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

On 4/4/24 10:55 AM, olcott wrote:

On 4/3/2024 11:47 PM, Ross Finlayson wrote:

On 04/03/2024 09:18 PM, olcott wrote:

On 4/3/2024 11:05 PM, Ross Finlayson wrote:

On 04/03/2024 07:35 PM, olcott wrote:

On 4/3/2024 9:11 PM, Ross Finlayson wrote:

On 04/03/2024 03:12 PM, Ben Bacarisse wrote:

Keith Thompson <Keith.S.Thompson+u@gmail.com> writes:
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olcott <polcott333@gmail.com> writes:

On 4/3/2024 12:23 PM, Keith Thompson wrote:

"Fred. Zwarts" <F.Zwarts@HetNet.nl> writes:
[...]

Olcott is unable to understand  what it says in the context of the
real number system, even when spelled out to him in great
detail. Therefore he sticks to his own (wrong) interpretation and
then
starts to fight it. Fighting windmills.

Might I suggest waiting to reply to olcott until he says something
*new*.  It could save a lot of time and effort.

>
0.999... everyone knows that this means infinitely repeating digits
that never reach 1.0 and lies about it. I am not going to start
lying
about it.

>
(I don't read everything olcott writes, but that *might* be something
new.)
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Nobody here is lying.  (I'm giving you the benefit of the doubt.)
Some people here are wrong.
>
You might take a moment to think about *why* so many people would be
motivated to lie about something like this.  Is it really plausible
that multiple people (a) know in their hearts that you're right,
but (b) deliberately pretend that you're wrong?

>
PO is in a genuine bind here.  He has almost no ability to understand
other people's mental states, let alone their reasoning.  He can't
begin
to comprehend what others think, and he struggles to understand what
they write, so he often thinks that people are lying or playing head
games.  He's accused me of this numerous times, and (the final
straw for
me) that I must be doing this deliberately and sadistically.  What
other
conclusion can he come to?
>
Every time PO paraphrases someone's reply to him he gets it wrong.  He
simply does not know what people are saying but since they disagree
with
something that is obvious to him, they must be stupid, lying or
playing
head games.
>
The classic technique in mediation where each person must reflect back
to the other what it is they believe the other is saying would,
were he
capable of it, be useful here.  But he would fail at every step.
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About the di-aletheic, ....
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https://www.youtube.com/watch?v=vbyFehrthIQ&list=PLb7rLSBiE7F4eHy5vT61UYFR7_BIhwcOY&index=23&t=1305
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About statements and fact and retraction, ....
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https://www.youtube.com/watch?v=tODnCZvVtLg&list=PLb7rLSBiE7F4eHy5vT61UYFR7_BIhwcOY&index=15
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Iota-values:  the word "iota" means "smallest non-zero value".
>
Real-values:  all the values between negative infinity and infinity.

>
So the geometric point immediately adjacent to 0.0 on the positive
side of the number line would be a real number.
>

>
That's kind of the idea where there's a sort of distinction
"real-valued" vis-a-vis just "real numbers", with the idea
that where there are more than one many models of the
linear continuum, a continuous domain, that they all live
in the same space, of real numbers, that they're real-valued.
>
That is, the linear continuum is complete already,
so any non-standard models live in the same space.
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Now, when you say real number, everybody's going to
think that it means the complete ordered field,
or at least everybody with the usual linear curriculum
and formal schooling and the formalism, so when you
say instead "real-valued", it sort of expresses that
if there _is_ a different continuous domain, then
the different models are treated differently and
they're not interchangeable except with regards to
various statements about particularly well-understood
points where they're the same in geometry, here 0 and 1,
these iota-values filling [0,1] empty to full, and
the usual real numbers as real values falling down
after the ordered field of rationals, getting axiomatized
their LUB and thus completion usually negative infinity
to infinity.
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There's another Katz been working on some revivals
of studies of infinitesimals, some years ago there
was a paper about the contradistinction of .999... = 1
and .999 < 1, and about for modular and clock arithmetic

>
I would say that 0.999... < 1.0 by an infinitesimal amount is
necessarily true because we know that infinite sequences never end.
>

that it's very natural that the notations read-out the
same under different meanings.
>
You can read Ehrlich for a sort of modern survey of
infinitesimals, yet, you might as well just look to
Cavalieri and MacLaurin, or you know, I wrote it up.
>
>
Here the point is that "real-valued" then makes for
it sort of suffices that there's a model of real numbers
with "integer part and non-integer part [0,1]" and a
model of real numbers "the ordered field of rationals
closed to least-upper-bound the complete ordered field",
then that it involves book-keeping so they don't get confused.
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Which keeps things simple while yet not blind, ....
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Just keep in mind that there's an entirely different model
of a continuous domain, zero to one empty to full, than
the usual model called R that is all the rationals plus
filling all the gaps, then there's also to learn about
how the Fourier-style analysis arrives at the signal domain,
so that there are at least three different models altogether,
in the sense of model theory's models, of continous domains,
real values and real-valued.
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I don't know about that. What if 1.000..., dot dot dot, ends -1?
>
It's like they say, don't be a quitter, the numbers do not quit,
even though we are only finite creatures, yet each has and is a
part of the continuum, where the world's numbers.
>
So, 1.000..., dot dot dot, +- ...01, it's not so different.

 Different enough to not me equal.
[0.0, 1.0] - [0.0, 1.0) = 0.0...1
0.000...2 - 0.000...1 = 0.000...1
*A good notational convention for infinitesimals*

Which means you are not working in the set of REALS as your subject says.
So, you are just admitting to being a LIAR, and also STUPID.

 0.999... + 0.000...1 = 1.0 > 0.999... + 0.000...2 = 1.000...1
0.999... + 0.000...3 = 1.000...2
0.999... + 0.000...n = 1.000...n-1
 

>
Consider Simon Stevin and the p-adics. Or, just that there
are both ends, big-end and little-end, infinite in the middle.
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This is why the infinite limit helps show, to prove, things like
iota-values being standard infinitesimals filling [0,1]
while the usual model of reals is the complete ordered field.
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Otherwise it's just matters or carry, roll, and book-keeping.
I.e., just because arithmetic is closed and carry and roll
may be free, it's not a joke. (The mathematician who's all
lazy about his trash-can fire, isn't necessarily the same as
the mathematician who always counts his beans.)
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