You explained how to identify a series of points on the number line 0, .5, .75, .875, and so on. Then you said “reverse the process,” but it is not clear what this means. I know how to identify the points in your series beginning from zero and proceeding to greater numbres. I do not know how to identify the points in your series beginning from one and proceeding to lesser numbers. The explanation of the former does not yield an explanation of the latter. Can you tell me how to identify the first point I will come upon after one when I “reverse the process”?
ETA: I assume you’re familiar with the ideas of closed and open intervals. The set of numbers you’ve defined (0, .5, .75, .875,…) is closed on the left, open at the right. By definition, the way we normally do math, this means there’s no “last” item in the set (i.e. rightmost as listed above) and so, if you reversed the process, by definition, there’d be no “first” item in the set.
To talk about a first one, then, you must be starting from different assumptions than those which yield “closed” and “open” sets as mentioned above. But that means your task is to tell us what those assumptions are, and how they help to identify a “last” member of the set listed above.
Like I wrote - you solve Zeno by putting the time intervals in a one to one relationship with the the distance intervals. If you can make the time intervals converge to the end time you can move. But this involves assuming the exact thing you are trying to disprove. The fact that motion is possible past the end of the line thus disproves your assertion. Why do you allow the time intervals to reach the time needed to end the interval but not the distance intervals?
Now wait, Allotrope, are you contradicting the generally accepted solution to Zeno’s paradox? Why are you saying he has to take an infinite amount of time to do it? Remember that each time interval is decreased by half as well as each space interval.
First I want to clarify my very roughly posed thought experiemnt hear.
There is no need to actually reverse the process. We just need to find the last (infinite numeral) which i am calling points. And being that one must cover at least an infinite number of points by the resolution of Zeno’s paradox in any movement which are noted as 1/2 + 1/4 + 1/8 … + (1/infinity)
1/infinity here equals the end of my walk. And where I may find the last 0 and last 9 of the two series:
1.000…
0.999…
which as unintuitive as it may seem to find the end of an infiite series, I never the less have described the process for. Seeing for your self when I reach the end merrily mark my 0 and my 9 respectively. The rest is but folly to caluculate since its all 0’s on top and 9’s on the bottom.
I wasn’t aware of that actually, but no, I was just fucking with him.
He seems to think that just because a line segment is discrete and contains an infinite number of dimensionless points that therefore the infinite set of points is also discrete. And I’m getting to the point were I think he is being willfully ignorant in many regards.
I mean how difficult is it to grasp that an infinite series has no last entry or member? I mean really. How difficult can that be?
No, with the words above you have just provided the inconsistency that proves that the proof is wrong. You are asserting that an infinite sequence has an end, indeed you are assuming that it has one. Since by definition it does not, the rest of the derivation is false. Either that, or you are simply assuming your result - which is also apparent.
Okay, I see that you are using a mathematical system in which 1/infinity is a defined quantity. In “regular math” (so to speak–I mean by this the math the average educated non-specialist knows) that is not a defined quantity.
You can certainly build mathematical systems in which 1/infinity is defined, and in which, as a result of this with other assumptions, 1.000… minus 0.999… is something other than zero.*
But you haven’t built it. You haven’t defined your assumptions, and you haven’t demonstrated consistency, etc. That will take a lot of work, but unfortunately, til you’ve done it, you can’t really have a fruitful discussion with those of us who are using “normal math” on the matter. We’re in a way speaking different languages–or at least, dialects so different it can be very hard to communicate.
*As I said in a previous post, this amounts to saying that your “0.999…” and/or your “1.000…” turn out not to refer to the same thing the rest of us mean by those two expressions. Because the rest of us use those two expressions to refer to one and the same object, not simply by assumption but as a result of valid reasoning from the assumptions we do have.
No, I gave you the solution. It is you that denies the solution by denying the convergence of the distance intervals. You allow time to reach the limit, but not the distance. So you are happy to allow limits in time, but not in distance. This is why the proof fails. You either let the sequence converge to the limit or not - but you can’t have it both ways - which you are trying to do. I can move happily, mostly because believe that the sum of 1/2 + 1/4 + 1/8 = 1, due to this I can move past the time when I reach the end of the line. You are not sure why you can move past the end of the line, just that you can.
You know, this might not be quite right. You may mean the same thing as us in your system by those two expressions, and it may instead be that you have a different way of dividing up numbers into “identity classes”–classes of numbers that are treated in your system as identical.
I could treat all even numbers as identical in some new system, for example, and so would say 2=4. But that wouldn’t mean I meant something different by “2” and “4” than “normal math” means by those expressions.
So, never mind what I said there in that last bit.
The words above are not part of my proof, mearly an afterthought and observation. Let’s discuss where my proof is wrong. I am happy to. I think there is likely an error, But i cannot see it.
I start with what is basically Zeno’s paradox, which we are able to solve thanks to Limits. This allows to reach the end of an infinite sequence of points does it not? Are there not an infinite number of 1/2s form point a to b? and we all know we can move therefore we must be able to reach those points and the end of those points most importantly. Which is all I want to find, because my points also represent the 0’s and 9’s in the simple subtraction of:
1.000…
0.999…
if wh know all the points up to and including the infinite’th one? then we can do our simple subrtaction and see the answer is:
0.000…1
as absurd you might think this?
how do you refute it?
And Limit theory, which thankfully allowed us to find this answer actually predicts:
0.000…0
how odd? How contradictory? Clearly there is an erro here, but where?
Unfortunately, I don’t have the expertise. And anyway, it’s not so much that there’s a flaw in your proof–rather it’s that no one can know if there’s a flaw in your proof until you’ve laid out all of your assumptions and all of your logical steps. That’s not something I can do for you because I don’t know what your assumptions are. If I had a background in math, I could help draw them out, but I don’t, so I can’t.
