brennan
Argumentative Brit
The result never varies from actually being zero. The interesting thing with 0.999... is that it is changing and what that means. No change: no paradox.
1=.999999...?
- Thread starter Mouthwash
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Leifmk
Deity
But 0.999... is not changing. It is the end result, if you will, after all the change is done.
brennan
Argumentative Brit
I feel your attempt at being obtuse is let down by your language. If it is a result then there must have been a process. 
Mise
isle of lucy
Again, so what? That's irrelevant to the argument you've been making all this time. You've been saying that that 0.999... doesn't equal 1 because the "..." represents a process, not an actual number. And since that process never terminates, any maths we do on it is just an approximation, not an equality or identity. That 0.000... happens to have the same value at every point in the process doesn't mean that 0.000... isn't a process. 0.111... is a process, 0.222... is a process, ... , 0.999... is a process. But 0.000... isn't? How can we know that the result of the process is exactly zero, if we don't evaluate it all? Surely anything we say about 0.000... is just an approximation too?
brennan
Argumentative Brit
Can you offer me an argument that any stage in iterating 0.000... will return a result other than plain ole zero?
And forgive me but I feel you are rather misrepresenting what I have said. Which is that because 0.999... is a series of infinite digits producible via an infinite series expression it seems to me to be a different sort of beast to something straightforward like 0.9 or 0.99. Not that it flat out isn't a number/identity or whatever you want to call it.
And forgive me but I feel you are rather misrepresenting what I have said. Which is that because 0.999... is a series of infinite digits producible via an infinite series expression it seems to me to be a different sort of beast to something straightforward like 0.9 or 0.99. Not that it flat out isn't a number/identity or whatever you want to call it.
Mise
isle of lucy
No, but I'm not talking about the intermediate steps, I'm talking about the end result. If I stop calculating at the 1,000th digit, then it's still just an approximation of the full, infinite process.
brennan
Argumentative Brit
Zero plus zero is an approximation of zero?
It is not that our instruments lack infinite precision, it is reality itself that lacks infinite precision. At the microscopic level, everything has not a true value, but a probability distribution of values. Therefore, it does not make sense to give anything real with infinite precision. Numbers thus are purely abstract concepts and we can make up rules as long as they are consistent.
So you never heard of he electric displacement field D? Whoever told you about B and H should have also told you about D and E.
brennan
Argumentative Brit
Well you can bring that up with my university and every physics textbook i've ever seen in the UK, none of which mention either H or D. H seems to be more of an engineering term and usage seems to vary across the Atlantic. In the UK it seems we have little use for two terms for either an electric or magnetic field.
Wait, i've found D in one book. It merits three sentences!
Wait, i've found D in one book. It merits three sentences!
Brennan, you still haven't answered why 0.999... would be rather a process than a number. Isn't 6 a process? It's 4+2. Why wouldn't a trivial process be process?
Also, the definition of the limit is:
So by definition a limit must be a real number. On the other hand, 0.999... was defined to be a limit of \sum_{k=1}^n 9/10^k. So 0.999... must be a real number.
Your intuition might be doing the trick in saying that a sum of a series is just like any other sum, it's just infinite. That's not the case. If you start summing up infinite amount of numbers, you'll never get there. That's why it's defined to be the limit of the finite sums. That you can determine in finite time.
Also, the definition of the limit is:
So by definition a limit must be a real number. On the other hand, 0.999... was defined to be a limit of \sum_{k=1}^n 9/10^k. So 0.999... must be a real number.
Your intuition might be doing the trick in saying that a sum of a series is just like any other sum, it's just infinite. That's not the case. If you start summing up infinite amount of numbers, you'll never get there. That's why it's defined to be the limit of the finite sums. That you can determine in finite time.
The Wikipedia article on this topic discusses some of the common misconceptions that have come up in this thread.
Mise
isle of lucy
That's what your argument implies, yes. (Well, more accurately, Zero is an approximation of 0 + 0 + 0 + ....)
Mise
isle of lucy
Well precisely - I'm not happy with it either. But if "..." represents a process that never terminates, and any maths we do to judge what might happen should we complete that process to infinity is merely an approximation, then 0.0 + 0.00 + 0.000 + ... merely approximates to 0.
Hm.
Well, if axioms are set arbitrarily (or non-intuitively) it is to be expected that people who do not work with those arbitrary axioms do not readily accept a conclusion that is merely following that closed up circuit.
A bit like having special axioms so as to go around the incomplete set theory.
Will have to look at the links for the axioms re the Real numbers and limits with the use of Epsilon, but 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.
And surely for any thing to be termed a process, there has to be at least something concretely juxtaposed (even in some level of examining both) to a process as a distinct object that is set. So if 0.999999... is a number, what is a process built up from in current axioms used re this issue?
Well, if axioms are set arbitrarily (or non-intuitively) it is to be expected that people who do not work with those arbitrary axioms do not readily accept a conclusion that is merely following that closed up circuit.
A bit like having special axioms so as to go around the incomplete set theory.
Will have to look at the links for the axioms re the Real numbers and limits with the use of Epsilon, but 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.
And surely for any thing to be termed a process, there has to be at least something concretely juxtaposed (even in some level of examining both) to a process as a distinct object that is set. So if 0.999999... is a number, what is a process built up from in current axioms used re this issue?
brennan
Argumentative Brit
Mise you are trying to compare a paradox based upon an infinite series whose result changes with every iteration with a single value to which you have attached an infinite series of non-changes. The inaptness of the comparison is obvious. Carry on repeating yourself if you wish. Zero does not behave like other numbers. You know this.
@Brennan:
By your logic still, 0 should be process and not a number, since it's 0+0+0+...
Also, you have disregarded completely what has been said above: that the sum on an infinite series is by definition a limit of a sequence of real numbers, and that a limit of a sequence of real numbers is by definition a real number.
If you insist on that 0.999.. would be "a process", can you elaborate what "processes" are, and how they differ from real numbers? What does count as real numbers according to you? What counts as a process?
It wouldn't be stupid though, to take time and learn the maths instead of arguing against it. It may take a few weeks to understand these things, and some people don't get it at all. But still, you obviously haven't even tried.
@Kyriakos:
The axioms of the real numbers aren't set arbitrarily nor counter intuitively. Quite the opposite. The point I was making before was that they are set. You can't say something is an axiom if it isn't.
Aside of that you can come up with your own things and set the axioms for them. If the axioms don't contradict each others, they need not to be intuitive to qualify as "real maths". A whole different thing is whether anyone will be interested on what you're creating then though.
All the rest of your post... Well, it shows that you really should rely on something else than your heritage. Sorry, but you know next to nothing about maths. Instead of people explaining it to you in this thread, you would be better of by borrowing an undergraduate book on analysis and reading it slowly, doing the exercises. Or better yet participate on some course, since the exercises can be difficult and the difficulties pile up when you progress.
That's a good hint for Onejayhawk too.
By your logic still, 0 should be process and not a number, since it's 0+0+0+...
Also, you have disregarded completely what has been said above: that the sum on an infinite series is by definition a limit of a sequence of real numbers, and that a limit of a sequence of real numbers is by definition a real number.
If you insist on that 0.999.. would be "a process", can you elaborate what "processes" are, and how they differ from real numbers? What does count as real numbers according to you? What counts as a process?
It wouldn't be stupid though, to take time and learn the maths instead of arguing against it. It may take a few weeks to understand these things, and some people don't get it at all. But still, you obviously haven't even tried.
@Kyriakos:
The axioms of the real numbers aren't set arbitrarily nor counter intuitively. Quite the opposite. The point I was making before was that they are set. You can't say something is an axiom if it isn't.
Aside of that you can come up with your own things and set the axioms for them. If the axioms don't contradict each others, they need not to be intuitive to qualify as "real maths". A whole different thing is whether anyone will be interested on what you're creating then though.
All the rest of your post... Well, it shows that you really should rely on something else than your heritage. Sorry, but you know next to nothing about maths. Instead of people explaining it to you in this thread, you would be better of by borrowing an undergraduate book on analysis and reading it slowly, doing the exercises. Or better yet participate on some course, since the exercises can be difficult and the difficulties pile up when you progress.
That's a good hint for Onejayhawk too.
^ ;_;
Ok. So now nothing before the late 19th century counts as math? Cause i am pretty sure i know a considerable amount of math stuff, even if i learned it while focusing on philosophy.
Not seeing why you are so confrontational. GO BACK TO ASIA!
Also try to imagine how good an argument it would be if someone was to not explain something in the context of an ongoing thread, and then just say 'well go read it, lol', cause that is what you do. I thought that actually knowing something well enables one to explain it in normal terms and present its logic.
Ok. So now nothing before the late 19th century counts as math? Cause i am pretty sure i know a considerable amount of math stuff, even if i learned it while focusing on philosophy.
Not seeing why you are so confrontational. GO BACK TO ASIA!
Also try to imagine how good an argument it would be if someone was to not explain something in the context of an ongoing thread, and then just say 'well go read it, lol', cause that is what you do. I thought that actually knowing something well enables one to explain it in normal terms and present its logic.
Mise
isle of lucy
That's super handwavey. If your argument works in nine out of ten cases, but doesn't work for the tenth, then it doesn't work. This isn't economics: if you have to add a "except in cases where it doesn't" get out clause in order to make the argument work, then the argument doesn't work. Zero is clearly a counter example to your argument.
Atticus' and Leifmk's argument requires no such get out clause; it works for 0.000... just as well as for 0.111..., 0.222..., 0.333..., 0.444..., 0.555..., 0.666..., 0.777..., 0.888..., and 0.999.... Why should anyone believe your "approximation" argument if you have to say "oh well zero is a special case so I don't have to worry about that". Atticus and Leifmk don't have this problem. You do. And yes, I'll continue to repeat your own arguments back at you as long as you continue to make them yourself.
Now note the second line: 9 x 1/9 = 9/9 = 1. The actual problem, of course, is with the first line, which assumes 1/9 equals 0.999... But the only thing that 1/9 equals is 1/9. As noted in the same article:
I've always learned that the first rule to attempt any kind of proof is proper definition. In this case, the very thing that needs proving (1=0.999...) is already assumed true before proving.
This continues with the second example given:
However, as anybody will recognize 3/9 = 1/3. This is not in the proof, however, as it would break the 'consistency proof'. Well, that it certainly does.
This one is better:
Which would then prove that 1=0.999. However, in line 1 x was already defined as 0.999... And the proof only results by removing 0.999... from the equation. (I would also leave line 5 out as superfluous.)
The problem, of course is if 1=0.999... why are they two different numbers? It's not so much counterintuitive, but rather illogical, as well as inconsistent.
(That's not to say, of course, that for all practical purposes 0.999... equals 1. Proof or no proof.)
