in
your view?

-- 
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 */

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#104153

wij <wyniijj5@gmail.com> writes:

> Got your idea.

It's not my idea.  It's a standard notation.

> I'll try use '|' exclusively. Thanks for the suggestions:
>
>      <fixed_point_number>::= [-] <wnum> [ . <frac> ]  // excluding "-0" case
>      <wnum>::= 0
>      <wnum>::= <nzd> { 0 | <nzd> }
>      <frac>::= { 0 | <nzd> } <nzd>
>      <nzd> ::= 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 // 'digit' varys depending on n-ary
>
>     Ex: 78, -12.345, 3.1414159

So what's the point of defining these strings that represent a subset of
the rationals?

-- 
Ben.

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#104163

On Wed, 2024-05-01 at 22:58 +0100, Ben Bacarisse wrote:
> wij <wyniijj5@gmail.com> writes:
> 
> > Got your idea.
> 
> It's not my idea.  It's a standard notation.
> 

There is no standard notation for formal grammar. I adopt the idea using either
',' or '|'. <fixed_point_number> is simple, average readers know what it should be.


> > I'll try use '|' exclusively. Thanks for the suggestions:
> > 
> >      <fixed_point_number>::= [-] <wnum> [ . <frac> ]  // excluding "-0" case
> >      <wnum>::= 0
> >      <wnum>::= <nzd> { 0 | <nzd> }
> >      <frac>::= { 0 | <nzd> } <nzd>
> >      <nzd> ::= 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 // 'digit' varys depending on n-ary
> > 
> >     Ex: 78, -12.345, 3.1414159
> 
> So what's the point of defining these strings that represent a subset of
> the rationals?
> 

<fixed_point_number> is a super set of rationals. So, what is the point you make a
false statement? My experience is that your response is generally very 'political',
if you want to influence other readers, good luck.

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#104164

wij <wyniijj5@gmail.com> writes:
> On Wed, 2024-05-01 at 22:58 +0100, Ben Bacarisse wrote:
>> wij <wyniijj5@gmail.com> writes:
[...]
>> >      <fixed_point_number>::= [-] <wnum> [ . <frac> ]  // excluding "-0" case
>> >      <wnum>::= 0
>> >      <wnum>::= <nzd> { 0 | <nzd> }
>> >      <frac>::= { 0 | <nzd> } <nzd>
>> >      <nzd> ::= 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 // 'digit' varys depending on n-ary
>> > 
>> >     Ex: 78, -12.345, 3.1414159
>> 
>> So what's the point of defining these strings that represent a subset of
>> the rationals?
>
> <fixed_point_number> is a super set of rationals.
[...]

An extraordinary claim.

Do you agree that 1/3 is a rational number?  How is 1/3 represented in
your <fixed_point_number> notation?

-- 
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 */

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#104166

On Wed, 2024-05-01 at 18:38 -0700, Keith Thompson wrote:
> wij <wyniijj5@gmail.com> writes:
> > On Wed, 2024-05-01 at 22:58 +0100, Ben Bacarisse wrote:
> > > wij <wyniijj5@gmail.com> writes:
> [...]
> > > >      <fixed_point_number>::= [-] <wnum> [ . <frac> ]  // excluding "-0" case
> > > >      <wnum>::= 0
> > > >      <wnum>::= <nzd> { 0 | <nzd> }
> > > >      <frac>::= { 0 | <nzd> } <nzd>
> > > >      <nzd> ::= 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 // 'digit' varys depending on n-ary
> > > > 
> > > >     Ex: 78, -12.345, 3.1414159
> > > 
> > > So what's the point of defining these strings that represent a subset of
> > > the rationals?
> > 
> > <fixed_point_number> is a super set of rationals.
> [...]
> 
> An extraordinary claim.
> 
> Do you agree that 1/3 is a rational number?  How is 1/3 represented in
> your <fixed_point_number> notation?
> 

I already told you: 1/3= 0.1 (3-ary <fixed_point_number>)
Substitute the n in n-ary with the q in p/q, every p/q is representable 
by <fixed_point_number>.
And, the rule of <frac> can generate infinitely long fractions, read it carefully!

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#104175

wij <wyniijj5@gmail.com> writes:
> On Wed, 2024-05-01 at 18:38 -0700, Keith Thompson wrote:
>> wij <wyniijj5@gmail.com> writes:
>> > On Wed, 2024-05-01 at 22:58 +0100, Ben Bacarisse wrote:
>> > > wij <wyniijj5@gmail.com> writes:
>> [...]
>> > > >      <fixed_point_number>::= [-] <wnum> [ . <frac> ]  // excluding "-0" case
>> > > >      <wnum>::= 0
>> > > >      <wnum>::= <nzd> { 0 | <nzd> }
>> > > >      <frac>::= { 0 | <nzd> } <nzd>
>> > > >      <nzd> ::= 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 // 'digit' varys depending on n-ary
>> > > > 
>> > > >     Ex: 78, -12.345, 3.1414159
>> > > 
>> > > So what's the point of defining these strings that represent a subset of
>> > > the rationals?
>> > 
>> > <fixed_point_number> is a super set of rationals.
>> [...]
>> 
>> An extraordinary claim.
>> 
>> Do you agree that 1/3 is a rational number?  How is 1/3 represented in
>> your <fixed_point_number> notation?
>> 
>
> I already told you: 1/3= 0.1 (3-ary <fixed_point_number>)
> Substitute the n in n-ary with the q in p/q, every p/q is representable 
> by <fixed_point_number>.
> And, the rule of <frac> can generate infinitely long fractions, read it carefully!

That kind of notation almost universally refers to *finite* sequences of
symbols.

If you intend it to be able to specify infinite sequences, that's fine,
but it's not inherent in the notation you've presented.  I also wonder
how an infinitely long <frac> can have <nzx> as its last element.

So <frac> can be infinitely long.  Can <wnum> be infinitely long?

I presume that the "n-ary" base can be any integer greater than or equal
to 2, and that the digits can range from 0 to n-1.  That means you'll
need arbitrarily many distinct symbols for the digits in large bases.
That's all fine, but it would be good to state all this explicitly.

There are already perfectly good mathematical methods for constructing
the integers, the rationals, and the reals.  Your method of using base-n
notation to *define* the reals and/or rationals seems superfluous.  It
can probably be done consistently, but I fail to see how it's useful.

-- 
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 */

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#104176

Keith Thompson <Keith.S.Thompson+u@gmail.com> writes:
> wij <wyniijj5@gmail.com> writes:
>> On Wed, 2024-05-01 at 18:38 -0700, Keith Thompson wrote:
>>> wij <wyniijj5@gmail.com> writes:
>>> > On Wed, 2024-05-01 at 22:58 +0100, Ben Bacarisse wrote:
>>> > > wij <wyniijj5@gmail.com> writes:
>>> [...]
>>> > > >      <fixed_point_number>::= [-] <wnum> [ . <frac> ]  // excluding "-0" case
>>> > > >      <wnum>::= 0
>>> > > >      <wnum>::= <nzd> { 0 | <nzd> }
>>> > > >      <frac>::= { 0 | <nzd> } <nzd>
>>> > > >      <nzd> ::= 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 // 'digit' varys depending on n-ary
>>> > > > 
>>> > > >     Ex: 78, -12.345, 3.1414159
>>> > > 
>>> > > So what's the point of defining these strings that represent a subset of
>>> > > the rationals?
>>> > 
>>> > <fixed_point_number> is a super set of rationals.
>>> [...]
>>> 
>>> An extraordinary claim.
>>> 
>>> Do you agree that 1/3 is a rational number?  How is 1/3 represented in
>>> your <fixed_point_number> notation?
>>> 
>>
>> I already told you: 1/3= 0.1 (3-ary <fixed_point_number>)
>> Substitute the n in n-ary with the q in p/q, every p/q is representable 
>> by <fixed_point_number>.
>> And, the rule of <frac> can generate infinitely long fractions, read it carefully!
>
> That kind of notation almost universally refers to *finite* sequences of
> symbols.
>
> If you intend it to be able to specify infinite sequences, that's fine,
> but it's not inherent in the notation you've presented.  I also wonder
> how an infinitely long <frac> can have <nzx> as its last element.
>
> So <frac> can be infinitely long.  Can <wnum> be infinitely long?
>
> I presume that the "n-ary" base can be any integer greater than or equal
> to 2, and that the digits can range from 0 to n-1.  That means you'll
> need arbitrarily many distinct symbols for the digits in large bases.
> That's all fine, but it would be good to state all this explicitly.
>
> There are already perfectly good mathematical methods for constructing
> the integers, the rationals, and the reals.  Your method of using base-n
> notation to *define* the reals and/or rationals seems superfluous.  It
> can probably be done consistently, but I fail to see how it's useful.

And something I thought of immediately after I posted the above:

You need to use different bases to represent all rational numbers, but
the base isn't part of your notation.  Your grammar matches "0.1", but
how do I know whether than's 1/10, 1/3, or 1/1729?

0.2 (base 10) and 0.1 (base 5) represent the same number.  0.2 (base 10)
and 0.1 (base 4) do not.  Your notation doesn't seem to have any way to
indicate this.  How can we know that 0.2 (base 10) and 0.1 (base 5) are
equal without using the real numbers that you're trying to *define*?

Or are you assuming that real numbers already exist, and you're defining
this notation on top of that?  If so, what's the point?

-- 
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 */

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#104199

On 05/01/2024 09:46 PM, wij wrote:
> On Wed, 2024-05-01 at 20:46 -0700, Keith Thompson wrote:
>> Keith Thompson <Keith.S.Thompson+u@gmail.com> writes:
>>> wij <wyniijj5@gmail.com> writes:
>>>> On Wed, 2024-05-01 at 18:38 -0700, Keith Thompson wrote:
>>>>> wij <wyniijj5@gmail.com> writes:
>>>>>> On Wed, 2024-05-01 at 22:58 +0100, Ben Bacarisse wrote:
>>>>>>> wij <wyniijj5@gmail.com> writes:
>>>>> [...]
>>>>>>>>       <fixed_point_number>::= [-] <wnum> [ . <frac> ]  // excluding "-0" case
>>>>>>>>       <wnum>::= 0
>>>>>>>>       <wnum>::= <nzd> { 0 | <nzd> }
>>>>>>>>       <frac>::= { 0 | <nzd> } <nzd>
>>>>>>>>       <nzd> ::= 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 // 'digit' varys depending on n-ary
>>>>>>>>
>>>>>>>>      Ex: 78, -12.345, 3.1414159
>>>>>>>
>>>>>>> So what's the point of defining these strings that represent a subset of
>>>>>>> the rationals?
>>>>>>
>>>>>> <fixed_point_number> is a super set of rationals.
>>>>> [...]
>>>>>
>>>>> An extraordinary claim.
>>>>>
>>>>> Do you agree that 1/3 is a rational number?  How is 1/3 represented in
>>>>> your <fixed_point_number> notation?
>>>>>
>>>>
>>>> I already told you: 1/3= 0.1 (3-ary <fixed_point_number>)
>>>> Substitute the n in n-ary with the q in p/q, every p/q is representable
>>>> by <fixed_point_number>.
>>>> And, the rule of <frac> can generate infinitely long fractions, read it carefully!
>>>
>>> That kind of notation almost universally refers to *finite* sequences of
>>> symbols.
>>>
>>> If you intend it to be able to specify infinite sequences, that's fine,
>>> but it's not inherent in the notation you've presented.  I also wonder
>>> how an infinitely long <frac> can have <nzx> as its last element.
>>>
>>> So <frac> can be infinitely long.  Can <wnum> be infinitely long?
>>>
>>> I presume that the "n-ary" base can be any integer greater than or equal
>>> to 2, and that the digits can range from 0 to n-1.  That means you'll
>>> need arbitrarily many distinct symbols for the digits in large bases.
>>> That's all fine, but it would be good to state all this explicitly.
>>>
>>> There are already perfectly good mathematical methods for constructing
>>> the integers, the rationals, and the reals.  Your method of using base-n
>>> notation to *define* the reals and/or rationals seems superfluous.  It
>>> can probably be done consistently, but I fail to see how it's useful.
>>
>> And something I thought of immediately after I posted the above:
>>
>> You need to use different bases to represent all rational numbers, but
>> the base isn't part of your notation.  Your grammar matches "0.1", but
>> how do I know whether than's 1/10, 1/3, or 1/1729?
>>
> Do you use different bases to represent all rational numbers?
>
>> 0.2 (base 10) and 0.1 (base 5) represent the same number.  0.2 (base 10)
>> and 0.1 (base 4) do not.  Your notation doesn't seem to have any way to
>> indicate this.  How can we know that 0.2 (base 10) and 0.1 (base 5) are
>> equal without using the real numbers that you're trying to *define*?
>>
> How should I know your numbers (1/10, 1/3, or 1/1729) are in base-12 or base-16
> if you also did not say it explicitly?
>
>> Or are you assuming that real numbers already exist, and you're defining
>> this notation on top of that?  If so, what's the point?
>>
>
> Your request is valid but not practically reasonable.
>

I've found there's at least three definitions of real numbers,
Aristotle's first, Aristotle's second, and Fourier's.

I've found there's at least three definitions of numeric continuum,
the integer continuum of the Scotists, the linear continuum of
the real numbers, and the long-line continuum of duBois-Reymond.

Of course these just naturally arise from thinking about numbers
then satisfy formalism and all these usual kinds of things.

The standard way today has one, it's one of Aristotle's, Eudoxus'.


Also geometry naturally arises from a theory of points and spaces,
or spaces and points, either way. It's Euclidean of course,
and Cartesian.


Of course all the usual transfinite cardinals and ordinals
are usual also, Cantorian.


It took about 30 years and tens of thousands of posts to
figure this out and establish it, while it's sort of
shaping up this way. Luckily it just points at dogmatic
canon for everything then just sort of caps it off.


Then, physics sort of is connected, also, and neatly.
(Naturally.)

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#104200

On 05/01/2024 09:52 PM, Ross Finlayson wrote:
> On 05/01/2024 09:46 PM, wij wrote:
>> On Wed, 2024-05-01 at 20:46 -0700, Keith Thompson wrote:
>>> Keith Thompson <Keith.S.Thompson+u@gmail.com> writes:
>>>> wij <wyniijj5@gmail.com> writes:
>>>>> On Wed, 2024-05-01 at 18:38 -0700, Keith Thompson wrote:
>>>>>> wij <wyniijj5@gmail.com> writes:
>>>>>>> On Wed, 2024-05-01 at 22:58 +0100, Ben Bacarisse wrote:
>>>>>>>> wij <wyniijj5@gmail.com> writes:
>>>>>> [...]
>>>>>>>>>       <fixed_point_number>::= [-] <wnum> [ . <frac> ]  //
>>>>>>>>> excluding "-0" case
>>>>>>>>>       <wnum>::= 0
>>>>>>>>>       <wnum>::= <nzd> { 0 | <nzd> }
>>>>>>>>>       <frac>::= { 0 | <nzd> } <nzd>
>>>>>>>>>       <nzd> ::= 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 // 'digit'
>>>>>>>>> varys depending on n-ary
>>>>>>>>>
>>>>>>>>>      Ex: 78, -12.345, 3.1414159
>>>>>>>>
>>>>>>>> So what's the point of defining these strings that represent a
>>>>>>>> subset of
>>>>>>>> the rationals?
>>>>>>>
>>>>>>> <fixed_point_number> is a super set of rationals.
>>>>>> [...]
>>>>>>
>>>>>> An extraordinary claim.
>>>>>>
>>>>>> Do you agree that 1/3 is a rational number?  How is 1/3
>>>>>> represented in
>>>>>> your <fixed_point_number> notation?
>>>>>>
>>>>>
>>>>> I already told you: 1/3= 0.1 (3-ary <fixed_point_number>)
>>>>> Substitute the n in n-ary with the q in p/q, every p/q is
>>>>> representable
>>>>> by <fixed_point_number>.
>>>>> And, the rule of <frac> can generate infinitely long fractions,
>>>>> read it carefully!
>>>>
>>>> That kind of notation almost universally refers to *finite*
>>>> sequences of
>>>> symbols.
>>>>
>>>> If you intend it to be able to specify infinite sequences, that's fine,
>>>> but it's not inherent in the notation you've presented.  I also wonder
>>>> how an infinitely long <frac> can have <nzx> as its last element.
>>>>
>>>> So <frac> can be infinitely long.  Can <wnum> be infinitely long?
>>>>
>>>> I presume that the "n-ary" base can be any integer greater than or
>>>> equal
>>>> to 2, and that the digits can range from 0 to n-1.  That means you'll
>>>> need arbitrarily many distinct symbols for the digits in large bases.
>>>> That's all fine, but it would be good to state all this explicitly.
>>>>
>>>> There are already perfectly good mathematical methods for constructing
>>>> the integers, the rationals, and the reals.  Your method of using
>>>> base-n
>>>> notation to *define* the reals and/or rationals seems superfluous.  It
>>>> can probably be done consistently, but I fail to see how it's useful.
>>>
>>> And something I thought of immediately after I posted the above:
>>>
>>> You need to use different bases to represent all rational numbers, but
>>> the base isn't part of your notation.  Your grammar matches "0.1", but
>>> how do I know whether than's 1/10, 1/3, or 1/1729?
>>>
>> Do you use different bases to represent all rational numbers?
>>
>>> 0.2 (base 10) and 0.1 (base 5) represent the same number.  0.2 (base 10)
>>> and 0.1 (base 4) do not.  Your notation doesn't seem to have any way to
>>> indicate this.  How can we know that 0.2 (base 10) and 0.1 (base 5) are
>>> equal without using the real numbers that you're trying to *define*?
>>>
>> How should I know your numbers (1/10, 1/3, or 1/1729) are in base-12
>> or base-16
>> if you also did not say it explicitly?
>>
>>> Or are you assuming that real numbers already exist, and you're defining
>>> this notation on top of that?  If so, what's the point?
>>>
>>
>> Your request is valid but not practically reasonable.
>>
>
> I've found there's at least three definitions of real numbers,
> Aristotle's first, Aristotle's second, and Fourier's.
>
> I've found there's at least three definitions of numeric continuum,
> the integer continuum of the Scotists, the linear continuum of
> the real numbers, and the long-line continuum of duBois-Reymond.
>
> Of course these just naturally arise from thinking about numbers
> then satisfy formalism and all these usual kinds of things.
>
> The standard way today has one, it's one of Aristotle's, Eudoxus'.
>
>
> Also geometry naturally arises from a theory of points and spaces,
> or spaces and points, either way. It's Euclidean of course,
> and Cartesian.
>
>
> Of course all the usual transfinite cardinals and ordinals
> are usual also, Cantorian.
>
>
> It took about 30 years and tens of thousands of posts to
> figure this out and establish it, while it's sort of
> shaping up this way. Luckily it just points at dogmatic
> canon for everything then just sort of caps it off.
>
>
> Then, physics sort of is connected, also, and neatly.
> (Naturally.)
>
>
>
>

It's kind of like Thomists and followers of Occam:

"I read the book of Aristotle."


Then Scotus is like:

"I read both books of Aristotle,
and read between the lines."

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#104205

On 02/05/2024 05:52, Ross Finlayson wrote:
[...]
[Keith:]
>>> Or are you assuming that real numbers already exist, and you're defining
>>> this notation on top of that?  If so, what's the point?
[Wij:]
>> Your request is valid but not practically reasonable.

	Yet without an answer to that request, this whole discussion is
pointless.  We have been given a start towards a notation [which is not
new, people have been writing numbers using strings of digits "forever"]
with no reason given for why we should pay attention.  Except as a
diversion from endless debates about "halting".

> I've found there's at least three definitions of real numbers,
> Aristotle's first, Aristotle's second, and Fourier's. [...]

	And the rest.  But every [sensible] new definition has come with
/either/ a proof that it is equivalent to an old definition, /or/ some
reason why it gives rise to a different collection of numbers [which is
then designated by some other name, to avoid confusion].  Wij refuses to
elaborate on his motivation, apart from apparently having some problem
with notions of limits.  He's not alone in that!

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

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#104227

On 05/02/2024 01:36 AM, Andy Walker wrote:
> On 02/05/2024 05:52, Ross Finlayson wrote:
> [...]
> [Keith:]
>>>> Or are you assuming that real numbers already exist, and you're
>>>> defining
>>>> this notation on top of that?  If so, what's the point?
> [Wij:]
>>> Your request is valid but not practically reasonable.
>
>      Yet without an answer to that request, this whole discussion is
> pointless.  We have been given a start towards a notation [which is not
> new, people have been writing numbers using strings of digits "forever"]
> with no reason given for why we should pay attention.  Except as a
> diversion from endless debates about "halting".
>
>> I've found there's at least three definitions of real numbers,
>> Aristotle's first, Aristotle's second, and Fourier's. [...]
>
>      And the rest.  But every [sensible] new definition has come with
> /either/ a proof that it is equivalent to an old definition, /or/ some
> reason why it gives rise to a different collection of numbers [which is
> then designated by some other name, to avoid confusion].  Wij refuses to
> elaborate on his motivation, apart from apparently having some problem
> with notions of limits.  He's not alone in that!
>

What I'm saying is that there is each of and all of these
"line-real, field-reals, signal-reals" of the "integer
continuum, linear continuum, long-line continuum", as
for Hardy and geometry's points in, or, or about and
around, the line, that there's Standard, or Sparse,
Cantor Space, and also Square Cantor Space, and,
Signal Cantor Space, and, these are the objects
of mathematics that represent the continuum.

These line-reals, field-reals, signal-reals are
each different sets in a usual axiomatic set
theory's descriptive set theory's model theory's
models of continuous domains, or "the real numbers".

They are alone in that.

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#104201

On Wed, 2024-05-01 at 20:46 -0700, Keith Thompson wrote:
> Keith Thompson <Keith.S.Thompson+u@gmail.com> writes:
> > wij <wyniijj5@gmail.com> writes:
> > > On Wed, 2024-05-01 at 18:38 -0700, Keith Thompson wrote:
> > > > wij <wyniijj5@gmail.com> writes:
> > > > > On Wed, 2024-05-01 at 22:58 +0100, Ben Bacarisse wrote:
> > > > > > wij <wyniijj5@gmail.com> writes:
> > > > [...]
> > > > > > >      <fixed_point_number>::= [-] <wnum> [ . <frac> ]  // excluding "-0" case
> > > > > > >      <wnum>::= 0
> > > > > > >      <wnum>::= <nzd> { 0 | <nzd> }
> > > > > > >      <frac>::= { 0 | <nzd> } <nzd>
> > > > > > >      <nzd> ::= 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 // 'digit' varys depending on n-ary
> > > > > > > 
> > > > > > >     Ex: 78, -12.345, 3.1414159
> > > > > > 
> > > > > > So what's the point of defining these strings that represent a subset of
> > > > > > the rationals?
> > > > > 
> > > > > <fixed_point_number> is a super set of rationals.
> > > > [...]
> > > > 
> > > > An extraordinary claim.
> > > > 
> > > > Do you agree that 1/3 is a rational number?  How is 1/3 represented in
> > > > your <fixed_point_number> notation?
> > > > 
> > > 
> > > I already told you: 1/3= 0.1 (3-ary <fixed_point_number>)
> > > Substitute the n in n-ary with the q in p/q, every p/q is representable 
> > > by <fixed_point_number>.
> > > And, the rule of <frac> can generate infinitely long fractions, read it carefully!
> > 
> > That kind of notation almost universally refers to *finite* sequences of
> > symbols.
> > 
> > If you intend it to be able to specify infinite sequences, that's fine,
> > but it's not inherent in the notation you've presented.  I also wonder
> > how an infinitely long <frac> can have <nzx> as its last element.
> > 
> > So <frac> can be infinitely long.  Can <wnum> be infinitely long?
> > 
> > I presume that the "n-ary" base can be any integer greater than or equal
> > to 2, and that the digits can range from 0 to n-1.  That means you'll
> > need arbitrarily many distinct symbols for the digits in large bases.
> > That's all fine, but it would be good to state all this explicitly.
> > 
> > There are already perfectly good mathematical methods for constructing
> > the integers, the rationals, and the reals.  Your method of using base-n
> > notation to *define* the reals and/or rationals seems superfluous.  It
> > can probably be done consistently, but I fail to see how it's useful.
> 
> And something I thought of immediately after I posted the above:
> 
> You need to use different bases to represent all rational numbers, but
> the base isn't part of your notation.  Your grammar matches "0.1", but
> how do I know whether than's 1/10, 1/3, or 1/1729?
> 
Do you use different bases to represent all rational numbers?

> 0.2 (base 10) and 0.1 (base 5) represent the same number.  0.2 (base 10)
> and 0.1 (base 4) do not.  Your notation doesn't seem to have any way to
> indicate this.  How can we know that 0.2 (base 10) and 0.1 (base 5) are
> equal without using the real numbers that you're trying to *define*?
> 
How should I know your numbers (1/10, 1/3, or 1/1729) are in base-12 or base-16
if you also did not say it explicitly?

> Or are you assuming that real numbers already exist, and you're defining
> this notation on top of that?  If so, what's the point?
> 

Your request is valid but not practically reasonable.

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#104228

wij <wyniijj5@gmail.com> writes:
> On Wed, 2024-05-01 at 20:46 -0700, Keith Thompson wrote:
>> Keith Thompson <Keith.S.Thompson+u@gmail.com> writes:
>> > wij <wyniijj5@gmail.com> writes:
>> > > On Wed, 2024-05-01 at 18:38 -0700, Keith Thompson wrote:
>> > > > wij <wyniijj5@gmail.com> writes:
>> > > > > On Wed, 2024-05-01 at 22:58 +0100, Ben Bacarisse wrote:
>> > > > > > wij <wyniijj5@gmail.com> writes:
>> > > > [...]
>> > > > > > >      <fixed_point_number>::= [-] <wnum> [ . <frac> ]  // excluding "-0" case
>> > > > > > >      <wnum>::= 0
>> > > > > > >      <wnum>::= <nzd> { 0 | <nzd> }
>> > > > > > >      <frac>::= { 0 | <nzd> } <nzd>
>> > > > > > >      <nzd> ::= 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 // 'digit' varys depending on n-ary
>> > > > > > > 
>> > > > > > >     Ex: 78, -12.345, 3.1414159
>> > > > > > 
>> > > > > > So what's the point of defining these strings that represent a subset of
>> > > > > > the rationals?
>> > > > > 
>> > > > > <fixed_point_number> is a super set of rationals.
>> > > > [...]
>> > > > 
>> > > > An extraordinary claim.
>> > > > 
>> > > > Do you agree that 1/3 is a rational number?  How is 1/3 represented in
>> > > > your <fixed_point_number> notation?
>> > > > 
>> > > 
>> > > I already told you: 1/3= 0.1 (3-ary <fixed_point_number>)
>> > > Substitute the n in n-ary with the q in p/q, every p/q is representable 
>> > > by <fixed_point_number>.
>> > > And, the rule of <frac> can generate infinitely long fractions, read it carefully!
>> > 
>> > That kind of notation almost universally refers to *finite* sequences of
>> > symbols.
>> > 
>> > If you intend it to be able to specify infinite sequences, that's fine,
>> > but it's not inherent in the notation you've presented.  I also wonder
>> > how an infinitely long <frac> can have <nzx> as its last element.
>> > 
>> > So <frac> can be infinitely long.  Can <wnum> be infinitely long?
>> > 
>> > I presume that the "n-ary" base can be any integer greater than or equal
>> > to 2, and that the digits can range from 0 to n-1.  That means you'll
>> > need arbitrarily many distinct symbols for the digits in large bases.
>> > That's all fine, but it would be good to state all this explicitly.
>> > 
>> > There are already perfectly good mathematical methods for constructing
>> > the integers, the rationals, and the reals.  Your method of using base-n
>> > notation to *define* the reals and/or rationals seems superfluous.  It
>> > can probably be done consistently, but I fail to see how it's useful.
>> 
>> And something I thought of immediately after I posted the above:
>> 
>> You need to use different bases to represent all rational numbers, but
>> the base isn't part of your notation.  Your grammar matches "0.1", but
>> how do I know whether than's 1/10, 1/3, or 1/1729?
>> 
> Do you use different bases to represent all rational numbers?

We're not talking about how *I* represent rational numbers.

You've presented a grammar for "<fixed_point_number>", and you claim (I
presume) that it's useful as a way to represent at least all rational
numbers.  A given string may or may not satisfy the grammar; if it does,
it represents some real number.  Is that correct so far?

The string "0.1" satisfies your grammar.  What number does it represent?
How is someone supposed to know what base "n" is being used?

The string "0.2" with n=10 and the string "0.1" with n=5 (assuming we
have some way of specifying the value of n) both represent the same
number.  Every rational number (perhaps other than 0) has infinitely
many representations in your notation, with different values of n.  Were
you aware of that?  Is it a deliberate feature of your system?

And I see you ignored another question: Given your assertion that <frac>
can be infinitely long, can <wnum> also be infinitely long?  Isn't this
the kind of thing that you need to state explicitly?

>> 0.2 (base 10) and 0.1 (base 5) represent the same number.  0.2 (base 10)
>> and 0.1 (base 4) do not.  Your notation doesn't seem to have any way to
>> indicate this.  How can we know that 0.2 (base 10) and 0.1 (base 5) are
>> equal without using the real numbers that you're trying to *define*?
>> 
> How should I know your numbers (1/10, 1/3, or 1/1729) are in base-12 or base-16
> if you also did not say it explicitly?
>
>> Or are you assuming that real numbers already exist, and you're defining
>> this notation on top of that?  If so, what's the point?
>
> Your request is valid but not practically reasonable.

I have no idea what you mean by that.  I did not make a request.
I asked several questions about the notation you're presenting.

How is your notation useful?  Do you believe it to be an improvement on
the other existing methods of defining the rational and real numbers?

-- 
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 */

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#104234

On Thu, 2024-05-02 at 13:51 -0700, Keith Thompson wrote:
> wij <wyniijj5@gmail.com> writes:
> > On Wed, 2024-05-01 at 20:46 -0700, Keith Thompson wrote:
> > > Keith Thompson <Keith.S.Thompson+u@gmail.com> writes:
> > > > wij <wyniijj5@gmail.com> writes:
> > > > > On Wed, 2024-05-01 at 18:38 -0700, Keith Thompson wrote:
> > > > > > wij <wyniijj5@gmail.com> writes:
> > > > > > > On Wed, 2024-05-01 at 22:58 +0100, Ben Bacarisse wrote:
> > > > > > > > wij <wyniijj5@gmail.com> writes:
> > > > > > [...]
> > > > > > > > >      <fixed_point_number>::= [-] <wnum> [ . <frac> ]  // excluding "-0" case
> > > > > > > > >      <wnum>::= 0
> > > > > > > > >      <wnum>::= <nzd> { 0 | <nzd> }
> > > > > > > > >      <frac>::= { 0 | <nzd> } <nzd>
> > > > > > > > >      <nzd> ::= 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 // 'digit' varys depending on n-ary
> > > > > > > > > 
> > > > > > > > >     Ex: 78, -12.345, 3.1414159
> > > > > > > > 
> > > > > > > > So what's the point of defining these strings that represent a subset of
> > > > > > > > the rationals?
> > > > > > > 
> > > > > > > <fixed_point_number> is a super set of rationals.
> > > > > > [...]
> > > > > > 
> > > > > > An extraordinary claim.
> > > > > > 
> > > > > > Do you agree that 1/3 is a rational number?  How is 1/3 represented in
> > > > > > your <fixed_point_number> notation?
> > > > > > 
> > > > > 
> > > > > I already told you: 1/3= 0.1 (3-ary <fixed_point_number>)
> > > > > Substitute the n in n-ary with the q in p/q, every p/q is representable 
> > > > > by <fixed_point_number>.
> > > > > And, the rule of <frac> can generate infinitely long fractions, read it carefully!
> > > > 
> > > > That kind of notation almost universally refers to *finite* sequences of
> > > > symbols.
> > > > 
> > > > If you intend it to be able to specify infinite sequences, that's fine,
> > > > but it's not inherent in the notation you've presented.  I also wonder
> > > > how an infinitely long <frac> can have <nzx> as its last element.
> > > > 
> > > > So <frac> can be infinitely long.  Can <wnum> be infinitely long?
> > > > 
> > > > I presume that the "n-ary" base can be any integer greater than or equal
> > > > to 2, and that the digits can range from 0 to n-1.  That means you'll
> > > > need arbitrarily many distinct symbols for the digits in large bases.
> > > > That's all fine, but it would be good to state all this explicitly.
> > > > 
> > > > There are already perfectly good mathematical methods for constructing
> > > > the integers, the rationals, and the reals.  Your method of using base-n
> > > > notation to *define* the reals and/or rationals seems superfluous.  It
> > > > can probably be done consistently, but I fail to see how it's useful.
> > > 
> > > And something I thought of immediately after I posted the above:
> > > 
> > > You need to use different bases to represent all rational numbers, but
> > > the base isn't part of your notation.  Your grammar matches "0.1", but
> > > how do I know whether than's 1/10, 1/3, or 1/1729?
> > > 
> > Do you use different bases to represent all rational numbers?
> 
> We're not talking about how *I* represent rational numbers.
> 
> You've presented a grammar for "<fixed_point_number>", and you claim (I
> presume) that it's useful as a way to represent at least all rational
> numbers.  A given string may or may not satisfy the grammar; if it does,
> it represents some real number.  Is that correct so far?
> 
> The string "0.1" satisfies your grammar.  What number does it represent?
> How is someone supposed to know what base "n" is being used?
> 
> The string "0.2" with n=10 and the string "0.1" with n=5 (assuming we
> have some way of specifying the value of n) both represent the same
> number.  Every rational number (perhaps other than 0) has infinitely
> many representations in your notation, with different values of n.  Were
> you aware of that?  Is it a deliberate feature of your system?
> 
> And I see you ignored another question: Given your assertion that <frac>
> can be infinitely long, can <wnum> also be infinitely long?  Isn't this
> the kind of thing that you need to state explicitly?
> 
> > > 0.2 (base 10) and 0.1 (base 5) represent the same number.  0.2 (base 10)
> > > and 0.1 (base 4) do not.  Your notation doesn't seem to have any way to
> > > indicate this.  How can we know that 0.2 (base 10) and 0.1 (base 5) are
> > > equal without using the real numbers that you're trying to *define*?
> > > 
> > How should I know your numbers (1/10, 1/3, or 1/1729) are in base-12 or base-16
> > if you also did not say it explicitly?
> > 
> > > Or are you assuming that real numbers already exist, and you're defining
> > > this notation on top of that?  If so, what's the point?
> > 
> > Your request is valid but not practically reasonable.
> 
> I have no idea what you mean by that.  I did not make a request.
> I asked several questions about the notation you're presenting.
> 
> How is your notation useful?  Do you believe it to be an improvement on
> the other existing methods of defining the rational and real numbers?
> 

Nothing is different from the math. you understand (except several corner cases
which you will never need to worry about).

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#104235

wij <wyniijj5@gmail.com> writes:
[snip]
> Nothing is different from the math. you understand (except several corner cases
> which you will never need to worry about).

So there's nothing novel in your notation, and I needn't waste any more
time asking questions about it that you're unwilling and/or unable to
answer.

Is that a fair summary?

(I'll ignore the "corner cases" you allude to.)

-- 
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 */

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#104236

On Thu, 2024-05-02 at 18:02 -0700, Keith Thompson wrote:
> wij <wyniijj5@gmail.com> writes:
> [snip]
> > Nothing is different from the math. you understand (except several corner cases
> > which you will never need to worry about).
> 
> So there's nothing novel in your notation, and I needn't waste any more
> time asking questions about it that you're unwilling and/or unable to
> answer.
> 
> Is that a fair summary?
> 
> (I'll ignore the "corner cases" you allude to.)
> 

No, nothing novel there changed the usual usage of 'fixed point number'.

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