Re: Definition of real number ℝ --infinitesimal--

On 3/28/24 9:56 PM, olcott wrote:

On 3/28/2024 7:07 PM, Richard Damon wrote:

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
 

 
Yet that is NOT what 0.999... actually says.
It says that it gets infinitely close to 1.0 without every actually
getting there. In other words it is infinitesimally less than 1.0.

 
But so close that no number exists between it and 1.0, so they are the
same number.
 
That comes out of the ACTUAL definitions of Real Numbers
 

 

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.