You aren't the only one who happens to know what a limit of a sequence is, nor what it is next to a sequence.
Well, if you think the talk about epsilons was cryptic, your notion of the limit is not very well founded. You may not know that, but I and many others in this thread do. That's why I've brought degrees into the discussion: it should give you a hint that instead of arguing against you might do yourself a service by listening.
Another thing that you may not notice is that half of what you write (and this goes to Onejayhawk and in a lesser dergree to Brennan too) is impossible to decipher. That's because you use words from your intuitive understanding, but haven't stopped to think what they mean, or whether that meaning could be consistent. Again, this isn't something you could expect from a layman. Just like you'd expect that I have lesser understanding of Platon's dialogues than you. This isn't any small phenomenon, but extremely overwhelming, for almost every sentence that you've written I could write a page as a response or questions. This is like learning a language or a game: you have to achieve a certain amount of knowledge before you can do anything of substance.
As an example, in the above quote, I can't even imagine what limit being "next to a sequence" could mean.
If I sound hostile, well, sorry, it's unintentional, but it's partly due to the frustration over this gap in communication. To make that gap smaller, you would have to study some maths. Just like I can't speak Greek to you without putting an effort into learning it.
Again, you are being confrontational. Also, what is "you don't seem to understand what Euclid is about" at all? Unless in your view that is another line which perfectly stands on its own without elaboration.
I thought it would be a line that would make you think: "Perhaps I should check into this one more time". The point was that this:
from the stuff posted in the thread it is pretty obvious that this is an issue of closed loop logic, ie it can be perfectly valid in the current math field, but it is not explainable without providing elaborate info on the axioms which close up this logic and limit the definitions used.
holds as much on Euclid's Elements as it does on this topic. Euclid's theorems are valid in his text, but aren't explainable without providing (less) elaborate info on the axioms which close up this logic and limit the definitions used. That's the whole point of the maths, that's why Euclid is considered to be as great as he is.
and then brush off the persistent notes on the argument being very counter-intuitive and indeed seemingly based on particularly set axioms, you should not really have accepted something other from the number of people here who honestly wrote down how it is not evidently consistent and asked for specific info on how it is set so as to be consistent in current axioms (and definitions) used.
What is not evidenetly consistent? What is set to be so as to be consistent in current axioms? I honestly don't understand a bit of this.
Also, I haven't seen anyone pointing out any inconsistencies in the proof. I have seen people stubbornly claiming that 0.999... is a process etc., but I haven't seen anyone point out any inconsistency.
So yeah, i am not at all seeing how you are in the right here. If anything you come across as needlessly arrogant, while in my view that you get asked questions should be regarded as evidence that people wanted to see what you (and i suppose current math systems re this issue) meant.
That's not at all how it looks to me. I don't think anyone posted any questions on it. It looks like no one even read it. On the contrary, I find it arrogant that some posters here claim superior knowledge over professionals without giving a single clue on where that knowledge comes from.
Do not take people who just do not have a uni math degree as inherently unable to follow what you say. I think we would if it was elaborated decently, and was legitimately consistent along with covering the notions discussed (Epsilon, Real numbers re fractions and other stuff, etc etc).
Yes, and for that there are books and courses. This thing can't be taught thoroughly in a forum post. That's just a thing you have to accept. I can't expect you to give me full disclosure on ancient Greek philosophy either. If I want to know, I read a book.
There's also been numerous easy access proofs here, that may not tell the thing exactly, but are enough to satisfy most laymen.
For example the thing of multiplying by ten and subtracting 0.999... is a very fine method.
(Similar method also btw, is useful to find out what a decimal with recurring digits is as a fraction. For example,
0.123123... *1000 = 123.123123...
so 999* 0.123123... = 123
and thus 0.123123... = 123/999.
This is especially funny, since there's been claims in this thread that 0.999... is a transcendent number, and this is an easy way to see that a number is rational).
0.999.. is NOT defined as the limit of a sequence.
Then how it is defined?
That's the minimum amount of information you should provide if you want to claim anything about the number.
0.999... is the sum of an infinite sequence, also called a series. That is 0.9+0.09+0.009+... This series is strictly less than 1, with an undefined difference. It is an element of the open segment (0,1), which 1 is not.
If you wish to show your greater knowledge, demonstrate that 0.9999... is not an element of (0,1) without resorting to limits. Since the limit of the series is not identical to the sum of the sequence, that is your often repeated fallacy.
1. An infinite sequence isn't called a series.
2. What is "an undefined difference"?
2.1. How can you do even the simplest of maths, if there's no guarantee that subtraction works.
For example, to solve x from x+y=1 you should take into account the possibility that y=0.999... Do you write in the margin those exceptions, do you check them at the end of the calculations. Yes, I know that zero does the same thing when you divide by it, but those things
are checked separately).
3. What exactly is strictly less than 1?
4. I have provided already an example of a sequence with elements strictly less than 1, but whose limit is 1.
5. Why shouldn't I resort to limits?
6. Series doesn't have limit.
7. Sum of a series is
defined to be the limit of the partial sums. If you have other knowledge, please tell us the definition. You're also welcome to give reference to some real mathematical work that uses your definition.
But most of all, tell us what do you think 0.999... means. How do you think it's defined?
).
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