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From: Richard Damon <richard@damon-family.org>
Newsgroups: comp.theory
Subject: =?UTF-8?Q?Re=3A_Definition_of_real_number_=E2=84=9D_--infinitesimal?=
 =?UTF-8?Q?--?=
Date: Thu, 28 Mar 2024 20:07:00 -0400
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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.
>>
>