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

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.)