I have looked for a precise definition of the “…” or ellipses notation as used here. Didn’t really find a formal one. Though maybe someone could. Anyway the notion I think as general understood is the same as what were refer to as the “countable” infinity. In more familiar terms this would be the infinity we think of with the set of natural numbers [0, 1, 2, 3, … ] One can always name a number bigger than any number given, it “never ends”, is infinite. There are larger infinities. Cantor showed that the set of rational numbers (all numbers that can be expressed as a ration of two integers n/m is actual equal to the set of natural numbers in size as unintuitive as that may seem. This is his famous proof by the diagonal method. He also show that the set of real numbers which include things like pi and 2^(1/2) and .555… is a larger infinity than those of the “countable”. In comparison, you could think of it this way… I showed how there is an infinite number of halfway points between any two given points. But if you were to consider ALL the points, not just the half way points in moving from point A to B, this would be the a larger “continuous” infinity of the reals. The mind blowing thing is to think about the fact that any given line (as taken to be infinitely long like the real number line), contains a number of line segments starting from any arbitrary point and ending at another arbitrary point, say 3.2455462… to 4.000.. which is uncountable infinite in itself, but then each those segments could be dividing into an uncountably infinite set of points as well. Anyway… if you want to learn more I would highly recommend the following link: THE SCIENCE AND PHILOSOPHY OF THE INFINITE
So back to the question of what infinite is .999… ? As far as I know we would normally consider this just the plain “countable” infinite. Note the word “countable” is somewhat misleading as to suggest you could actually count them in the normal sense of counting. If you were to say, that .999… should extend beyond that “countable” infinity, then I would say you would now really only be talking about infinitesimal numbers, you be progressing through mind boggling layers of countable infinities onto higher and higher infinities. The analysis of that is beyond my grasp to explain and certainly one that I am sure is as debatable as anything could be. I see no reason to go beyond countable infinity here for our description of .999… , but if you wanted to I suppose you could. But I could also then extend the infinite series of half way points in my proof above to those infinities as well and continue my 1-1-1 mapping. How math works at those levels… I don’t know. Is it possible the math doesn’t work like ordinary math and breaks down and thus my proof falls apart, possibly. But it wouldn’t necessarily prove the anti-thesis either that .999… = 1. If I say all cats are dogs and all dogs are mammals. Therefor all cats are mammals. This proof is false because all cats are not dogs. But that does not mean the conclusion is wrong.
In short I think it makes no sense to take the concept of .999… beyond the “countable” infinity. One could, I guess… if you do go that way, please let me know when you get back and how things turned out.
Rambing away here. 6:30am, cold and evil morning outside.
As suggested above - this is confusing the definition of “end”.
Given we already know that you can move and Zeno’s paradox is fallacious, it doesn’t seem sensible to persist with this particular line.
I can construct an infinite series that converges to each and every one of the intermediate points in your base infinite series. This means that you come to an “end” of an infinite series at every point in your series. So what? As I wrote earlier - there are an infinite number of infinite series that converge to B. (And since they may have real terms, there are uncountably many of these.) When I reach B it appears that I have reached the “end” of every one of the these series. So there are uncountably many “end” points , right at the end of the interval. Again - So what? There are uncountably many points past B. Each of these can have an uncountably many infinite series converging to them. I can also have an uncountable number of series that converge from above. So what? The existence of infinite series that converge to numbers has no relationship to anything else. It is a mathematical curiosity as far as we are concerned. When I move I pass though an uncountable number of points. There is absolutely nothing special about these points and the manner in which it is possible to construct a geometric series that converges to them. There is most certainly nothing special about the set of points that forms a specific geometric series that converges to just one of the points on the line. That is just a countable subset of points of an uncountable set. And there are an uncountable number of ways of constructing these subsets. So why is any of them special?
When I move from A to B I cover an uncountable number of points. Every one of these points has an uncountable number of series that converge to it, both from above and from below, starting in the interval (A,B). So what?
If you apply the Zeno converging to B, you should probably also apply the Zeno that proves you can’t even begin moving. The series that converge to A from above. All the above applies again.
Our ability to generate expressions that select points from the uncountable set of reals on a line has no bearing on movement, or much else. An expression that selects the members of a single geometric series from this set even less so. The OP’s question is simply whether B is a member of the set so selected.
I am sorry, but I think you confusing the Limit of a converging series with some known proof on the matter. The solution to Zen’s paradox does not require limits it requires the acceptance that one can and does complete (come to an end) of an infinite set.
Let me put more plainly what do you mean by converging without the context of a Limit?
Sorry you seem to have missed the point of my post totally. The entire point is to note that there is absolutely nothing special about the numbers. Nothing. Zeno simply is irrelevant. Limits or not. There are an infinitude, an uncountable infinitude, of series that work just like the 1/2 + 1/4 … or 0.99999… And they cover the entire interval.
You don’t need to solve Zeno, because it doesn’t need solving. It simply is one of an uncountable number of ways a way of selecting points from an uncountable set. The question of whether 0.9999 = 1 is simply whether 1 is a member of the subset or not. But Zeno simply has no applicability.
Call it shorthand for the OP problem. For every point of the line it is possible to construct an uncountable number of questions of the form “Is the point a member of the subset of points of the geometric series that the sum of a geometric series would suggest that point is the end value of?”
[QUOTE=erik150x;15368986
You asked me what’s the second to last number in the sequence? Okay, what about the 2nd to last? Or 3rd to last, or (10^1000000000) last number just to pick one ridiculously large number (in comparison to most we usually speak of), but yet even I could ask what’s the 10^(number of particles in the known universe), would this still be 1? By your logic it must be?
[/QUOTE]
Yes, this is correct. The problem is that there is a significant gap between finite and infinity. They are fundamentally different, it isn’t just that infinity is a really really large number, it is a whole different class. In the Zeno paradox problem or the 0.999… problem you note that each successive step gets you closer to 1, but you always end up less than 1. This is true for any finite, and so you conclude that it must be true for infinite as well. But the truth is that you make a jump as you go from the finite to the infinite, and in the process of making that jump you hit the value one. Working backwards from the infinity will still keep you in the realm of infinity and no matter how many steps you take back, you will never track your way back to finite, and never track yourself away from 1. The … is just a notation for I have made that jump between finite and infinite.
You are at point A literally, physically. You want to go to point B which is lets say 1 mile away.
In order to reach point B you would have to cross the half way point? Yes or No?
Once you reach (arrive at are existing at) the half way point between point A and B. Now simply have the same issue only of and admittedly shorter distance. But you can always divide the distance in half for ever and ever and ever without and end. How can you ever complete the journey? You can get closer and closer and closer, but you never get to arrive at are existing at point B. Now you can shorten the original distance A to B of 1 mile as short as you like unit the problem becomes being able to move any distance at all.
Now not using limits, how do you solve this problem of end endless set of halfway points which must be completed, to arrive at to reach it end?
I would like to know your solution WITHOUT limits.
After that I think we can think continue the discussion of whether Zeno is relevant.
“Never” doesn’t validly follow here. It’s true that at no point during the process of reaching the halfway^n points in succession do you arrive at B. But that does not mean you “never” arrive at B. There is a point later than all those halfway^n points, and at that point, you arrive at B.
Indeed your interpretation COULD be right. But one could also argue that the jump to 1 requires at least an additional step of 1/infinity. Or potentially one must progress though even higher levels all the way the end of all infinities. There is no evidence to support either that I know of. So I have never said that it is impossible that .999… = 1, only that there is no proof. One can easily take the position that .999… literally gets you infinitely close to 1 without reaching it.
I think one of the biggest problems you’re having here, erik150x, is that you think you can do arithmetic with ‘infinity’ as if it were a number. It is not. It only has meaning, in the math we are talking about, in the context of limits. The reason that 1/0 and 1/infinity are undefined is that arithmetic cannot be done with infinity without introducing strange results that would destroy the usefulness of all the other things we use math for.
So if you don’t accept limits, you should forget about using infinity.
So its okay to just merrily skip though a multitude of imponderable infinities in the real world, as you suggest you just do it. But in math one can not achieve the same and skip though a multitude of infinite 9s?
“Infinitesimals have a long and colourful history. They make an early appearance in the mathematics of the Greek atomist philosopher Democritus (c. 450 B.C.E.), only to be banished by the mathematician Eudoxus (c. 350 B.C.E.) in what was to become official “Euclidean” mathematics. Taking the somewhat obscure form of “indivisibles,” they reappear in the mathematics of the late middle ages and later played an important role in the development of the calculus. Their doubtful logical status led in the nineteenth century to their abandonment and replacement by the limit concept. In recent years, however, the concept of infinitesimal has been refounded on a rigorous basis.” - Continuity and Infinitesimals, First published Wed Jul 27, 2005; substantive revision Mon Jul 20, 2009
"inasmuch as the number of terms in nature is infinite, the infinitesimal exists ipso facto.” - Johann Bernoulli
I suppose one way to view the argument of .999… = 1 or not is to say it is an argument over how far the 9s go. If you say it’s “just” the countable infinite, then it could still leave us a little short of 1, like 1/infinity short. If you want to define .999… as progressing though all possible levels of infinity then I might have to accept that it does = 1. Simply describing it as an endless series of 9s may not be definitive enough (as even the countable infinite is usually considered “endless”).
I’m not sure if this is a question or a statement, or a rhetorical question.
You still didn’t answer my questions above - what does “completed” mean, and why must we perform this completion?
Swap the viewpoint about. As I walk I cover an uncountable number of points. Any question about labelling or otherwise selecting from this set of points as I travel isn’t part of my travel. Perhaps some Platonic universe keeps track, but it has nothing to do with me or my movement. Given that it is possible to construct specifications for an uncountable number of these subset selections, I guess this Platonic universe simply has an uncountable set of subsets of the points I pass though. Makes no difference to me. It is just way of specifying the selection of points. I am not required to know about these selections in order to move. So I can’t see how they stop me from moving.
“…in which the use of limits in defining the basic notions of the calculus is replaced by nilpotent infinitesimals, that is, of quantities so small (but not actually zero) that some power—most usefully, the square—vanishes.”
I can’t understand how you can accept a nonzero number so small that its square is zero, but still have trouble with 0.9…=1.
