Re: Provably unprovable eliminates incompleteness (Cantor was wrong about cardinality) Deceptive Strawman by Ben

[peteolcott]

On 9/12/2019 8:00 PM, Ben Bacarisse wrote:
> peteolcott <Here@Home> writes:
>
>> On 9/12/2019 10:33 AM, Ben Bacarisse wrote:
>>> peteolcott <Here@Home> writes:
>>>>>>> peteolcott <Here@Home> writes:
>>>>>>>>> peteolcott <Here@Home> writes:
>>> <cut>
>>>>>>>>>> Now I have to correct my prior reasoning. All of the geometric points on
>>>>>>>>>> the number line correspond to all of the numbers that exist. Some of these
>>>>>>>>>> are not real numbers.
>>>>>>>>>>
>>>>>>>>>> Since we can count all of these numbers with integers using the
>>>>>>>>>> infinitesimal number system we have proven that the set of real
>>>>>>>>>> numbers is not larger than the set of integers.
>>> <cut>
>>>>>>>>>> Real_Part[Infinitesimal_Offset]
>>>>>>>>>>
>>>>>>>>>> 0.0[1 to ∞] forms a bijection between the positive integers and all of
>>>>>>>>>> the positive numbers through all of the positive points on the number
>>>>>>>>>> line.
>>> <cut>
>>>>>>>> (1) Do you understand that all of the points on a number line are all
>>>>>>>> of the numbers?
>>>>>>>>
>>>>>>>> (2) Do you understand that a bijection between the integers and these
>>>>>>>> points would make these points countable?
>>> <cut>
>>>> I will answer any good faith questions in good faith.
>>>
>>> Oh, good. Here are a few questions to clarify what you mean.
>>>
>>> Q1. Did you mean to exclude a zero infinitesimal offset?
>>>
>> No.
>>
>>> Q1a: If you did mean to exclude r[0], do you write the point at, say,
>>> exactly 1 as 1.0 with no [...] part?
>>>
>>> Q1b: Again, if so, would you mind writing that as 1[0] to give all
>>> points the same form?
>>
>> 1[0] is the same as 1.
>>>
>>> Q2. What numbers can go in the "Real_Part"? All the examples you have
>>> given use a "whole number". Can it be any real as the name
>>> implies?
>>
>> Any identifiable number can go into the "real" part.
>
> That's unfortunately unclear. What you mean by "identifiable number"
> needs to be clarified. I have a guess, but that's no way to proceed.
>
> My guess: lbh zrna nal ahzore gung pna or jevggra qbja va fbzr svavgr
> sbez yvxr "7.1" naq "fdeg(2)". Vs fb, gur frg bs CB-trbzrgevp cbvagf vf
> vaqrrq pbhagnoyr.
>
>> π[1] is one geometric point to the right of π.
>
>>> Q3. Given two arbitrary points r1[i1] and r2[i2], is r1[i1] = r2[i2] if
>>> and only if r1 = r2 and i1 = i2?
>>>
>> I think so. I can't see any exception right now.
>
>>> Q4. You say
>>>
>>> "0.0[1 to ∞] forms a bijection between the positive integers and
>>> all of the positive numbers through all of the positive points on
>>> the number line"
>>>
>>> What n makes 0.0[n] equal to the point I would call 1?
>> There is no bijection there.
>
> Hm... that is not a use of words that makes sense. But Q4 is answered
> by your answer to Q3: there is no n that makes 0[n] equal to 1[0]. Can
> you confirm this?
>
> The text I quote in Q4 speaks of a bijection but in terms that are
> non-standard. (It's the "through" in "a bijection between X and Y
> though Z" that is odd.) Can you state, clearly, what sets are in
> bijection? Both "the positive numbers" and "the positive points on the
> number line" are ambiguous terms. The only clear set is first: { 0[n] |
> n ∈ Z+ }.
>
>>> Q5. If all the points on the number line can written in the form 0.0[n]
>>> for some n, what is the point of having your "Real_Part"?
>> To identify it as an infinitesimal number of the infinitesimal number system.
>
> This seems like an odd reply because your answer to Q3 says that one
> can't write 1[0] as some offset from 0, so you should be rejecting the
> premise of my question. And, because there is no n for which 0[n] =
> 1[0], different "Real_Part"s are essential simply to be able to write
> all the points.
>
>> To specifically label the base part of the base/offset because there is
>> no bijection between reals and infinitesimals having an offset.
>
> Again, can you state clearly what two sets do /not/ biject? One is
> clear: it's just R, yes?, but what "infinitesimals having an offset" are
> is not, in part because Q2 is not yet answered.
>
>>> Q6. If, say, 0.0[23] and 0.0[24] are both points on the number line,
>>> how do you represent the point half way between them?
>> As you already know this is defined as non existent.
>> There is no one half of infinitesimal.
>
> I thought you were questioning Cantor's results about the continuum?
> The continuum has the property that between every two points, there are
> infinitely many more points. If your points are a subset of the
> continuum, they may very well be countable, but no one will care about
> them.
>
>>> Q7. One can compare and do arithmetic with real numbers. Are any of
>>> the usual relations and/or operations defined for yours? If so,
>>> how are these defined:
>>>
>>> Q7a. r1[i1] < r2[i2] ?
>>> Q7b. r1[i1] + r2[i2] ?
>>> Q7c. r1[i1] - r2[i2] ?
>>> Q7d. r1[i1] * r2[i2] ?
>>> Q7e. r1[i1] / r2[i2] ?
>>>
>>> (I covered r1[i1] = r2[i2] in Q3.)
>>
>> I laid this all out previously.
>
> Where did you define r1[i1] < r2[i2]? I did not see it. In fact I did
> not see the definitions of any of these. Just cut and paste the
> definitions so they are all on one place.
>
>> There is no bijection between reals and infinitesimals having an
>> offset. Single infinitesimal quantities are indivisible.
>>
>> Relations and operations between infinitesimals
>> generally operate on their base part and offset part separately.
>
> Can you give the definitions, please?
>
>> Likewise with mixed type relations and operations.
>
> What are mixed type relations and operations? The obvious meaning
> (operations between your point and other numbers) makes no sense because
> of the "likewise".
>
>>>> It really seems to me that your good faith answer to my
>>>> above questions would necessarily prove that I am correct.
>>>
>>> But your (1) and (2) combine to make a hypothetical. The answers can't
>>> prove anything, so my answering them would be as pointless and your
>>> asking them. Sure, if there /were/ a bijection between all the points
>>> and the integers, they (all the points) would be countable. However,
>>> either your points are not all the points, or there is not a bijection
>>> between your points and the integers.
>>
>> It seems that I did specify a notational system that does form a
>> bijection between integers and the set of points on a number line.
>
> Answering my follow-up questions will clarify what sets do and do not
> biject.
>
>> The common notational convention used for intervals make it clear
>> that the key notion of my system: [immediately adjacent points on
>> a number line] do exist, and can be uniquely identified.
>
> No. The common notational convention used for intervals shows that
> adjacent points do not exist. While [0, 1] has a least member, (0, 1]
> does not. You can remove the first number in [0, 1] to get (0, 1] but
> you can't remove the first and second.
>

https://en.wikipedia.org/wiki/Straw_man
Intentional deceptive straw_man WHY LIE? Does lying give you kicks?

IT IS CLEAR THAT I HAVE BEEN ALWAYS USING REAL ENDPOINTS.

YOUR SWITCH TO INTEGERS IS PRETTY DAMN STUPID.
DID YOU THINK YOU WOULD GET AWAY WITH THAT?

The first point in this interval: (0.0, 1.0] very obviously
comes immediately after the first point in this interval: [0.0, 1.0].




--
Copyright 2019 Pete Olcott All rights reserved

"Great spirits have always encountered violent
opposition from mediocre minds." Albert Einstein