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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?=
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Date: Fri, 29 Mar 2024 09:13:13 -0400
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On 3/28/24 11:50 PM, olcott wrote:
> On 3/28/2024 10:36 PM, Keith Thompson wrote:
>> olcott <polcott2@gmail.com> writes:
>> [...]
>>> It seems dead obvious that 0.999... is infinitesimally less than 1.0.
>>
>> Yes, it *seems* dead obvious.  That doesn't make it true, and in fact it
>> isn't.
>>
> 
> 0.999... means that is never reaches 1.0.
> and math simply stipulates that it does even though it does not.


0.999... isn't a "number" in the Real Number system, just an alternate 
representation for the number 1.

> 
>> 0.999... denotes a *limit*.  In particular, it's the limit of the value
>> as the number of 9s increases without bound.  That's what the notation
> 
> That is how it has been misinterpreted yet it has always meant
> infinitesimally less than 1.0.

But "infintesimally" doesn't exist in the Real Number System, it deals 
onloy with FINITE numbers.

> 
>> "0.999..." *means*.  (There are more precise notations for the same
>> thing, such as "0.9̅" (that's a 9 with an overbar, or "vinculum") or
>> "0.(9)".
>>
> 
> I already know all that.
> 
>> You have a sequence of numbers:
>>
>>      0.9
>>      0.99
>>      0.999
>>      0.9999
>>      0.99999
>>      ...
>>
>> Each member of that sequence is strictly less than 1.0, but the *limit*
>> is exactly 1.0.  The limit of a sequence doesn't have to be a member of
>> the sequence.  The limit is, informally, the value that members of the
>> sequence approach arbitrarily closely.
>>
> 
> Yet never reaching.
> 
>> <https://en.wikipedia.org/wiki/Limit_of_a_sequence>
>>
>>> That we can say this in English yet not say this in conventional
>>> number systems proves the need for another number system that can
>>> say this.
>>
>> Then I have good news for you.  There are several such systems, for
>> example <https://en.wikipedia.org/wiki/Hyperreal_number>.
>>
> 
> Infinitesimally less than 1.0 means one single geometric point
> on the number line less than 1.0.

Nope.

> 
>> If your point is that you personally like hyperreals better than you
>> like reals, that's fine, as long as you're clear which number system
>> you're using. 
> 
> The Infinitesimal number system that I created.

So you are lying about talking about the Reals.

> 
>> If you talk about things like "0.999..." without
>> qualification, everyone will assume you're talking about real numbers.
>>
> 
> It is already the case that 0.999...
> specifies Infinitesimally less than 1.0.
> 
>> And if you're going to play with hyperreal numbers, or surreal numbers,
>> or any of a number of other extensions to the real numbers, I suggest
>> that understanding the real numbers is a necessary prerequisite.  That
>> includes understanding that no real number is either infinitesimal or
>> infinite.
>>
>> Disclaimer: I'm not a mathematician.  I welcome corrections.
>>
>