I think a reasonable position is “I personally do not know how to make sense of this notation,” but are you claiming that “nobody knows how to make sense of this notation”? Would you be willing to make a bet to that effect, and at what odds, for how much money?
Thomas, I think you may be misunderstanding what Qiaochu_Yuan is trying to do here, which is not to argue that 0.999… actually might (for all he knows, or for all you know) be something other than 1, nor to argue that any particular other non-standard[1] construction actually might (for all he or you know) have a coherent meaning.
Rather, he is saying: someone who hasn’t come across this stuff before might reasonably not see any important difference between these constructions (if the difference seems obvious to you, it’s only because you have seen it before; it took mathematicians a long time to figure out how to think correctly about these things) and adopt parallel attitudes to them. This would be reasonable, and rational in any sense that doesn’t require something like logical omniscience.
It seems as if you are arguing against the first sort of claim (which I believe QY is not making) rather than the second (which I believe he is making).
[1] In the sense of “not usually used in mathematics”, not that of “model of real analysis with infinities, infinitesimals and a transfer principle”.
What I am trying to achieve here is to present the current official math position. Not that I agree with it—it’s too generous to infinities for my taste, but who am I to judge—but I still want to explain this official math position.
It is possible that I am somehow wrong doing that, but still, I do try.
What I am basically saying is that because there is no finite positive real epsilon, such that 1-0.9999… would be equal to that epsilon, therefore those two should be equal.
If they weren’t equal, there would be such a positive epsilon, which would be equal to their difference. But there isn’t. If you postulate one such an epsilon, a FINITE number of 9s already yields to a smaller difference—therefore contradicts your assumption.
This is the official math position as I understand it. I might be wrong about that, but I don’t think I am.
Qiauchu_Yuan is a mathematician and I’m quite sure he’s familiar with the “current official math position”. I don’t think your presentation of it is wrong, but I think it’s unnecessary in this particular discussion :-). When you say there are no real numbers between 0.999… and 1 and therefore the two are equal, you are not disagreeing with QY but with a hypothetical person he’s postulated, whose knowledge of mathematics is much less than either yours or QY’s.
I think a reasonable position is “I personally do not know how to make sense of this notation,” but are you claiming that “nobody knows how to make sense of this notation”? Would you be willing to make a bet to that effect, and at what odds, for how much money?
I am saying you cannot write …9990 - the decimal point, then an infinite number of 9s and then the last zero!
Okay, perhaps you can in some other axiomatic system. But not for the ordinary real numbers.
Sure. What is different about the situation with 0.999...? How do you know that that is a sensible name for a real number?
0.999… is the limit of 9/10+9/100+9/1000+...
...9990 is what?
Thomas, I think you may be misunderstanding what Qiaochu_Yuan is trying to do here, which is not to argue that 0.999… actually might (for all he knows, or for all you know) be something other than 1, nor to argue that any particular other non-standard[1] construction actually might (for all he or you know) have a coherent meaning.
Rather, he is saying: someone who hasn’t come across this stuff before might reasonably not see any important difference between these constructions (if the difference seems obvious to you, it’s only because you have seen it before; it took mathematicians a long time to figure out how to think correctly about these things) and adopt parallel attitudes to them. This would be reasonable, and rational in any sense that doesn’t require something like logical omniscience.
It seems as if you are arguing against the first sort of claim (which I believe QY is not making) rather than the second (which I believe he is making).
[1] In the sense of “not usually used in mathematics”, not that of “model of real analysis with infinities, infinitesimals and a transfer principle”.
I see your angle now. Perhaps his angle, too.
What I am trying to achieve here is to present the current official math position. Not that I agree with it—it’s too generous to infinities for my taste, but who am I to judge—but I still want to explain this official math position.
It is possible that I am somehow wrong doing that, but still, I do try.
What I am basically saying is that because there is no finite positive real epsilon, such that 1-0.9999… would be equal to that epsilon, therefore those two should be equal.
If they weren’t equal, there would be such a positive epsilon, which would be equal to their difference. But there isn’t. If you postulate one such an epsilon, a FINITE number of 9s already yields to a smaller difference—therefore contradicts your assumption.
This is the official math position as I understand it. I might be wrong about that, but I don’t think I am.
Qiauchu_Yuan is a mathematician and I’m quite sure he’s familiar with the “current official math position”. I don’t think your presentation of it is wrong, but I think it’s unnecessary in this particular discussion :-). When you say there are no real numbers between 0.999… and 1 and therefore the two are equal, you are not disagreeing with QY but with a hypothetical person he’s postulated, whose knowledge of mathematics is much less than either yours or QY’s.