Path: ...!weretis.net!feeder6.news.weretis.net!feeder8.news.weretis.net!eternal-september.org!feeder3.eternal-september.org!news.eternal-september.org!.POSTED!not-for-mail
From: olcott <polcott2@gmail.com>
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 22:50:49 -0500
Organization: A noiseless patient Spider
Lines: 81
Message-ID: <uu5dqp$2tti$2@dont-email.me>
References: <bebe16f4f02eed7ac4e4d815dc0e1e98f9f0f2a0.camel@gmail.com>
 <uu3qk7$3jc94$1@dont-email.me> <uu444a$3lnuc$1@dont-email.me>
 <uu44k2$3lrph$1@dont-email.me> <uu50n4$3ca7i$6@i2pn2.org>
 <uu573n$3tt5t$7@dont-email.me> <uu58nh$3ca7j$2@i2pn2.org>
 <uu59t9$3ubje$2@dont-email.me> <8734s9u2tl.fsf@nosuchdomain.example.com>
MIME-Version: 1.0
Content-Type: text/plain; charset=UTF-8; format=flowed
Content-Transfer-Encoding: 8bit
Injection-Date: Fri, 29 Mar 2024 03:50:50 +0100 (CET)
Injection-Info: dont-email.me; posting-host="bfd65a280c18a2165003beacad9b3410";
	logging-data="96178"; mail-complaints-to="abuse@eternal-september.org";	posting-account="U2FsdGVkX18DWMYAMHM4j5n0NNkcd9P0"
User-Agent: Mozilla Thunderbird
Cancel-Lock: sha1:oWadltWecxWWGCoShzEgwQDXi1o=
Content-Language: en-US
In-Reply-To: <8734s9u2tl.fsf@nosuchdomain.example.com>
Bytes: 4010

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... 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.

> "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.

> 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.

> 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.
> 

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