On 5/22/2022 5:15 PM, Chris M. Thomasson wrote:
> On 5/22/2022 2:42 PM, sergi o wrote:
>> On 5/22/2022 4:14 PM, Chris M. Thomasson wrote:
>>> On 5/22/2022 12:54 PM, sergi o wrote:
>>>> On 5/22/2022 2:28 PM, Chris M. Thomasson wrote:
>>>>> On 5/22/2022 3:20 AM, FromTheRafters wrote:
>>>>>> Chris M. Thomasson used his keyboard to write :
>>>>>>> On 5/21/2022 7:09 PM, FromTheRafters wrote:
>>>>>>>> Dan joyce presented the following explanation :
>>>>>>>>> On Thursday, May 19, 2022 at 1:25:44 PM UTC-4, mitchr...@gmail.com wrote:
>>>>>>>>>> and you get the first integer.
>>>>>>>>>
>>>>>>>>> Think of it this way Mitch.
>>>>>>>>> When .999... becomes so close to being 1,
>>>>>>>>
>>>>>>>> That number being represented by this symbol is not 'becoming' anything though. It simply is a representation of a number (the natural number
>>>>>>>> one in this case) as embedded in the real number system.
>>>>>>>
>>>>>>> I wonder if he thinks "becoming" just might be:
>>>>>>>
>>>>>>> [0] = 0
>>>>>>> [1] = .9
>>>>>>> [2] = .99
>>>>>>> [3] = .999
>>>>>>> [4] = .9999
>>>>>>> [5] = .99999
>>>>>>> ...
>>>>>>>
>>>>>>> If one takes each individual step into account... The limit is one.
>>>>>>>
>>>>>>> However, wrt the step-by-step accounting, there is no step where 1 - [n] = 0
>>>>>>>
>>>>>>> Just wondering here.
>>>>>>
>>>>>> It is the mistake of thinking that summation is a step by step process. Just because there are indices for each member does not mean you have to
>>>>>> add them in any particular order.
>>>>>
>>>>> True. Humm... I am wondering if this is kosher wrt the step-by-step process... If the index of the process is infinity, then it is equal to one.
>>>>
>>>>
>>>> "step by step" implies you are at a step number, came from previous step, and have not gone to the next step.  so you are stuck at k. it is just one
>>>> line in the code and needs the outside loop to drive all the steps
>>>>
>>>> s=.9
>>>> for i = 1 to oo
>>>> s= s + s *.1    ; the step by step process dependent upon i
>>>> next i
>>>> Print s
>>>>
>>>> Ill wait till it prints s
>>>
>>> We can define a limit? Perhaps even an epsilon that terminates the loop when things get "close enough", so to speak... ?
>
> Wait a minute here... You s gets larger than one right?
>
> s[0] = .9
> s[1] = .9 + .9 *.1 = .99
> s[2] = .99 + .99 * .1 = 1.089
>
> Oops!
>
> s[3] = 1.089 + 1.089 * .1 = 1.1979
>
> Oh shit... Does it even have a limit?
>
> What am I missing here?
>
>

s[4] = 1.1979 + 1.1979 * .1 = 1.31769

its going up in flames!!

and that means <e wont stop it either!

so with a =.9 and r = 1.1 the nth term should be .9*(1.1^(n-1))/(.1)

but with |r| > 1 there is no finite infinite sum

>
>>
>>
>> could use Mitches infinitesimal, call it e, where e is really very very super tiny smallish, 10^(-20) or less ...
>>
>> if s*.1 is less than e    or
>> if s - 1 is less than e   or
>> if i > 200
>>
>> Input e
>> s=.9
>> for i = 1 to oo
>> s= s + s *.1    ; the step by step process dependent upon i
>> if s*.1 is less than e
>>   if s - 1 is less than e
>>    if i > 200
>>     bail out to Print s
>>    Endif
>>   Endif
>> Endif
>> next i
>> Print s
>>
>> but you need a program that deals with huge number precision, I know a guys website, he had a calculator, used to post in here too, professor from
>> south America, I guess you could make this "step by step process" by spitting out the s*.1 factor, then have the operator hit continue, and have some
>> external guy add it up. (yes using punch tape from the 60's)
>>
>> anyhow, How do they get the precision on fractals and zooming ?
>
> It depends on far you want to zoom in. Shaders use 32-bit floats and they are okay. However, if you want to get a really deep zoom... Well, we are going
> to have to use arbitrary precision....
>
> ;^)
>
>

amazing technology out there now! a lot of it free too

>>
>>>
>>>
>>>>
>>>>>
>>>>> [infinity] = 1 because the limit is one for the process as a whole?
>>>>>
>>>>> Is that just, moronic?
>>>>>
>>>>> I know that .999... is just a representation of one in base 10.
>>>>
>>>>
>>>>
>>>
>>
>

On 5/22/2022 5:33 PM, Chris M. Thomasson wrote:
> On 5/22/2022 2:42 PM, sergi o wrote:
>> On 5/22/2022 4:14 PM, Chris M. Thomasson wrote:
>>> On 5/22/2022 12:54 PM, sergi o wrote:
>>>> On 5/22/2022 2:28 PM, Chris M. Thomasson wrote:
>>>>> On 5/22/2022 3:20 AM, FromTheRafters wrote:
>>>>>> Chris M. Thomasson used his keyboard to write :
>>>>>>> On 5/21/2022 7:09 PM, FromTheRafters wrote:
>>>>>>>> Dan joyce presented the following explanation :
>>>>>>>>> On Thursday, May 19, 2022 at 1:25:44 PM UTC-4, mitchr...@gmail.com wrote:
>>>>>>>>>> and you get the first integer.
>>>>>>>>>
>>>>>>>>> Think of it this way Mitch.
>>>>>>>>> When .999... becomes so close to being 1,
>>>>>>>>
>>>>>>>> That number being represented by this symbol is not 'becoming' anything though. It simply is a representation of a number (the natural number
>>>>>>>> one in this case) as embedded in the real number system.
>>>>>>>
>>>>>>> I wonder if he thinks "becoming" just might be:
>>>>>>>
>>>>>>> [0] = 0
>>>>>>> [1] = .9
>>>>>>> [2] = .99
>>>>>>> [3] = .999
>>>>>>> [4] = .9999
>>>>>>> [5] = .99999
>>>>>>> ...
>>>>>>>
>>>>>>> If one takes each individual step into account... The limit is one.
>>>>>>>
>>>>>>> However, wrt the step-by-step accounting, there is no step where 1 - [n] = 0
>>>>>>>
>>>>>>> Just wondering here.
>>>>>>
>>>>>> It is the mistake of thinking that summation is a step by step process. Just because there are indices for each member does not mean you have to
>>>>>> add them in any particular order.
>>>>>
>>>>> True. Humm... I am wondering if this is kosher wrt the step-by-step process... If the index of the process is infinity, then it is equal to one.
>>>>
>>>>
>>>> "step by step" implies you are at a step number, came from previous step, and have not gone to the next step.  so you are stuck at k. it is just one
>>>> line in the code and needs the outside loop to drive all the steps
>>>>
>>>> s=.9
>>>> for i = 1 to oo
>>>> s= s + s *.1    ; the step by step process dependent upon i
>>>> next i
>>>> Print s
>>>>
>>>> Ill wait till it prints s
> [...]
>
> Did you mean something like:
>
> [0] = 0
> [1] = 0 + .9 * .1^0 = .9
> [2] = .9 + .9 * .1^1 = .99
> [3] = .99 + .9 * .1^2 = .999
> [4] = .999 + .9 * .1^3 = .9999
> [5] = .9999 + .9 * .1^4 = .99999
>
> on and on, getting closer to one...
>
> [0] = 0
> [n + 1] = [n] + .9 * .1^n
>
> This converges on 1 at infinity... right?

Correctomundo, I found my mistake, first one this year, was going by 1.1^n

On Sunday, May 22, 2022 at 2:42:38 PM UTC-7, sergi o wrote:
> On 5/22/2022 4:14 PM, Chris M. Thomasson wrote:
> > On 5/22/2022 12:54 PM, sergi o wrote:
> >> On 5/22/2022 2:28 PM, Chris M. Thomasson wrote:
> >>> On 5/22/2022 3:20 AM, FromTheRafters wrote:
> >>>> Chris M. Thomasson used his keyboard to write :
> >>>>> On 5/21/2022 7:09 PM, FromTheRafters wrote:
> >>>>>> Dan joyce presented the following explanation :
> >>>>>>> On Thursday, May 19, 2022 at 1:25:44 PM UTC-4, mitchr...@gmail.com wrote:
> >>>>>>>> and you get the first integer.
> >>>>>>>
> >>>>>>> Think of it this way Mitch.
> >>>>>>> When .999... becomes so close to being 1,
> >>>>>>
> >>>>>> That number being represented by this symbol is not 'becoming' anything though. It simply is a representation of a number (the natural number one
> >>>>>> in this case) as embedded in the real number system.
> >>>>>
> >>>>> I wonder if he thinks "becoming" just might be:
> >>>>>
> >>>>> [0] = 0
> >>>>> [1] = .9
> >>>>> [2] = .99
> >>>>> [3] = .999
> >>>>> [4] = .9999
> >>>>> [5] = .99999
> >>>>> ...
> >>>>>
> >>>>> If one takes each individual step into account... The limit is one.
> >>>>>
> >>>>> However, wrt the step-by-step accounting, there is no step where 1 - [n] = 0
> >>>>>
> >>>>> Just wondering here.
> >>>>
> >>>> It is the mistake of thinking that summation is a step by step process. Just because there are indices for each member does not mean you have to add
> >>>> them in any particular order.
> >>>
> >>> True. Humm... I am wondering if this is kosher wrt the step-by-step process... If the index of the process is infinity, then it is equal to one.
> >>
> >>
> >> "step by step" implies you are at a step number, came from previous step, and have not gone to the next step. so you are stuck at k. it is just one
> >> line in the code and needs the outside loop to drive all the steps
> >>
> >> s=.9
> >> for i = 1 to oo
> >> s= s + s *.1 ; the step by step process dependent upon i
> >> next i
> >> Print s
> >>
> >> Ill wait till it prints s
> >
> > We can define a limit? Perhaps even an epsilon that terminates the loop when things get "close enough", so to speak... ?
> could use Mitches infinitesimal, call it e, where e is really very very super tiny smallish, 10^(-20) or less ...
>
> if s*.1 is less than e or
> if s - 1 is less than e or
> if i > 200
>
> Input e
> s=.9
> for i = 1 to oo
> s= s + s *.1 ; the step by step process dependent upon i
> if s*.1 is less than e
> if s - 1 is less than e
> if i > 200
> bail out to Print s
> Endif
> Endif
> Endif
> next i
> Print s
>
> but you need a program that deals with huge number precision, I know a guys website, he had a calculator, used to post in here too, professor from south
> America, I guess you could make this "step by step process" by spitting out the s*.1 factor, then have the operator hit continue, and have some external
> guy add it up. (yes using punch tape from the 60's)
>
> anyhow, How do they get the precision on fractals and zooming ?
> >
> >
> >>
> >>>
> >>> [infinity] = 1 because the limit is one for the process as a whole?
> >>>
> >>> Is that just, moronic?
> >>>
> >>> I know that .999... is just a representation of one in base 10.
> >>
> >>
> >>
> >

"pyramidal resolution"

On Sunday, May 22, 2022 at 2:35:11 PM UTC-7, Ben wrote:
> "Chris M. Thomasson" <chris.m.t...@gmail.com> writes:
>
> > On 5/22/2022 12:54 PM, sergi o wrote:
> >> On 5/22/2022 2:28 PM, Chris M. Thomasson wrote:
> >>> On 5/22/2022 3:20 AM, FromTheRafters wrote:
> >>>> Chris M. Thomasson used his keyboard to write :
> >>>>> On 5/21/2022 7:09 PM, FromTheRafters wrote:
> >>>>>> Dan joyce presented the following explanation :
> >>>>>>> On Thursday, May 19, 2022 at 1:25:44 PM UTC-4, mitchr...@gmail.com wrote:
> >>>>>>>> and you get the first integer.
> >>>>>>>
> >>>>>>> Think of it this way Mitch.
> >>>>>>> When .999... becomes so close to being 1,
> >>>>>>
> >>>>>> That number being represented by this symbol is not 'becoming' anything though. It simply is a representation of a number (the
> >>>>>> natural number one in this case) as embedded in the real number system.
> >>>>>
> >>>>> I wonder if he thinks "becoming" just might be:
> >>>>>
> >>>>> [0] = 0
> >>>>> [1] = .9
> >>>>> [2] = .99
> >>>>> [3] = .999
> >>>>> [4] = .9999
> >>>>> [5] = .99999
> >>>>> ...
> >>>>>
> >>>>> If one takes each individual step into account... The limit is one.
> >>>>>
> >>>>> However, wrt the step-by-step accounting, there is no step where 1 - [n] = 0
> >>>>>
> >>>>> Just wondering here.
> >>>>
> >>>> It is the mistake of thinking that summation is a step by step process. Just because there are indices for each member does not mean
> >>>> you have to add them in any particular order.
> >>>
> >>> True. Humm... I am wondering if this is kosher wrt the step-by-step process... If the index of the process is infinity, then it is equal
> >>> to one.
> >>
> >> "step by step" implies you are at a step number, came from previous step, and have not gone to the next step. so you are stuck at k. it is
> >> just one line in the code and needs the outside loop to drive all the steps
> >> s=.9
> >> for i = 1 to oo
> >> s= s + s *.1 ; the step by step process dependent upon i
> >> next i
> >> Print s
> >> Ill wait till it prints s
> >
> > We can define a limit?
> Of course we can. Look at any text on real analysis.
>
> The reason that 0.999... = 1 is that the sequence of partial sums is not
> bounded by any r less than one. Every real >=1 is an upper bound of the
> sequence of partial sums, but 1 is the least upper bound.
>
> The existence of least upper bounds (for the right kinds of convergent
> sequences) is no accident. It's how the reals are defined and why the
> reals are useful for doing analysis (essentially calculus).
>
> --
> Ben.

There's that 1.0 is special.

Derivatives for infinitesimals like dx have dy/dx
equal ratios in x and y, i.e., for 1.0 in y and 1.0 in x.

And, when you multiply something by greater than
1.0: it's greater, while, less than 1.0: less.

The idea for dual representation is of course for
after .333... and that "there is only one representation
of one-third as a decimal, an infinite sequence of 3's",
where representations are simply limited to the space [0,1].

The amazing thing about Cantor space, is, that,
half of the sequences have equal zero and one densities.
This is what I call "Borel vs. Combinatorics",
that instead that "they most all should be normal"
or "they most all should be not normal", the sequences
between what Borel and Combinatorics "say".
(Not what I say, but according to Borel and Combinatorics resp.)

Having half this way is a usual bisector routine -
1.0, half, usual terms of things.

Of course 0 = 0.0000..., while though with significance
that .00 * .00 = .00, .00 + .0000 = .00, .00 * .0 = .0, ....

Then, Borel vs. Combinatorics into "this Cantor space
which is effectively all the infinite sequences" gets to
where the alternate of sampling holds: it's interesting
the "Factorial/Exponential Identity", that builds up a model
of statistical expectations, is for where "half", of the
sequences "somewhere between zero and one",
is the usual middle.

Of course, 0 = .0000.... So, it's easy to start writing numbers,
..0 (000) ...
but the next number would start writing:
..00000000000.., ..., "1".
which is much like the consideration that
the "next" number down from 1 is .999..., "0".
(A notation in effect since the above five lines.)

Reading off Cantor space as a, "square",
is kind of where, "it's a 1.0 x 1.0 square."

Of course, Cantor space is much more a column:
and greater than as of 2 to the omega. The "space"
that reads off as it as a square, the space members
as a sequence each or the rows as ordinals:
is for 1.0 and specifically 2^w or b^p.

Here the point about Cantor space is that if you read
down a row it is a sequence, that also when you read
down a column: it is also a sequence. (In a space full
of all of them, each.)

So, this kind of thing is simple among real functions.

On Sunday, May 22, 2022 at 9:53:03 PM UTC-7, zelos...@gmail.com wrote:
> torsdag 19 maj 2022 kl. 19:25:44 UTC+2 skrev mitchr...@gmail.com:
> > and you get the first integer.
> There are no infinitesimals in real numbers.
>
> And 1=9/9=0.999...
>
> wrong as always

Are there infinite numbers in infinite numbers?

If there are infinite numbers, they are infinite numbers
in infinite numbers.

Instead it's "for any large number, finite, there's
a larger one (also finite)" besides "for any large number,
finite, there's a large infinite, larger", from that
infinite numbers exist.

This simply keeps what is quantitative there,
with respect to qualitative.

On 5/22/2022 11:23 PM, Ross A. Finlayson wrote:
> On Sunday, May 22, 2022 at 2:35:11 PM UTC-7, Ben wrote:
>> "Chris M. Thomasson" <chris.m.t...@gmail.com> writes:
>>
>>> On 5/22/2022 12:54 PM, sergi o wrote:
>>>> On 5/22/2022 2:28 PM, Chris M. Thomasson wrote:
>>>>> On 5/22/2022 3:20 AM, FromTheRafters wrote:
>>>>>> Chris M. Thomasson used his keyboard to write :
>>>>>>> On 5/21/2022 7:09 PM, FromTheRafters wrote:
>>>>>>>> Dan joyce presented the following explanation :
>>>>>>>>> On Thursday, May 19, 2022 at 1:25:44 PM UTC-4, mitchr...@gmail.com wrote:
>>>>>>>>>> and you get the first integer.
>>>>>>>>>
>>>>>>>>> Think of it this way Mitch.
>>>>>>>>> When .999... becomes so close to being 1,
>>>>>>>>
>>>>>>>> That number being represented by this symbol is not 'becoming' anything though. It simply is a representation of a number (the
>>>>>>>> natural number one in this case) as embedded in the real number system.
>>>>>>>
>>>>>>> I wonder if he thinks "becoming" just might be:
>>>>>>>
>>>>>>> [0] = 0
>>>>>>> [1] = .9
>>>>>>> [2] = .99
>>>>>>> [3] = .999
>>>>>>> [4] = .9999
>>>>>>> [5] = .99999
>>>>>>> ...
>>>>>>>
>>>>>>> If one takes each individual step into account... The limit is one.
>>>>>>>
>>>>>>> However, wrt the step-by-step accounting, there is no step where 1 - [n] = 0
>>>>>>>
>>>>>>> Just wondering here.
>>>>>>
>>>>>> It is the mistake of thinking that summation is a step by step process. Just because there are indices for each member does not mean
>>>>>> you have to add them in any particular order.
>>>>>
>>>>> True. Humm... I am wondering if this is kosher wrt the step-by-step process... If the index of the process is infinity, then it is equal
>>>>> to one.
>>>>
>>>> "step by step" implies you are at a step number, came from previous step, and have not gone to the next step. so you are stuck at k. it is
>>>> just one line in the code and needs the outside loop to drive all the steps
>>>> s=.9
>>>> for i = 1 to oo
>>>> s= s + s *.1 ; the step by step process dependent upon i
>>>> next i
>>>> Print s
>>>> Ill wait till it prints s
>>>
>>> We can define a limit?
>> Of course we can. Look at any text on real analysis.
>>
>> The reason that 0.999... = 1 is that the sequence of partial sums is not
>> bounded by any r less than one. Every real >=1 is an upper bound of the
>> sequence of partial sums, but 1 is the least upper bound.
>>
>> The existence of least upper bounds (for the right kinds of convergent
>> sequences) is no accident. It's how the reals are defined and why the
>> reals are useful for doing analysis (essentially calculus).
>>
>> --
>> Ben.
>
> There's that 1.0 is special.
>
> Derivatives for infinitesimals like dx have dy/dx
> equal ratios in x and y, i.e., for 1.0 in y and 1.0 in x.
>
> And, when you multiply something by greater than
> 1.0: it's greater, while, less than 1.0: less.
>
> The idea for dual representation is of course for
> after .333... and that "there is only one representation
> of one-third as a decimal, an infinite sequence of 3's",
> where representations are simply limited to the space [0,1].
>
> The amazing thing about Cantor space, is, that,
> half of the sequences have equal zero and one densities.
> This is what I call "Borel vs. Combinatorics",
> that instead that "they most all should be normal"
> or "they most all should be not normal", the sequences
> between what Borel and Combinatorics "say".
> (Not what I say, but according to Borel and Combinatorics resp.)
>
> Having half this way is a usual bisector routine -
> 1.0, half, usual terms of things.
>
> Of course 0 = 0.0000..., while though with significance
> that .00 * .00 = .00, .00 + .0000 = .00, .00 * .0 = .0, ....
>
> Then, Borel vs. Combinatorics into "this Cantor space
> which is effectively all the infinite sequences" gets to
> where the alternate of sampling holds: it's interesting
> the "Factorial/Exponential Identity", that builds up a model
> of statistical expectations, is for where "half", of the
> sequences "somewhere between zero and one",
> is the usual middle.
>
> Of course, 0 = .0000.... So, it's easy to start writing numbers,
> .0 (000) ...
> but the next number would start writing:
> .00000000000.., ..., "1".
> which is much like the consideration that
> the "next" number down from 1 is .999..., "0".
> (A notation in effect since the above five lines.)
>
> Reading off Cantor space as a, "square",
> is kind of where, "it's a 1.0 x 1.0 square."
>
> Of course, Cantor space is much more a column:
> and greater than as of 2 to the omega. The "space"
> that reads off as it as a square, the space members
> as a sequence each or the rows as ordinals:
> is for 1.0 and specifically 2^w or b^p.
>
> Here the point about Cantor space is that if you read
> down a row it is a sequence, that also when you read
> down a column: it is also a sequence. (In a space full
> of all of them, each.)
>
> So, this kind of thing is simple among real functions.
>
>
>

Borel combinatorics;

A fundamental theorem of Kechris, Solecki, and Todorcevic [10, Proposition 4.6]
states that every Borel graph G of degree at most d has Borel chromatic number χB (G) ≤
d + 1, where the Borel chromatic number χB (G) of G is the least cardinality of a Polish
space Y so that there is a Borel Y -coloring of G.

https://www.math.cmu.edu/~clintonc/onlinepapers/hyp_games.pdf

Just because in theory an infinite number of steps is required doesn't
mean the limit cannot be reached. Consider Zeno's Paradox where Achilles
races a tortoise with a head start. Each time Achilles reaches a point
where the tortoise was, the tortoise advances somewhat. When Achilles
reaches that point, the tortoise advances more. And so on for an
infinite number of steps. Yet Achilles catches up to the tortoise and
passes it and wins the race, despite taking an infinite number of steps
to catch up to the tortoise.

On 5/22/2022 3:33 PM, Chris M. Thomasson wrote:
> On 5/22/2022 2:42 PM, sergi o wrote:
>> On 5/22/2022 4:14 PM, Chris M. Thomasson wrote:
>>> On 5/22/2022 12:54 PM, sergi o wrote:
>>>> On 5/22/2022 2:28 PM, Chris M. Thomasson wrote:
>>>>> On 5/22/2022 3:20 AM, FromTheRafters wrote:
>>>>>> Chris M. Thomasson used his keyboard to write :
>>>>>>> On 5/21/2022 7:09 PM, FromTheRafters wrote:
>>>>>>>> Dan joyce presented the following explanation :
>>>>>>>>> On Thursday, May 19, 2022 at 1:25:44 PM UTC-4,
>>>>>>>>> mitchr...@gmail.com wrote:
>>>>>>>>>> and you get the first integer.
>>>>>>>>>
>>>>>>>>> Think of it this way Mitch.
>>>>>>>>> When .999... becomes so close to being 1,
>>>>>>>>
>>>>>>>> That number being represented by this symbol is not 'becoming'
>>>>>>>> anything though. It simply is a representation of a number (the
>>>>>>>> natural number one in this case) as embedded in the real number
>>>>>>>> system.
>>>>>>>
>>>>>>> I wonder if he thinks "becoming" just might be:
>>>>>>>
>>>>>>> [0] = 0
>>>>>>> [1] = .9
>>>>>>> [2] = .99
>>>>>>> [3] = .999
>>>>>>> [4] = .9999
>>>>>>> [5] = .99999
>>>>>>> ...
>>>>>>>
>>>>>>> If one takes each individual step into account... The limit is one.
>>>>>>>
>>>>>>> However, wrt the step-by-step accounting, there is no step where
>>>>>>> 1 - [n] = 0
>>>>>>>
>>>>>>> Just wondering here.
>>>>>>
>>>>>> It is the mistake of thinking that summation is a step by step
>>>>>> process. Just because there are indices for each member does not
>>>>>> mean you have to add them in any particular order.
>>>>>
>>>>> True. Humm... I am wondering if this is kosher wrt the step-by-step
>>>>> process... If the index of the process is infinity, then it is
>>>>> equal to one.
>>>>
>>>>
>>>> "step by step" implies you are at a step number, came from previous
>>>> step, and have not gone to the next step.  so you are stuck at k. it
>>>> is just one line in the code and needs the outside loop to drive all
>>>> the steps
>>>>
>>>> s=.9
>>>> for i = 1 to oo
>>>> s= s + s *.1    ; the step by step process dependent upon i
>>>> next i
>>>> Print s
>>>>
>>>> Ill wait till it prints s
> [...]
>
> Did you mean something like:
>
> [0] = 0
> [1] = 0 + .9 * .1^0 = .9
> [2] = .9 + .9 * .1^1 = .99
> [3] = .99 + .9 * .1^2 = .999
> [4] = .999 + .9 * .1^3 = .9999
> [5] = .9999 + .9 * .1^4 = .99999
>
> on and on, getting closer to one...
>
> [0] = 0
> [n + 1] = [n] + .9 * .1^n
>
> This converges on 1 at infinity... right?

This is another way to get the same result:

[0] = 0
[1] = 0 + 9 / 10^1 = .9
[2] = .9 + 9 / 10^2 = .99
[3] = .99 + 9 / 10^3 = .999
[4] = .999 + 9 / 10^4 = .9999
[...]

[0] = 0
[n + 1] = [n] + 9 / 10^(n + 1)

[...]

[0] = 0
[0 + 1] = 0 + 9 / 10^(0 + 1) = .9
[1 + 1] = .9 + 9 / 10^(1 + 1) = .99
[2 + 1] = .99 + 9 / 10^(2 + 1) = .999
[3 + 1] = .999 + 9 / 10^(3 + 1) = .9999
[4 + 1] = .9999 + 9 / 10^(4 + 1) = .99999

[...]

The limit is one, so the index:

[infinity] = 1

Sounds alright to me. Humm...

On 5/19/2022 10:25 AM, mitchr...@gmail.com wrote:
> and you get the first integer.

Is your infinitesimal something like:

..1^(infinity) ?

..1^5 = .00001
..1^6 = .000001
....

It gets pretty damn small! Run for infinity the limit is zero, right? So
if zero is the limit of the infinitesimal .000...1, and one is the limit
of .999..., then that just means:

1 + 0 = 1

Right?

On 5/23/2022 1:25 PM, Michael Moroney wrote:
> Just because in theory an infinite number of steps is required doesn't
> mean the limit cannot be reached. Consider Zeno's Paradox where Achilles
> races a tortoise with a head start. Each time Achilles reaches a point
> where the tortoise was, the tortoise advances somewhat. When Achilles
> reaches that point, the tortoise advances more.  And so on for an
> infinite number of steps.  Yet Achilles catches up to the tortoise and
> passes it and wins the race, despite taking an infinite number of steps
> to catch up to the tortoise.

If Achilles strictly plays by the tortoises rules on a step-by-step basis:

step 1: tortoise moves one meter; Achilles moves one meter. The tortoise
is ahead because of the head start.

step 2: tortoise moves one meter; Achilles moves one meter. The tortoise
is still ahead because of the head start.

on and on. The turtle will cross the finish line before Achilles.

Now, if Achilles tells the tortoise to f-off and just starts running, he
will quickly pass the tortoise...

;^)

On 5/19/2022 10:25 AM, mitchr...@gmail.com wrote:
> and you get the first integer.

Get brain soon.

On Monday, May 23, 2022 at 10:19:05 PM UTC-7, zelos...@gmail.com wrote:
> måndag 23 maj 2022 kl. 18:21:35 UTC+2 skrev Ross A. Finlayson:
> > On Sunday, May 22, 2022 at 9:53:03 PM UTC-7, zelos...@gmail.com wrote:
> > > torsdag 19 maj 2022 kl. 19:25:44 UTC+2 skrev mitchr...@gmail.com:
> > > > and you get the first integer.
> > > There are no infinitesimals in real numbers.
> > >
> > > And 1=9/9=0.999...
> > >
> > > wrong as always
> > Are there infinite numbers in infinite numbers?
> >
> > If there are infinite numbers, they are infinite numbers
> > in infinite numbers.
> There are no "infinite numbers" in real numbers, real numbers are archimedian.
> >
> > Instead it's "for any large number, finite, there's
> > a larger one (also finite)" besides "for any large number,
> > finite, there's a large infinite, larger", from that
> > infinite numbers exist.
> >
> > This simply keeps what is quantitative there,
> > with respect to qualitative.

Also it is like an existence result itself,
that there are infinitely many
there are infintely grand.

"Having the Archimedean property" is
often read two ways,
for the unbounded (not finitely many)
and the unbounded (not infinitely grand).

It's kind of like Goedelian completeness:
reminding people of both the completeness
theorems, and the incompleteness theorems.

These days non-Archimedean fields are
a usual introduction to "non" standard ("extra" standard).

Then, yes, I am talking about a logical consequence
of there being infinitely many that there are infinitesimals
in the reals and that besides there are infinites in integers.

Then, what is the "standard" is just as above in matters of
"representation theory", here model theory for a function theory
for a space of values: it's standard and well-defined but not
complete, the space of representations those of the field reals,
Archimedean field reals, made replete with a space of
representations of those of line reals, or signal reals.

That field reals, line reals, and signal reals, each in the
spaces of real values like usual vector spaces, are each
models of real numbers with IVT and resultingly the FTCs,
and otherwise real character: is central and important.

(In mathematics.)

It's kind of like "Burali-Forti's largest ordinal, that would
contain itself", or "Russell's set-of-all-sets-that-don't-contain-
themselves contains itself": starting with that the only ordinals
are finite and Archimedean as you advise, that immediately any
"infinite" including omega or otherwise actual infinite:
includes itself. I.e. without "defining" omega all well-founded
and regular: it ("omega, an inductive set") would be "derived"
from the "paradoxes of Burali-Forti and Russell in an Archimedean
universe", as _not_ well-founded, regular, ordinary, ....

Then, for infinitesimals and the long line, which usually enough
abstractly includes infinitesimals, a usual enough notion of
the real line, partitions any segment into infinitely-many
equal-size pieces.

Of course calculus was called "infinitesimal analysis" for
some hundreds of years, and that's what was meant, also.

These days of course everybody knows Cauchy/Weierstrass as
the formalism after Riemann/Lebesgue the formalism, knowing
most all of a development of the complete ordered field (Archimedean),
besides usual graphical notions of the points that in their space
mark (draw) a line. (Point-sets, ..., in what are real-valued systems.)

Then, line reals, field reals, and signal reals are three _different_
models of real numbers, connected and having the gaplessness
property, least upper bound property, measure(s), ..., here of
course that line reals are modeled as "unboundedly many,
and, vanishingly small, and equal, values what sum to 1" .

Then where "the curriculum includes _only_ the field reals,
stop", it is short, because usual models of line reals and signal
reals besides field reals are everywhere and central, in all sorts
models in mathematics. So, the curriculum is short because
there are at least three _different_ models of "real numbers".

Chris M. Thomasson wrote:

>> and you get the first integer.
>
> Is your infinitesimal something like: .1^(infinity) ?
> .1^5 = .00001 .1^6 = .000001 ...
> It gets pretty damn small! Run for infinity the limit is zero, right? So
> if zero is the limit of the infinitesimal .000...1, and one is the limit
> of .999..., then that just means: 1 + 0 = 1

idiot, there is no "damn small" in mathematics. If your region of interest
is bellow that limit, that number can be huge, not small.

tisdag 24 maj 2022 kl. 09:42:59 UTC+2 skrev Ross A. Finlayson:
> On Monday, May 23, 2022 at 10:19:05 PM UTC-7, zelos...@gmail.com wrote:
> > måndag 23 maj 2022 kl. 18:21:35 UTC+2 skrev Ross A. Finlayson:
> > > On Sunday, May 22, 2022 at 9:53:03 PM UTC-7, zelos...@gmail.com wrote:
> > > > torsdag 19 maj 2022 kl. 19:25:44 UTC+2 skrev mitchr...@gmail.com:
> > > > > and you get the first integer.
> > > > There are no infinitesimals in real numbers.
> > > >
> > > > And 1=9/9=0.999...
> > > >
> > > > wrong as always
> > > Are there infinite numbers in infinite numbers?
> > >
> > > If there are infinite numbers, they are infinite numbers
> > > in infinite numbers.
> > There are no "infinite numbers" in real numbers, real numbers are archimedian.
> > >
> > > Instead it's "for any large number, finite, there's
> > > a larger one (also finite)" besides "for any large number,
> > > finite, there's a large infinite, larger", from that
> > > infinite numbers exist.
> > >
> > > This simply keeps what is quantitative there,
> > > with respect to qualitative.
> Also it is like an existence result itself,
> that there are infinitely many
> there are infintely grand.
>
> "Having the Archimedean property" is
> often read two ways,
> for the unbounded (not finitely many)
> and the unbounded (not infinitely grand).
>
> It's kind of like Goedelian completeness:
> reminding people of both the completeness
> theorems, and the incompleteness theorems.
>
>
> These days non-Archimedean fields are
> a usual introduction to "non" standard ("extra" standard).

They are not standard.

>
> Then, yes, I am talking about a logical consequence
> of there being infinitely many that there are infinitesimals
> in the reals and that besides there are infinites in integers.

There are infinitely many objects in the set of real numbers, but there are no infinitesimals or infinities in the set.

>
> Then, what is the "standard" is just as above in matters of
> "representation theory", here model theory for a function theory
> for a space of values: it's standard and well-defined but not
> complete, the space of representations those of the field reals,
> Archimedean field reals, made replete with a space of
> representations of those of line reals, or signal reals.
>
> That field reals, line reals, and signal reals, each in the
> spaces of real values like usual vector spaces, are each
> models of real numbers with IVT and resultingly the FTCs,
> and otherwise real character: is central and important.
>
> (In mathematics.)
>
> It's kind of like "Burali-Forti's largest ordinal, that would
> contain itself", or "Russell's set-of-all-sets-that-don't-contain-
> themselves contains itself": starting with that the only ordinals
> are finite and Archimedean as you advise, that immediately any
> "infinite" including omega or otherwise actual infinite:
> includes itself. I.e. without "defining" omega all well-founded
> and regular: it ("omega, an inductive set") would be "derived"
> from the "paradoxes of Burali-Forti and Russell in an Archimedean
> universe", as _not_ well-founded, regular, ordinary, ....
>
>
> Then, for infinitesimals and the long line, which usually enough
> abstractly includes infinitesimals, a usual enough notion of
> the real line, partitions any segment into infinitely-many
> equal-size pieces.
>
> Of course calculus was called "infinitesimal analysis" for
> some hundreds of years, and that's what was meant, also.
>
> These days of course everybody knows Cauchy/Weierstrass as
> the formalism after Riemann/Lebesgue the formalism, knowing
> most all of a development of the complete ordered field (Archimedean),
> besides usual graphical notions of the points that in their space
> mark (draw) a line. (Point-sets, ..., in what are real-valued systems.)
>
> Then, line reals, field reals, and signal reals are three _different_
> models of real numbers, connected and having the gaplessness
> property, least upper bound property, measure(s), ..., here of
> course that line reals are modeled as "unboundedly many,
> and, vanishingly small, and equal, values what sum to 1" .
>
>
> Then where "the curriculum includes _only_ the field reals,
> stop", it is short, because usual models of line reals and signal
> reals besides field reals are everywhere and central, in all sorts
> models in mathematics. So, the curriculum is short because
> there are at least three _different_ models of "real numbers".

On Tuesday, May 24, 2022 at 3:03:28 AM UTC-7, zelos...@gmail.com wrote:
> tisdag 24 maj 2022 kl. 09:42:59 UTC+2 skrev Ross A. Finlayson:
> > On Monday, May 23, 2022 at 10:19:05 PM UTC-7, zelos...@gmail.com wrote:
> > > måndag 23 maj 2022 kl. 18:21:35 UTC+2 skrev Ross A. Finlayson:
> > > > On Sunday, May 22, 2022 at 9:53:03 PM UTC-7, zelos...@gmail.com wrote:
> > > > > torsdag 19 maj 2022 kl. 19:25:44 UTC+2 skrev mitchr...@gmail.com:
> > > > > > and you get the first integer.
> > > > > There are no infinitesimals in real numbers.
> > > > >
> > > > > And 1=9/9=0.999...
> > > > >
> > > > > wrong as always
> > > > Are there infinite numbers in infinite numbers?
> > > >
> > > > If there are infinite numbers, they are infinite numbers
> > > > in infinite numbers.
> > > There are no "infinite numbers" in real numbers, real numbers are archimedian.
> > > >
> > > > Instead it's "for any large number, finite, there's
> > > > a larger one (also finite)" besides "for any large number,
> > > > finite, there's a large infinite, larger", from that
> > > > infinite numbers exist.
> > > >
> > > > This simply keeps what is quantitative there,
> > > > with respect to qualitative.
> > Also it is like an existence result itself,
> > that there are infinitely many
> > there are infintely grand.
> >
> > "Having the Archimedean property" is
> > often read two ways,
> > for the unbounded (not finitely many)
> > and the unbounded (not infinitely grand).
> >
> > It's kind of like Goedelian completeness:
> > reminding people of both the completeness
> > theorems, and the incompleteness theorems.
> >
> >
> > These days non-Archimedean fields are
> > a usual introduction to "non" standard ("extra" standard).
> They are not standard.
> >
> > Then, yes, I am talking about a logical consequence
> > of there being infinitely many that there are infinitesimals
> > in the reals and that besides there are infinites in integers.
> There are infinitely many objects in the set of real numbers, but there are no infinitesimals or infinities in the set.
> >
> > Then, what is the "standard" is just as above in matters of
> > "representation theory", here model theory for a function theory
> > for a space of values: it's standard and well-defined but not
> > complete, the space of representations those of the field reals,
> > Archimedean field reals, made replete with a space of
> > representations of those of line reals, or signal reals.
> >
> > That field reals, line reals, and signal reals, each in the
> > spaces of real values like usual vector spaces, are each
> > models of real numbers with IVT and resultingly the FTCs,
> > and otherwise real character: is central and important.
> >
> > (In mathematics.)
> >
> > It's kind of like "Burali-Forti's largest ordinal, that would
> > contain itself", or "Russell's set-of-all-sets-that-don't-contain-
> > themselves contains itself": starting with that the only ordinals
> > are finite and Archimedean as you advise, that immediately any
> > "infinite" including omega or otherwise actual infinite:
> > includes itself. I.e. without "defining" omega all well-founded
> > and regular: it ("omega, an inductive set") would be "derived"
> > from the "paradoxes of Burali-Forti and Russell in an Archimedean
> > universe", as _not_ well-founded, regular, ordinary, ....
> >
> >
> > Then, for infinitesimals and the long line, which usually enough
> > abstractly includes infinitesimals, a usual enough notion of
> > the real line, partitions any segment into infinitely-many
> > equal-size pieces.
> >
> > Of course calculus was called "infinitesimal analysis" for
> > some hundreds of years, and that's what was meant, also.
> >
> > These days of course everybody knows Cauchy/Weierstrass as
> > the formalism after Riemann/Lebesgue the formalism, knowing
> > most all of a development of the complete ordered field (Archimedean),
> > besides usual graphical notions of the points that in their space
> > mark (draw) a line. (Point-sets, ..., in what are real-valued systems.)
> >
> > Then, line reals, field reals, and signal reals are three _different_
> > models of real numbers, connected and having the gaplessness
> > property, least upper bound property, measure(s), ..., here of
> > course that line reals are modeled as "unboundedly many,
> > and, vanishingly small, and equal, values what sum to 1" .
> >
> >
> > Then where "the curriculum includes _only_ the field reals,
> > stop", it is short, because usual models of line reals and signal
> > reals besides field reals are everywhere and central, in all sorts
> > models in mathematics. So, the curriculum is short because
> > there are at least three _different_ models of "real numbers".

There are at least three models of reals, different sets,
and one of them is infinitesimals ("iota-values") zero to one.

And, in their own way, the line reals are "Regular" and "Standard":
for example standardly modelling the Equivalency Function as a limit
of real functions.

So, mathematics writ large is missing out from that the
properties of "real" infinites/infinitesimals: are direct and
consequent from the properties of the objects the numbers
their values themselves.

If we can agree that "retro-finitism": is backward, then I hope
that you can see that ignoring these features, is also.

I.e. for theories where these things exist, consistently of
course, not knowing them or ignoring them: is as bad as
retro-finitism.

Which points all sorts arguments/rhetoric strongly back around - ....

On Tuesday, May 24, 2022 at 9:02:15 AM UTC-7, Ross A. Finlayson wrote:
> On Tuesday, May 24, 2022 at 3:03:28 AM UTC-7, zelos...@gmail.com wrote:
> > tisdag 24 maj 2022 kl. 09:42:59 UTC+2 skrev Ross A. Finlayson:
> > > On Monday, May 23, 2022 at 10:19:05 PM UTC-7, zelos...@gmail.com wrote:
> > > > måndag 23 maj 2022 kl. 18:21:35 UTC+2 skrev Ross A. Finlayson:
> > > > > On Sunday, May 22, 2022 at 9:53:03 PM UTC-7, zelos...@gmail.com wrote:
> > > > > > torsdag 19 maj 2022 kl. 19:25:44 UTC+2 skrev mitchr...@gmail.com:
> > > > > > > and you get the first integer.
> > > > > > There are no infinitesimals in real numbers.
> > > > > >
> > > > > > And 1=9/9=0.999...
> > > > > >
> > > > > > wrong as always
> > > > > Are there infinite numbers in infinite numbers?
> > > > >
> > > > > If there are infinite numbers, they are infinite numbers
> > > > > in infinite numbers.
> > > > There are no "infinite numbers" in real numbers, real numbers are archimedian.
> > > > >
> > > > > Instead it's "for any large number, finite, there's
> > > > > a larger one (also finite)" besides "for any large number,
> > > > > finite, there's a large infinite, larger", from that
> > > > > infinite numbers exist.
> > > > >
> > > > > This simply keeps what is quantitative there,
> > > > > with respect to qualitative.
> > > Also it is like an existence result itself,
> > > that there are infinitely many
> > > there are infintely grand.
> > >
> > > "Having the Archimedean property" is
> > > often read two ways,
> > > for the unbounded (not finitely many)
> > > and the unbounded (not infinitely grand).
> > >
> > > It's kind of like Goedelian completeness:
> > > reminding people of both the completeness
> > > theorems, and the incompleteness theorems.
> > >
> > >
> > > These days non-Archimedean fields are
> > > a usual introduction to "non" standard ("extra" standard).
> > They are not standard.
> > >
> > > Then, yes, I am talking about a logical consequence
> > > of there being infinitely many that there are infinitesimals
> > > in the reals and that besides there are infinites in integers.
> > There are infinitely many objects in the set of real numbers, but there are no infinitesimals or infinities in the set.
> > >
> > > Then, what is the "standard" is just as above in matters of
> > > "representation theory", here model theory for a function theory
> > > for a space of values: it's standard and well-defined but not
> > > complete, the space of representations those of the field reals,
> > > Archimedean field reals, made replete with a space of
> > > representations of those of line reals, or signal reals.
> > >
> > > That field reals, line reals, and signal reals, each in the
> > > spaces of real values like usual vector spaces, are each
> > > models of real numbers with IVT and resultingly the FTCs,
> > > and otherwise real character: is central and important.
> > >
> > > (In mathematics.)
> > >
> > > It's kind of like "Burali-Forti's largest ordinal, that would
> > > contain itself", or "Russell's set-of-all-sets-that-don't-contain-
> > > themselves contains itself": starting with that the only ordinals
> > > are finite and Archimedean as you advise, that immediately any
> > > "infinite" including omega or otherwise actual infinite:
> > > includes itself. I.e. without "defining" omega all well-founded
> > > and regular: it ("omega, an inductive set") would be "derived"
> > > from the "paradoxes of Burali-Forti and Russell in an Archimedean
> > > universe", as _not_ well-founded, regular, ordinary, ....
> > >
> > >
> > > Then, for infinitesimals and the long line, which usually enough
> > > abstractly includes infinitesimals, a usual enough notion of
> > > the real line, partitions any segment into infinitely-many
> > > equal-size pieces.
> > >
> > > Of course calculus was called "infinitesimal analysis" for
> > > some hundreds of years, and that's what was meant, also.
> > >
> > > These days of course everybody knows Cauchy/Weierstrass as
> > > the formalism after Riemann/Lebesgue the formalism, knowing
> > > most all of a development of the complete ordered field (Archimedean),
> > > besides usual graphical notions of the points that in their space
> > > mark (draw) a line. (Point-sets, ..., in what are real-valued systems..)
> > >
> > > Then, line reals, field reals, and signal reals are three _different_
> > > models of real numbers, connected and having the gaplessness
> > > property, least upper bound property, measure(s), ..., here of
> > > course that line reals are modeled as "unboundedly many,
> > > and, vanishingly small, and equal, values what sum to 1" .
> > >
> > >
> > > Then where "the curriculum includes _only_ the field reals,
> > > stop", it is short, because usual models of line reals and signal
> > > reals besides field reals are everywhere and central, in all sorts
> > > models in mathematics. So, the curriculum is short because
> > > there are at least three _different_ models of "real numbers".
> There are at least three models of reals, different sets,
> and one of them is infinitesimals ("iota-values") zero to one.
>
> And, in their own way, the line reals are "Regular" and "Standard":
> for example standardly modelling the Equivalency Function as a limit
> of real functions.
>
> So, mathematics writ large is missing out from that the
> properties of "real" infinites/infinitesimals: are direct and
> consequent from the properties of the objects the numbers
> their values themselves.
>
> If we can agree that "retro-finitism": is backward, then I hope
> that you can see that ignoring these features, is also.
>
> I.e. for theories where these things exist, consistently of
> course, not knowing them or ignoring them: is as bad as
> retro-finitism.
>
> Which points all sorts arguments/rhetoric strongly back around - ....

If you add zero to .999 repeating you get .999 repeating
but if you add an infinitesimal you get the first integer.

On 5/24/2022 3:48 PM, mitchr...@gmail.com wrote:
> On Tuesday, May 24, 2022 at 9:02:15 AM UTC-7, Ross A. Finlayson wrote:
>> On Tuesday, May 24, 2022 at 3:03:28 AM UTC-7, zelos...@gmail.com wrote:
>>> tisdag 24 maj 2022 kl. 09:42:59 UTC+2 skrev Ross A. Finlayson:
>>>> On Monday, May 23, 2022 at 10:19:05 PM UTC-7, zelos...@gmail.com wrote:
>>>>> måndag 23 maj 2022 kl. 18:21:35 UTC+2 skrev Ross A. Finlayson:
>>>>>> On Sunday, May 22, 2022 at 9:53:03 PM UTC-7, zelos...@gmail.com wrote:
>>>>>>> torsdag 19 maj 2022 kl. 19:25:44 UTC+2 skrev mitchr...@gmail.com:
>>>>>>>> and you get the first integer.
>>>>>>> There are no infinitesimals in real numbers.
>>>>>>>
>>>>>>> And 1=9/9=0.999...
>>>>>>>
>>>>>>> wrong as always
>>>>>> Are there infinite numbers in infinite numbers?
>>>>>>
>>>>>> If there are infinite numbers, they are infinite numbers
>>>>>> in infinite numbers.
>>>>> There are no "infinite numbers" in real numbers, real numbers are archimedian.
>>>>>>
>>>>>> Instead it's "for any large number, finite, there's
>>>>>> a larger one (also finite)" besides "for any large number,
>>>>>> finite, there's a large infinite, larger", from that
>>>>>> infinite numbers exist.
>>>>>>
>>>>>> This simply keeps what is quantitative there,
>>>>>> with respect to qualitative.
>>>> Also it is like an existence result itself,
>>>> that there are infinitely many
>>>> there are infintely grand.
>>>>
>>>> "Having the Archimedean property" is
>>>> often read two ways,
>>>> for the unbounded (not finitely many)
>>>> and the unbounded (not infinitely grand).
>>>>
>>>> It's kind of like Goedelian completeness:
>>>> reminding people of both the completeness
>>>> theorems, and the incompleteness theorems.
>>>>
>>>>
>>>> These days non-Archimedean fields are
>>>> a usual introduction to "non" standard ("extra" standard).
>>> They are not standard.
>>>>
>>>> Then, yes, I am talking about a logical consequence
>>>> of there being infinitely many that there are infinitesimals
>>>> in the reals and that besides there are infinites in integers.
>>> There are infinitely many objects in the set of real numbers, but there are no infinitesimals or infinities in the set.
>>>>
>>>> Then, what is the "standard" is just as above in matters of
>>>> "representation theory", here model theory for a function theory
>>>> for a space of values: it's standard and well-defined but not
>>>> complete, the space of representations those of the field reals,
>>>> Archimedean field reals, made replete with a space of
>>>> representations of those of line reals, or signal reals.
>>>>
>>>> That field reals, line reals, and signal reals, each in the
>>>> spaces of real values like usual vector spaces, are each
>>>> models of real numbers with IVT and resultingly the FTCs,
>>>> and otherwise real character: is central and important.
>>>>
>>>> (In mathematics.)
>>>>
>>>> It's kind of like "Burali-Forti's largest ordinal, that would
>>>> contain itself", or "Russell's set-of-all-sets-that-don't-contain-
>>>> themselves contains itself": starting with that the only ordinals
>>>> are finite and Archimedean as you advise, that immediately any
>>>> "infinite" including omega or otherwise actual infinite:
>>>> includes itself. I.e. without "defining" omega all well-founded
>>>> and regular: it ("omega, an inductive set") would be "derived"
>>>> from the "paradoxes of Burali-Forti and Russell in an Archimedean
>>>> universe", as _not_ well-founded, regular, ordinary, ....
>>>>
>>>>
>>>> Then, for infinitesimals and the long line, which usually enough
>>>> abstractly includes infinitesimals, a usual enough notion of
>>>> the real line, partitions any segment into infinitely-many
>>>> equal-size pieces.
>>>>
>>>> Of course calculus was called "infinitesimal analysis" for
>>>> some hundreds of years, and that's what was meant, also.
>>>>
>>>> These days of course everybody knows Cauchy/Weierstrass as
>>>> the formalism after Riemann/Lebesgue the formalism, knowing
>>>> most all of a development of the complete ordered field (Archimedean),
>>>> besides usual graphical notions of the points that in their space
>>>> mark (draw) a line. (Point-sets, ..., in what are real-valued systems.)
>>>>
>>>> Then, line reals, field reals, and signal reals are three _different_
>>>> models of real numbers, connected and having the gaplessness
>>>> property, least upper bound property, measure(s), ..., here of
>>>> course that line reals are modeled as "unboundedly many,
>>>> and, vanishingly small, and equal, values what sum to 1" .
>>>>
>>>>
>>>> Then where "the curriculum includes _only_ the field reals,
>>>> stop", it is short, because usual models of line reals and signal
>>>> reals besides field reals are everywhere and central, in all sorts
>>>> models in mathematics. So, the curriculum is short because
>>>> there are at least three _different_ models of "real numbers".
>> There are at least three models of reals, different sets,
>> and one of them is infinitesimals ("iota-values") zero to one.
>>
>> And, in their own way, the line reals are "Regular" and "Standard":
>> for example standardly modelling the Equivalency Function as a limit
>> of real functions.
>>
>> So, mathematics writ large is missing out from that the
>> properties of "real" infinites/infinitesimals: are direct and
>> consequent from the properties of the objects the numbers
>> their values themselves.
>>
>> If we can agree that "retro-finitism": is backward, then I hope
>> that you can see that ignoring these features, is also.
>>
>> I.e. for theories where these things exist, consistently of
>> course, not knowing them or ignoring them: is as bad as
>> retro-finitism.
>>
>> Which points all sorts arguments/rhetoric strongly back around - ....
>
> If you add zero to .999 repeating you get .999 repeating
> but if you add an infinitesimal you get the first integer.
>

Mitch you keep dropping your infinitesimals all over the floor!

On 5/24/2022 1:43 AM, Reef Kubo wrote:
> Chris M. Thomasson wrote:
>
>>> and you get the first integer.
>>
>> Is your infinitesimal something like: .1^(infinity) ?
>> .1^5 = .00001 .1^6 = .000001 ...
>> It gets pretty damn small! Run for infinity the limit is zero, right? So
>> if zero is the limit of the infinitesimal .000...1, and one is the limit
>> of .999..., then that just means: 1 + 0 = 1
>
> idiot, there is no "damn small" in mathematics. If your region of interest
> is bellow that limit, that number can be huge, not small.

Well, this does get infinitely small with a step-by-step process:

..1^1 = .1
..1^2 = .01
..1^3 = .001
..1^4 = .0001
..1^5 = .00001
..1^6 = .000001
....

Agreed?

On Tuesday, May 24, 2022 at 2:49:26 PM UTC-7, Chris M. Thomasson wrote:
> On 5/24/2022 1:43 AM, Reef Kubo wrote:
> > Chris M. Thomasson wrote:
> >
> >>> and you get the first integer.
> >>
> >> Is your infinitesimal something like: .1^(infinity) ?
> >> .1^5 = .00001 .1^6 = .000001 ...
> >> It gets pretty damn small! Run for infinity the limit is zero, right? So
> >> if zero is the limit of the infinitesimal .000...1, and one is the limit
> >> of .999..., then that just means: 1 + 0 = 1
> >
> > idiot, there is no "damn small" in mathematics. If your region of interest
> > is bellow that limit, that number can be huge, not small.
> Well, this does get infinitely small with a step-by-step process:
>
> .1^1 = .1
> .1^2 = .01
> .1^3 = .001
> .1^4 = .0001
> .1^5 = .00001
> .1^6 = .000001
> ...
> Agreed?

That completes the infinitesimal that is nonzero.
Add that to .999 repeating and you get one.
Add zero and it stays .999 repeating.

On 5/24/2022 4:49 PM, Chris M. Thomasson wrote:
> On 5/24/2022 1:43 AM, Reef Kubo wrote:
>> Chris M. Thomasson wrote:
>>
>>>> and you get the first integer.
>>>
>>> Is your infinitesimal something like: .1^(infinity) ?
>>> .1^5 = .00001 .1^6 = .000001 ...
>>> It gets pretty damn small! Run for infinity the limit is zero, right? So
>>> if zero is the limit of the infinitesimal .000...1, and one is the limit
>>> of .999..., then that just means: 1 + 0 = 1
>>
>> idiot, there is no "damn small" in mathematics. If your region of interest
>> is bellow that limit, that number can be huge, not small.
>
> Well, this does get infinitely small with a step-by-step process:
>
> .1^1 = .1
> .1^2 = .01
> .1^3 = .001
> .1^4 = .0001
> .1^5 = .00001
> .1^6 = .000001
> ...
>
> Agreed?

..1 = .1
..1 * .1 = .01
..1 * .01 = .001
..1 * .001 = .0001
..1 * .0001 = .00001
....

this way is faster (?)

Chris M. Thomasson wrote :
> On 5/24/2022 1:43 AM, Reef Kubo wrote:
>> Chris M. Thomasson wrote:
>>
>>>> and you get the first integer.
>>>
>>> Is your infinitesimal something like: .1^(infinity) ?
>>> .1^5 = .00001 .1^6 = .000001 ...
>>> It gets pretty damn small! Run for infinity the limit is zero, right? So
>>> if zero is the limit of the infinitesimal .000...1, and one is the limit
>>> of .999..., then that just means: 1 + 0 = 1
>>
>> idiot, there is no "damn small" in mathematics. If your region of interest
>> is bellow that limit, that number can be huge, not small.
>
> Well, this does get infinitely small with a step-by-step process:
>
> .1^1 = .1
> .1^2 = .01
> .1^3 = .001
> .1^4 = .0001
> .1^5 = .00001
> .1^6 = .000001
> ...
>
> Agreed?

It gets smaller with each of infinitely many steps.

What's the series (sum)?

On 5/24/2022 4:37 PM, FromTheRafters wrote:
> Chris M. Thomasson wrote :
>> On 5/24/2022 1:43 AM, Reef Kubo wrote:
>>> Chris M. Thomasson wrote:
>>>
>>>>> and you get the first integer.
>>>>
>>>> Is your infinitesimal something like: .1^(infinity) ?
>>>> .1^5 = .00001 .1^6 = .000001 ...
>>>> It gets pretty damn small! Run for infinity the limit is zero,
>>>> right? So
>>>> if zero is the limit of the infinitesimal .000...1, and one is the
>>>> limit
>>>> of .999..., then that just means: 1 + 0 = 1
>>>
>>> idiot, there is no "damn small" in mathematics. If your region of
>>> interest
>>> is bellow that limit, that number can be huge, not small.
>>
>> Well, this does get infinitely small with a step-by-step process:
>>
>> .1^1 = .1
>> .1^2 = .01
>> .1^3 = .001
>> .1^4 = .0001
>> .1^5 = .00001
>> .1^6 = .000001
>> ...
>>
>> Agreed?
>
> It gets smaller with each of infinitely many steps.
>
> What's the series (sum)?

Ahh, the sum... Well, what about this:

..1 + .01 = .11

Okay? What about:

..11 + .001 = .111

Humm... There is a pattern. .111... forever?

..111 + .0001 = .1111

Ahhh... .1111 * 9 = .9999 so getting bigger, yet confined within a limit.

tisdag 24 maj 2022 kl. 18:02:15 UTC+2 skrev Ross A. Finlayson:
> On Tuesday, May 24, 2022 at 3:03:28 AM UTC-7, zelos...@gmail.com wrote:
> > tisdag 24 maj 2022 kl. 09:42:59 UTC+2 skrev Ross A. Finlayson:
> > > On Monday, May 23, 2022 at 10:19:05 PM UTC-7, zelos...@gmail.com wrote:
> > > > måndag 23 maj 2022 kl. 18:21:35 UTC+2 skrev Ross A. Finlayson:
> > > > > On Sunday, May 22, 2022 at 9:53:03 PM UTC-7, zelos...@gmail.com wrote:
> > > > > > torsdag 19 maj 2022 kl. 19:25:44 UTC+2 skrev mitchr...@gmail.com:
> > > > > > > and you get the first integer.
> > > > > > There are no infinitesimals in real numbers.
> > > > > >
> > > > > > And 1=9/9=0.999...
> > > > > >
> > > > > > wrong as always
> > > > > Are there infinite numbers in infinite numbers?
> > > > >
> > > > > If there are infinite numbers, they are infinite numbers
> > > > > in infinite numbers.
> > > > There are no "infinite numbers" in real numbers, real numbers are archimedian.
> > > > >
> > > > > Instead it's "for any large number, finite, there's
> > > > > a larger one (also finite)" besides "for any large number,
> > > > > finite, there's a large infinite, larger", from that
> > > > > infinite numbers exist.
> > > > >
> > > > > This simply keeps what is quantitative there,
> > > > > with respect to qualitative.
> > > Also it is like an existence result itself,
> > > that there are infinitely many
> > > there are infintely grand.
> > >
> > > "Having the Archimedean property" is
> > > often read two ways,
> > > for the unbounded (not finitely many)
> > > and the unbounded (not infinitely grand).
> > >
> > > It's kind of like Goedelian completeness:
> > > reminding people of both the completeness
> > > theorems, and the incompleteness theorems.
> > >
> > >
> > > These days non-Archimedean fields are
> > > a usual introduction to "non" standard ("extra" standard).
> > They are not standard.
> > >
> > > Then, yes, I am talking about a logical consequence
> > > of there being infinitely many that there are infinitesimals
> > > in the reals and that besides there are infinites in integers.
> > There are infinitely many objects in the set of real numbers, but there are no infinitesimals or infinities in the set.
> > >
> > > Then, what is the "standard" is just as above in matters of
> > > "representation theory", here model theory for a function theory
> > > for a space of values: it's standard and well-defined but not
> > > complete, the space of representations those of the field reals,
> > > Archimedean field reals, made replete with a space of
> > > representations of those of line reals, or signal reals.
> > >
> > > That field reals, line reals, and signal reals, each in the
> > > spaces of real values like usual vector spaces, are each
> > > models of real numbers with IVT and resultingly the FTCs,
> > > and otherwise real character: is central and important.
> > >
> > > (In mathematics.)
> > >
> > > It's kind of like "Burali-Forti's largest ordinal, that would
> > > contain itself", or "Russell's set-of-all-sets-that-don't-contain-
> > > themselves contains itself": starting with that the only ordinals
> > > are finite and Archimedean as you advise, that immediately any
> > > "infinite" including omega or otherwise actual infinite:
> > > includes itself. I.e. without "defining" omega all well-founded
> > > and regular: it ("omega, an inductive set") would be "derived"
> > > from the "paradoxes of Burali-Forti and Russell in an Archimedean
> > > universe", as _not_ well-founded, regular, ordinary, ....
> > >
> > >
> > > Then, for infinitesimals and the long line, which usually enough
> > > abstractly includes infinitesimals, a usual enough notion of
> > > the real line, partitions any segment into infinitely-many
> > > equal-size pieces.
> > >
> > > Of course calculus was called "infinitesimal analysis" for
> > > some hundreds of years, and that's what was meant, also.
> > >
> > > These days of course everybody knows Cauchy/Weierstrass as
> > > the formalism after Riemann/Lebesgue the formalism, knowing
> > > most all of a development of the complete ordered field (Archimedean),
> > > besides usual graphical notions of the points that in their space
> > > mark (draw) a line. (Point-sets, ..., in what are real-valued systems..)
> > >
> > > Then, line reals, field reals, and signal reals are three _different_
> > > models of real numbers, connected and having the gaplessness
> > > property, least upper bound property, measure(s), ..., here of
> > > course that line reals are modeled as "unboundedly many,
> > > and, vanishingly small, and equal, values what sum to 1" .
> > >
> > >
> > > Then where "the curriculum includes _only_ the field reals,
> > > stop", it is short, because usual models of line reals and signal
> > > reals besides field reals are everywhere and central, in all sorts
> > > models in mathematics. So, the curriculum is short because
> > > there are at least three _different_ models of "real numbers".
> There are at least three models of reals, different sets,
> and one of them is infinitesimals ("iota-values") zero to one.
>
> And, in their own way, the line reals are "Regular" and "Standard":
> for example standardly modelling the Equivalency Function as a limit
> of real functions.
>
> So, mathematics writ large is missing out from that the
> properties of "real" infinites/infinitesimals: are direct and
> consequent from the properties of the objects the numbers
> their values themselves.
>
> If we can agree that "retro-finitism": is backward, then I hope
> that you can see that ignoring these features, is also.
>
> I.e. for theories where these things exist, consistently of
> course, not knowing them or ignoring them: is as bad as
> retro-finitism.
>
> Which points all sorts arguments/rhetoric strongly back around - ....

reals are by definition archimedian. There are no infinitesimals. If there are, they are not the real numbers.

tisdag 24 maj 2022 kl. 22:48:35 UTC+2 skrev mitchr...@gmail.com:
> On Tuesday, May 24, 2022 at 9:02:15 AM UTC-7, Ross A. Finlayson wrote:
> > On Tuesday, May 24, 2022 at 3:03:28 AM UTC-7, zelos...@gmail.com wrote:
> > > tisdag 24 maj 2022 kl. 09:42:59 UTC+2 skrev Ross A. Finlayson:
> > > > On Monday, May 23, 2022 at 10:19:05 PM UTC-7, zelos...@gmail.com wrote:
> > > > > måndag 23 maj 2022 kl. 18:21:35 UTC+2 skrev Ross A. Finlayson:
> > > > > > On Sunday, May 22, 2022 at 9:53:03 PM UTC-7, zelos...@gmail.com wrote:
> > > > > > > torsdag 19 maj 2022 kl. 19:25:44 UTC+2 skrev mitchr...@gmail.com:
> > > > > > > > and you get the first integer.
> > > > > > > There are no infinitesimals in real numbers.
> > > > > > >
> > > > > > > And 1=9/9=0.999...
> > > > > > >
> > > > > > > wrong as always
> > > > > > Are there infinite numbers in infinite numbers?
> > > > > >
> > > > > > If there are infinite numbers, they are infinite numbers
> > > > > > in infinite numbers.
> > > > > There are no "infinite numbers" in real numbers, real numbers are archimedian.
> > > > > >
> > > > > > Instead it's "for any large number, finite, there's
> > > > > > a larger one (also finite)" besides "for any large number,
> > > > > > finite, there's a large infinite, larger", from that
> > > > > > infinite numbers exist.
> > > > > >
> > > > > > This simply keeps what is quantitative there,
> > > > > > with respect to qualitative.
> > > > Also it is like an existence result itself,
> > > > that there are infinitely many
> > > > there are infintely grand.
> > > >
> > > > "Having the Archimedean property" is
> > > > often read two ways,
> > > > for the unbounded (not finitely many)
> > > > and the unbounded (not infinitely grand).
> > > >
> > > > It's kind of like Goedelian completeness:
> > > > reminding people of both the completeness
> > > > theorems, and the incompleteness theorems.
> > > >
> > > >
> > > > These days non-Archimedean fields are
> > > > a usual introduction to "non" standard ("extra" standard).
> > > They are not standard.
> > > >
> > > > Then, yes, I am talking about a logical consequence
> > > > of there being infinitely many that there are infinitesimals
> > > > in the reals and that besides there are infinites in integers.
> > > There are infinitely many objects in the set of real numbers, but there are no infinitesimals or infinities in the set.
> > > >
> > > > Then, what is the "standard" is just as above in matters of
> > > > "representation theory", here model theory for a function theory
> > > > for a space of values: it's standard and well-defined but not
> > > > complete, the space of representations those of the field reals,
> > > > Archimedean field reals, made replete with a space of
> > > > representations of those of line reals, or signal reals.
> > > >
> > > > That field reals, line reals, and signal reals, each in the
> > > > spaces of real values like usual vector spaces, are each
> > > > models of real numbers with IVT and resultingly the FTCs,
> > > > and otherwise real character: is central and important.
> > > >
> > > > (In mathematics.)
> > > >
> > > > It's kind of like "Burali-Forti's largest ordinal, that would
> > > > contain itself", or "Russell's set-of-all-sets-that-don't-contain-
> > > > themselves contains itself": starting with that the only ordinals
> > > > are finite and Archimedean as you advise, that immediately any
> > > > "infinite" including omega or otherwise actual infinite:
> > > > includes itself. I.e. without "defining" omega all well-founded
> > > > and regular: it ("omega, an inductive set") would be "derived"
> > > > from the "paradoxes of Burali-Forti and Russell in an Archimedean
> > > > universe", as _not_ well-founded, regular, ordinary, ....
> > > >
> > > >
> > > > Then, for infinitesimals and the long line, which usually enough
> > > > abstractly includes infinitesimals, a usual enough notion of
> > > > the real line, partitions any segment into infinitely-many
> > > > equal-size pieces.
> > > >
> > > > Of course calculus was called "infinitesimal analysis" for
> > > > some hundreds of years, and that's what was meant, also.
> > > >
> > > > These days of course everybody knows Cauchy/Weierstrass as
> > > > the formalism after Riemann/Lebesgue the formalism, knowing
> > > > most all of a development of the complete ordered field (Archimedean),
> > > > besides usual graphical notions of the points that in their space
> > > > mark (draw) a line. (Point-sets, ..., in what are real-valued systems.)
> > > >
> > > > Then, line reals, field reals, and signal reals are three _different_
> > > > models of real numbers, connected and having the gaplessness
> > > > property, least upper bound property, measure(s), ..., here of
> > > > course that line reals are modeled as "unboundedly many,
> > > > and, vanishingly small, and equal, values what sum to 1" .
> > > >
> > > >
> > > > Then where "the curriculum includes _only_ the field reals,
> > > > stop", it is short, because usual models of line reals and signal
> > > > reals besides field reals are everywhere and central, in all sorts
> > > > models in mathematics. So, the curriculum is short because
> > > > there are at least three _different_ models of "real numbers".
> > There are at least three models of reals, different sets,
> > and one of them is infinitesimals ("iota-values") zero to one.
> >
> > And, in their own way, the line reals are "Regular" and "Standard":
> > for example standardly modelling the Equivalency Function as a limit
> > of real functions.
> >
> > So, mathematics writ large is missing out from that the
> > properties of "real" infinites/infinitesimals: are direct and
> > consequent from the properties of the objects the numbers
> > their values themselves.
> >
> > If we can agree that "retro-finitism": is backward, then I hope
> > that you can see that ignoring these features, is also.
> >
> > I.e. for theories where these things exist, consistently of
> > course, not knowing them or ignoring them: is as bad as
> > retro-finitism.
> >
> > Which points all sorts arguments/rhetoric strongly back around - ....
> If you add zero to .999 repeating you get .999 repeating
> but if you add an infinitesimal you get the first integer.
0.999...+0=1

On 5/24/2022 2:55 PM, sergi o wrote:
> On 5/24/2022 4:49 PM, Chris M. Thomasson wrote:
>> On 5/24/2022 1:43 AM, Reef Kubo wrote:
>>> Chris M. Thomasson wrote:
>>>
>>>>> and you get the first integer.
>>>>
>>>> Is your infinitesimal something like: .1^(infinity) ?
>>>> .1^5 = .00001 .1^6 = .000001 ...
>>>> It gets pretty damn small! Run for infinity the limit is zero,
>>>> right? So
>>>> if zero is the limit of the infinitesimal .000...1, and one is the
>>>> limit
>>>> of .999..., then that just means: 1 + 0 = 1
>>>
>>> idiot, there is no "damn small" in mathematics. If your region of
>>> interest
>>> is bellow that limit, that number can be huge, not small.
>>
>> Well, this does get infinitely small with a step-by-step process:
>>
>> .1^1 = .1
>> .1^2 = .01
>> .1^3 = .001
>> .1^4 = .0001
>> .1^5 = .00001
>> .1^6 = .000001
>> ...
>>
>> Agreed?
>
> .1 = .1
> .1 * .1 = .01
> .1 * .01 = .001
> .1 * .001 = .0001
> .1 * .0001 = .00001
> ...
>
> this way is faster (?)

I would say so. :^)