Re: Collatz Conjecture proved.

From	wij <wyniijj5@gmail.com>
Newsgroups	comp.lang.c
Subject	Re: Collatz Conjecture proved.
Date	2026-01-25 13:28 +0800
Organization	A noiseless patient Spider
Message-ID	<975e829c29ff11e83681aa87e347c7ef1ffe43dc.camel@gmail.com> (permalink)
References	<96ed450bdb96454829f94b79519afa93595b27c1.camel@gmail.com> <5dbb5d96b3bb34ac21754a4f59617b572059e857.camel@gmail.com> <10l44rd$u7or$1@dont-email.me>

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On Sat, 2026-01-24 at 23:06 -0500, James Kuyper wrote:
> On Sun, 25 Jan 2026 00:37:38 +0800, wij wrote:
> 
> > [Reference] Real number and infinity. Recurring decimals are
> > irrational numbers.
> 
> So x = 0.1212.... is irrational?
> 100x = 12.1212...
> -  x = -0.1212...
>  ===   ==========
>  99x = 12
> x = 4/33
> 
> This generalizes to work with any recurring decimal. When the recurrence
> is n digits long (in the above example, n=2), just multiply by 10^n.

Such arithmetic is called approximation, or magic proof simply because
no such axiom/theorem exists (it will cause contradiction).

Take Prop2 for example: "0.1212..." should mean 12/100 +12/10000 +12/1000000 +....
Infinite addition of rational numbers cannot be rational number the same reason
as infinite addition of natural numbers cannot be a natural number.
Bunches of reasons are there.

-----------------
https://sourceforge.net/projects/cscall/files/MisFiles/RealNumber2-en.txt/download
.....[cut]

Prop 1: The statement ℕ+ℕ=ℕ (or, let a,b∈ℕ (natural numbers), then a+b∈ℕ) holds
        only for a finite number of addition steps.
  Proof: Non-zero natural numbers are strictly increasing when added together.
         Adding with an infinite number of steps will yield a number of
         infinite length. By Axiom 1, such a number is not a natural number.

Prop 2: ℚ+ℚ=ℚ (the sum of a rational number and a rational number is still a
        rational number) The statement holds only for a finite number of
        addition steps.
  Proof: If the positive rational number q is strictly increasing by
         accumulation, and the infinite term q = a/b = q₁+q₂+q₃..., then either
         a or b must be infinitely large.

  The above propositions raise a question:
   a+b = (a+b) ... leads to the dubious conclusion a,b∈N => a+b∈N (or the
                   question of the number of applications requires explanation).
   a/b+c/d = (ad+cb)/bd ... leads to the dubious conclusion a,b∈ℚ => a+b∈ℚ (not
                            always true).
  Thus, formal logic cannot handle propositions involving the concept of
  infinite procedures.

Prop 3: Recurring decimals are irrational numbers.
  Proof: Let x be a recurring decimal, 0<x<1, x has a recurring period S, and is
  written in the base system as x=0.S₁S₂S₃...S∞. Then by definition x= S₁/b
  +S₂/b² +S₃/b³ +... +S∞/b^∞ = q₁ +q₂ +q₃ +...+q∞ (b∈ℕ).
  That is, recurring decimals are a special case of "ℚ+ℚ". According to Prop 2,
  when adding terms infinitely, x is not a rational number (i.e., irrational).
         
  Note: By the way, a brief explanation of common algebraic magic:
       (1) x= 0.999...
       (2{ 10x= 9.999... // may implicitly define 0.999... in (1)
       (3) 10x= 9+x
       (4) 9x=9
       (5) x=1
       Answer: There is no axiom or theorem to prove that (1) <=> (3).
       (3) is one of the infinite interpretations of (1), or (3) is the
       'introduction' definition of 0.999..., etc. In short, there is no
       necessary relationship between (3) and (1), or it still needs to be
       proved. For example, 0.999... formed by 1/2+1/4+1/8+... does not have the
       property of (3).

       There are actually many other examples, for example:
         1. If 0.999... = 1 holds, then it can be proved that the unique prime
            factorization theorem of positive integers does not hold:
            0.999... = 999.../1000... = 9*(111...)/(5*2)*... =1
            <=> 3*3*(111...) = (5*2)*...
            The integer to the left of the equal sign contains the prime number
            3, but the integer to the right cannot contain 3... Prime
            factorization is not unique.
         2. Are the intervals [0,1) and [0,1] equal? 0.999...∉[0,1)? What number
            is to the left of 1 (infinitely close)? Can we define a number x
            that can be infinitely close to c but not equal to c (e.g.,
            0.999...*c)?...
         3. 1/3 = 0.333... + non-zero-remainder (True identity. How to deny?)
.... [cut]
------------------

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Re: Collatz Conjecture proved. wij <wyniijj5@gmail.com> - 2026-01-25 00:37 +0800
  Re: Collatz Conjecture proved. Lawrence D’Oliveiro <ldo@nz.invalid> - 2026-01-25 00:23 +0000
  Re: Collatz Conjecture proved. James Kuyper <jameskuyper@alumni.caltech.edu> - 2026-01-24 23:06 -0500
    Re: Collatz Conjecture proved. wij <wyniijj5@gmail.com> - 2026-01-25 13:28 +0800
      Re: Collatz Conjecture proved. Lawrence D’Oliveiro <ldo@nz.invalid> - 2026-01-25 08:15 +0000
        Re: Collatz Conjecture proved. wij <wyniijj5@gmail.com> - 2026-01-25 16:46 +0800
          Re: Collatz Conjecture proved. David Brown <david.brown@hesbynett.no> - 2026-01-25 10:38 +0100
            Re: Collatz Conjecture proved. wij <wyniijj5@gmail.com> - 2026-01-25 18:55 +0800
            Re: Collatz Conjecture proved. wij <wyniijj5@gmail.com> - 2026-01-25 19:06 +0800
              Re: Collatz Conjecture proved. David Brown <david.brown@hesbynett.no> - 2026-01-25 12:47 +0100
                Re: Collatz Conjecture proved. wij <wyniijj5@gmail.com> - 2026-01-25 23:44 +0800
                Re: Collatz Conjecture proved. James Kuyper <jameskuyper@alumni.caltech.edu> - 2026-01-25 12:22 -0500
          [OT] Proofs.  Was: Collatz Conjecture proved. Ben Bacarisse <ben@bsb.me.uk> - 2026-01-25 11:33 +0000
            Re: [OT] Proofs.  Was: Collatz Conjecture proved. richard@cogsci.ed.ac.uk (Richard Tobin) - 2026-01-25 13:11 +0000
          Re: Collatz Conjecture proved. Lawrence D’Oliveiro <ldo@nz.invalid> - 2026-01-25 18:52 +0000
            Re: Collatz Conjecture proved. wij <wyniijj5@gmail.com> - 2026-01-26 03:58 +0800
      Re: Collatz Conjecture proved. James Kuyper <jameskuyper@alumni.caltech.edu> - 2026-01-25 11:25 -0500
        Re: Collatz Conjecture proved. wij <wyniijj5@gmail.com> - 2026-01-26 01:20 +0800
          Re: Collatz Conjecture proved. Janis Papanagnou <janis_papanagnou+ng@hotmail.com> - 2026-01-26 01:25 +0100
            Re: Collatz Conjecture proved. wij <wyniijj5@gmail.com> - 2026-01-26 23:51 +0800
              Re: Collatz Conjecture proved. wij <wyniijj5@gmail.com> - 2026-01-27 00:07 +0800
                Re: Collatz Conjecture proved. David Brown <david.brown@hesbynett.no> - 2026-01-26 21:05 +0100
              Re: Collatz Conjecture proved. David Brown <david.brown@hesbynett.no> - 2026-01-26 21:07 +0100
                Re: Collatz Conjecture proved. wij <wyniijj5@gmail.com> - 2026-01-27 04:34 +0800
                Re: Collatz Conjecture proved. David Brown <david.brown@hesbynett.no> - 2026-01-27 09:21 +0100
                Re: Collatz Conjecture proved. antispam@fricas.org (Waldek Hebisch) - 2026-01-27 16:31 +0000
                Re: Collatz Conjecture proved. Janis Papanagnou <janis_papanagnou+ng@hotmail.com> - 2026-01-27 18:24 +0100
                Re: Collatz Conjecture proved. antispam@fricas.org (Waldek Hebisch) - 2026-01-28 15:17 +0000
                Re: Collatz Conjecture proved. David Brown <david.brown@hesbynett.no> - 2026-01-27 18:44 +0100
                Re: Collatz Conjecture proved. Lawrence D’Oliveiro <ldo@nz.invalid> - 2026-01-27 22:52 +0000
                Re: Collatz Conjecture proved. David Brown <david.brown@hesbynett.no> - 2026-01-28 08:29 +0100
                Re: Collatz Conjecture proved. Janis Papanagnou <janis_papanagnou+ng@hotmail.com> - 2026-01-28 10:27 +0100
                Re: Collatz Conjecture proved. "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> - 2026-01-28 12:59 -0800
                Re: Collatz Conjecture proved. Janis Papanagnou <janis_papanagnou+ng@hotmail.com> - 2026-01-30 06:33 +0100
                Re: Collatz Conjecture proved. "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> - 2026-01-30 11:59 -0800
                Re: Collatz Conjecture proved. wij <wyniijj5@gmail.com> - 2026-01-28 04:08 +0800
                Re: Collatz Conjecture proved. Keith Thompson <Keith.S.Thompson+u@gmail.com> - 2026-01-27 15:11 -0800
                Re: Collatz Conjecture proved. Ben Bacarisse <ben@bsb.me.uk> - 2026-01-28 17:34 +0000
                Re: Collatz Conjecture proved. richard@cogsci.ed.ac.uk (Richard Tobin) - 2026-01-28 18:23 +0000
                Re: Collatz Conjecture proved. David Brown <david.brown@hesbynett.no> - 2026-01-29 08:39 +0100
                Re: Collatz Conjecture proved. "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> - 2026-01-28 13:02 -0800
              Re: Collatz Conjecture proved. James Kuyper <jameskuyper@alumni.caltech.edu> - 2026-01-26 21:18 -0500
                Re: Collatz Conjecture proved. wij <wyniijj5@gmail.com> - 2026-01-28 04:01 +0800
          Re: Collatz Conjecture proved. wij <wyniijj5@gmail.com> - 2026-01-30 08:29 +0800
      Re: Collatz Conjecture proved. Tim Rentsch <tr.17687@z991.linuxsc.com> - 2026-01-27 19:46 -0800
        Re: Collatz Conjecture proved. wij <wyniijj5@gmail.com> - 2026-01-28 12:34 +0800
          Re: Collatz Conjecture proved. Tim Rentsch <tr.17687@z991.linuxsc.com> - 2026-02-03 04:16 -0800
        Re: Collatz Conjecture proved. "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> - 2026-01-28 13:04 -0800
  Re: Collatz Conjecture proved. Mike Terry <news.dead.person.stones@darjeeling.plus.com> - 2026-01-29 16:50 +0000
    Re: Collatz Conjecture proved. wij <wyniijj5@gmail.com> - 2026-01-30 05:40 +0800
      Re: Collatz Conjecture proved. Mike Terry <news.dead.person.stones@darjeeling.plus.com> - 2026-01-30 02:20 +0000
        Re: Collatz Conjecture proved. wij <wyniijj5@gmail.com> - 2026-01-30 11:03 +0800
          Re: Collatz Conjecture proved. Mike Terry <news.dead.person.stones@darjeeling.plus.com> - 2026-01-30 04:22 +0000
          Re: Collatz Conjecture proved. Keith Thompson <Keith.S.Thompson+u@gmail.com> - 2026-01-29 20:38 -0800
            Re: Collatz Conjecture proved. wij <wyniijj5@gmail.com> - 2026-01-31 05:30 +0800
              Re: Collatz Conjecture proved. wij <wyniijj5@gmail.com> - 2026-02-06 14:16 +0800
        Re: Collatz Conjecture proved. wij <wyniijj5@gmail.com> - 2026-01-30 11:52 +0800

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