On 2024-04-04 13:57, olcott wrote:
> On 4/4/2024 1:08 PM, André G. Isaak wrote:
>> On 2024-04-04 08:55, olcott wrote:
>>
>>
>>> 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*
>>
>> And that's supposed to mean what exactly? That you take an unending
>> sequence of zeros and once that unending sequence ends you tack on a 1?

Your complete lack of response is noted. What does ... mean in the
above? I know what it means in standard decimal notation, but can't make
heads or tails of it in your notation.

>>> 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
>>
>>
>> So what's 0.999... + 0.000...05 ? Is that half an infinitesimal shy of 1?
>>
>
> That is 5 infinitesimals.

So then what's 0.999 + 000...5 ?

> Do you have a better way to encode them?

It's your proposal, not mine. I'm working with the reals which do not
admit infinitesimals, so I have no reason to encode them.

You're the one who wants infinitesimals. But if you want to propose a
system which uses them, you need to actually define it. Just inventing a
dubious and unexplained notation is not enough.

How are numbers constructed in this system?

How do infinitesimals work with respect to addition, subtraction,
multiplication, and division? What, for example is 0.000...1 × 0.000...1 ?

If you can't explain the above, you don't have a system.

André

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On 4/4/24 3:57 PM, olcott wrote:
> On 4/4/2024 1:08 PM, André G. Isaak wrote:
>> On 2024-04-04 08:55, olcott wrote:
>>
>>
>>> 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*
>>
>> And that's supposed to mean what exactly? That you take an unending
>> sequence of zeros and once that unending sequence ends you tack on a 1?
>>
>>> 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
>>
>>
>> So what's 0.999... + 0.000...05 ? Is that half an infinitesimal shy of 1?
>>
>
> That is 5 infinitesimals.

That would be 0.000...5

putting the 0 in front makes it smaller since we are below the decimal
point.

After all 0.5 is bigger then 0.05 by a factor of 10, so ...5 should be a
factor of 10 larger than ...05

> Do you have a better way to encode them?

A normal notation is as multiple of a symbol for the infinitesimal, like
iota.

>
>> André
>>
>

On 04/04/2024 11:08 AM, André G. Isaak wrote:
> On 2024-04-04 08:55, olcott wrote:
>
>
>> 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*
>
> And that's supposed to mean what exactly? That you take an unending
> sequence of zeros and once that unending sequence ends you tack on a 1?
>
>> 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
>
>
> So what's 0.999... + 0.000...05 ? Is that half an infinitesimal shy of 1?
>
> André
>

What it seems is that iota-values don't have one or the other of
products or sums, in a deconstructive approach to arithmetic,
with defining arithmetic as addition and division completing
toward each other instead of higher-order increment as reciprocal.

So, arithmetic thusly doesn't make much sense, necessarily,
without instead "iota-sums" and "iota-multiples" as that
the multiples are defined as inverse division, instead,
because of the way they're defined as division of numerator
by denominator for all values of the numerator through
the denominator, as the denominator goes to infinity,
so that it's well defined as an infinite limit.

This way iota-values arrive at their own definition of
a continuous domain, extent density completeness measure,
and just happen to share 0 and 1 with the integers as [0,1].

Otherwise the sort of Big-End/Little-End stuff of expansions,
with an infinity in the middle, gets very well into these
sorts of deconstructive accounts of arithmetic and algebra,
sort of two groups meeting in the middle as one field,
algebra-wise, that as they fall apart either way,
ranges through in continuous differences, as that
there's a total ordering at either end.

The nilpotent and nilsquare are about the most usual
ways to describe whether infinitesimal is zero or non.
This is a usual reflection on the differential and
the integration as the quadrature. It's not the only way.

The iota-values demonstrate extent density completeness measure
all by themselves with the integers. So, have some respect
for them as you would other theories in their vacuums.

Or, you know, don't.

It happens to be similar with the rationals being
"as dense as" the irrationals, for themselves, on
their own, in the theory where they're logical.

One might aver "most people don't think so", yet,
maybe they just never thought so.

On 04/04/2024 05:09 AM, Richard Damon wrote:
> On 4/3/24 10:35 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:
>>>>
>>>>> 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.)
>>>>>
>>>>> 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.
>>>>
>>>
>>>
>>> About the di-aletheic, ....
>>>
>>> https://www.youtube.com/watch?v=vbyFehrthIQ&list=PLb7rLSBiE7F4eHy5vT61UYFR7_BIhwcOY&index=23&t=1305
>>>
>>>
>>> About statements and fact and retraction, ....
>>>
>>> https://www.youtube.com/watch?v=tODnCZvVtLg&list=PLb7rLSBiE7F4eHy5vT61UYFR7_BIhwcOY&index=15
>>>
>>>
>>>
>>> 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.
>>
>
> A Point "Immediately adjacent" doesn't exist.
>
> The problem is that points, like Real Numbers, are "dense" and between
> ANY two of them, are an infinite number of other points.

It's mostly a usual idea that a function from discrete domain
the infinitely many integers to a continuous domain [0,1] is
a usual sort of "scalar" consideration, in terms of the
infinite-divisibility of [0,1] and the infinitely-many integers.

It sort of happens they're all divided at once, see,
equi-partitioned, not just with a farther and farther
inverse to lever it on in.

It's not the same "set", as with regards to the "set" of
elements of the complete ordered field modeling real values,
it's a different "set", iota-values, and only has very
particular functions, those also being models that happen
to be sets in a set theory and the usual set theory.

Having thusly multiple and replete definitions of
complete, being continuous, is just a fact.

And: draw a line from zero to one, pencil and paper,
tools of geometry, however you so care: they're in order.