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From: Richard Damon <richard@damon-family.org>
Newsgroups: comp.theory
Subject: Re: Cantor Diagonal Proof
Date: Fri, 11 Apr 2025 13:53:42 -0400
Organization: i2pn2 (i2pn.org)
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On 4/11/25 1:35 PM, wij wrote:
> On Fri, 2025-04-11 at 12:42 -0400, Richard Damon wrote:
>> On 4/11/25 11:43 AM, wij wrote:
>>> On Fri, 2025-04-11 at 09:07 -0400, Richard Damon wrote:
>>>> On 4/11/25 7:32 AM, wij wrote:
>>>>> On Fri, 2025-04-11 at 09:50 +0000, Alan Mackenzie wrote:
>>>>>> wij <wyniijj5@gmail.com> wrote:
>>>>>>> On Thu, 2025-04-10 at 17:23 -0700, Keith Thompson wrote:
>>>>>>>> wij <wyniijj5@gmail.com> writes:
>>>>>>>> [...]
>>>>>>>>> "lim(x->c) f(x)=L" means the limit of f approaching c is L, not
>>>>>>>>> f(c)=L 'eventually'.  f at c is not defined (handled) in limit.
>>>>>>
>>>>>>>> Correct.
>>>>>>
>>>>>>>>> lim 0.333...=1/3    ... The *limit* is 1/3, not 0.333...=1/3
>>>>>>>>> 0.3+0.33+0.333+...  ... The sequence converges to 1/3
>>>>>>>>> Σ(n=1,∞) 3/10^n     ... The sum converges to 1/3 (or you can use lim)
>>>>>>
>>>>>>>> The limit as the number of 3s increases without bound *is exactly what
>>>>>>>> we mean* by the notation "0.333...".  Once you understand that, it's
>>>>>>>> obvious that 0.333... is exactly equal to 1/3, and that 0.333... is a
>>>>>>>> rational number.
>>>>>>
>>>>>>> You agree "f at c is not defined (handled) in limit", yet, on the other hand
>>>>>>> ASSERTING 0.333... is 'exactly' 1/3 from limit? Are you nut?
>>>>>>
>>>>>> No, Keith Thompson is simply correct, here.  It is you who are nuts,
>>>>>> making unfounded claims about mathematics without having studied it.
>>>>>>
>>>>>> The sentence ....
>>>>>>>> The limit as the number of 3s increases without bound *is exactly what
>>>>>>>> we mean* by the notation "0.333...".
>>>>>> .... is entirely correct.
>>>>>>
>>>>>>> As usual, you need to prove what you say. Or you are just showing yourself
>>>>>>> another olcott, just blink belief, nothing else.
>>>>>>
>>>>>> No, one doesn't need continually to prove standard mathematical
>>>>>> definitions and results.  One could spend the whole day, every day, doing
>>>>>> nothing else.
>>>>>>
>>>>>> It is _you_ who needs to prove your remarkable assertions.  You can't, of
>>>>>> course, because they're false.  What you could do, of course, is to show
>>>>>> a bit of respect for those who have studied and learnt mathematics.
>>>>>
>>>>> I am not interesting to blind beliefs.
>>>>> As I may guess from your posts, your knowledge is essentially 'what people say'
>>>>> without knowing the meaning of words.
>>>>> You may say it is 'standard', 'mainstream'...,etc. But whatever it is, simply
>>>>> no logical proof.
>>>>>
>>>>> Remind you, the so called 'standard', 'mainstream' is on the side of logical proof.
>>>>> They may evolve/change from errors. It is not a static thing and not the source of fact.
>>>>>
>>>>> To save garbage talks, provide your logical proof (as usual, I believe NONE).
>>>>>
>>>>>
>>>>
>>>> Remmeber, the claim is that 0.33333... is 1/3 in the limit, i.e. that
>>>> for any possible epsilon, no matter how small, but still positive, there
>>>> is a point in the sequence of generatation of 0.3333... that all points
>>>> after that will be closer to the limit then epsilon.
>>>>
>>>> We can compute that point, and thus show the limit is that value.
>>>>
>>>> We do that by taking the log base 10 of epsilon, taking its floor (the
>>>> largest integer that is less than or equal to the value), negate it, and
>>>> use that many 3's (but at least one if we start with a big epsilon).
>>>>
>>>> For instance, an epsilon of 0.001 has a log base 10 of -3, so we say
>>>> that all number of the pattern with at least 3 3's are that close.
>>>>
>>>> we can show the example as 1/3 - 0.333 will be 0.0003333... which is
>>>> less than 0.0004 which is less than 0.001, and adding more 3s to the
>>>> number just makes us closer.
>>>
>>> Firstly, we are now talking about limit, nothing to do with "repeating decimals
>>> are irrational".
>>
>> But the limit is what DEFINES what a repeating decimal represents, at
>> least within the normal Real Number System.
>>
>>> Your statement above is sloppy, cannot be verified or refuted. It contains too
>>> many concepts to be defined. So, you just jump to the conclusion (or assertion)
>>> you like. So, no valid proof is taken.
>>
>> Of course it can be verified and thus refuted if it was wrong.
>>
>> What concepts used were not defined in normal mathematics?
>>
>>>
>>> limit only says the *limit* is 1/3, all others are your wishes (or most people,
>>> 'standard', 'mainstream',... whatever you like, it does not matter).
>>> 1. No one disagree that the sequence 0.3,0,33..... 0.3333 can go on forever.
>>
>> Right, and the question is, what value does that sequence, when taken
>> "to the end" become. Since we can't actually do the infinite operation
>> to the end, we define it "in the limit".
>>
>> Of course, in some hyper-mathematics which uses trans-finite values,
>> like the infintesimals, we might be able to come up with other
>> definitions, but then we are not working with what are called the "Real
>> Numbers", but some Hyper-Real number system, which you claim not to be
>> doing,
>>
>>> 2. No one disagree that we can choose an arbitrary epsolon/delta whatever,
>>>      to make the error arbitrarily close to the limit (i.e L or 1/3).
>>
>> Which means that the limit of the sequence, which is the definition of
>> what that notation means, is defined and found.
>>
>>> So, don't make implications that I disagrees with these basics. (your are just
>>> slight, others may even imply I claim 1/3 is irrational. Smear as proof?).
>>
>> But if the value of 0.333... but the definition of the representation
>> *IS* 1/3, and you claim that 0.333... is irrational, you are claiming
>> that 1/3 is irrational. That or  your logic doesn't support the axiom of
>> equivalence (if A = B and B = C, thus A = C)
>>
>>>
>>> In logic language and point of view, the premise (i.e. the sequence 0.3, 0.33...)
>>> does not contain 1/3 (also whatever epsilon/delta you like), therefore,
>>> no possibility a valid logical proof can yield the conclusion 1/3 (QED).
>>
>> But no one says that the series contains the limit, just that the
>> "value" of the series is that limit.
> 
> OK, I will make one exception for you.
> 
> "lim(x->c) f(x)=L" means the limit of f approaching c is L, not f(c)=L.
> f at c is not defined (handled) in limit.
> 
> Do you agree?

Yes, but that is the definition of the limit of a function, not the 
limit of a sequence.

To apply that to a sequence, we need to define the f to be a function of 
an integer, which says which term of the sequence we are talking about, 
and the limit is as the integer reaches infinity.

thus, your f(1) = 0.3, f(2) = 0.33, f(3) = 0.333, and then we have that

0.333... is defined to be the lim(n-> infinity) f(n), which has been 
shown to be 1/3

> 
>> It seems that you problem is that you want to try to say you are using
>> mathematics at least compatible with the "standard" mathematics, but you
>> want to try to define somethings differently, but can't find the axioms
>> you need, so you try to just define the results, which isn't how
>> mathematics works.
>>
>> If you want to be working in a Hyper-Real number system, you need to
>> admit that, and then you can likely find a lot of formulations that get
>> what you want to do, they just admit they are not working in the "Real
>> Number System" as standardly defined.
>>
>>>
>>>>
> 
> Don't be that fast. You will have problem to eat your own words.

So, what is wrong with what I said?
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