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

On 4/1/2024 6:11 PM, Keith Thompson wrote:

olcott <polcott333@gmail.com> writes:
[...]

Since PI is represented by a single geometric point on the number line
then 0.999... would be correctly represented by the geometric point
immediately to the left of 1.0 on the number line or the RHS of this
interval [0,0, 1.0). If there is no Real number at that point then
there is no Real number that exactly represents 0.999...

[...]
 In the following I'm talking about real numbers, and only real
numbers -- not hyperreals, or surreals, or any other extension to the
real numbers.
 You assert that there is a geometric point immediately to the left of
1.0 on the number line.  (I disagree, but let's go with it for now.)
 Am I correct in assuming that this means that that point corresponds to
a real number that is distinct from, and less than, 1.0?
 

IDK, probably not. I am saying that 0.999... exactly equals this number.

More generally, does each real number correspond to a point on the
number line, and does each point on the number line correspond to a real
number?  (The real numbers can be formally defined without reference to
geometry, but let's go with your geometric model for now.)
 

The line segment [0.0, 1.0] is exactly one geometric point longer than
[0.0, 1.10), having all points in common besides the last point.

If so, let's call that real number (immediately to the left of 1.0) x.
 Consider ((x + 1.0)/2.0).  Let's call that number y.  (The intent is to
construct a real number that is exactly halfway between x and 1.0.)
 Is y a real number?  (If not, the real numbers are, unexpectedly, not
closed under common arithmetic operations.)
 Is y less than, equal to, or greater than x?
 Is y less than, equal to, or greater than 1.0?
 Again, I am talking *only* about real numbers.
 Given your past history, I do not expect straight answers to these
questions, but I'm prepared to be pleasantly surprised.
 

--
Copyright 2024 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer