erik150x, I will point to this again because you did not respond to it:
ugh… And dear, Sir, since there is no end to the sequence A, AA, AAA, AAAA, … how do you propose I finish reading those in order to get to AB?
Ugh, you guys are killing me.
The point of the paradox is not to convince any one that we cannot move or rulers don’t exists. So why are you asking me if I buy it? It’s irrelevant.
Who said anything about reading them?
I am pointing out how an endless sequence can nonetheless be followed by something which comes after each of its items. In the case of movement across the room, there is an endless sequence of events which, nonetheless, is followed by another event (your having reached the other side of the room). And there is no problem with this. That is all I am noting.
What would you say was the problem with the reasoning in the context of rulers? I would say the problem is that it is nonsense; there is no rule of logic which will take us from the premise to the conclusion. That’s all I have to say. If the argument does not justify itself, then I don’t have to listen to it.
You don’t show how the reasoning is flawed. You simply state that the conclusion is wrong, and granted it is wrong, but it follows valid from the premise. In order to resolve the paradox one must either show how the premises are false or how the conclusion does not follow from the premisses.
Here would be possible resolutions:
- distance in the real word is not infinitely (endlessly) divisible.
or - it is not impossible to complete an endless series of tasks
Who is claiming there cannot be another infinite set beyond an initial infinite set? why do you bother to point that out, what is the point of you statement?
I would like to know how you ever completed writing the A, AA, AAA, AAA… section of your dictionary since it endless. I do not doubt the AB section exists, but just wondering how you managed to get to it writing it since there would be no end to writing your A, AA, AAA, … section?
No one here thinks that is the point of the paradox, and no one here thinks you buy its conclusion. We are showing that the reasoning you gave (which, again, we know is not your reasoning) is flawed. Since it is flawed, the paradox is thereby resolved.
Indistinguishable is showing that the reasoning is flawed by giving an exactly parallel argument, and pointing out that the parallel argument is obviously invalid. (Hence the “would you buy that?” comment.) (And, in case you don’t already know, I’ll note that this is a standard move called "giving a counterexample to the form of the argument.) Since the parallel argument is invalid, so also the original argument is invalid. And since the original argument is invalid (and, again, we know it’s not your argument), the paradox is resolved.
For a paradox to exist here, there needs to be good reasoning on hand for the conclusion that motion is impossible. The reasoning you’ve given for that conclusion, however, is not good reasoning. It has prima facie plausibility, but Indistinguishable and I have explained why that plausibility is only apparent.
No. This is utterly incorrect. I have not once in this thread attempted to refute the argument by simply disagreeing with the conclusion. I have in fact explicitly repudiated the idea of refuting the argument in that way. Instead, I have stated the flaw in the argument–explained why its conclusion does not follow from its premises–now three times. (Post 610 being the most recent.)
You should probably state what assumption you’re objecting to. I don’t see anything missing. If our system of mathematics states that you can’t do arithmetic with infinity as if it were just some number, then 1/infinity cannot be equal to anything in that system.
However, try responding to my point: If you think it’s possible to have a non-zero value x such that x[sup]2[/sup]=0, why isn’t it possible for 0.9… to equal 1? To me, the first is a very abstract concept, while the second is just an artifact of decimal notation. So if you claim to understand the philosophy of infinitesimals (which I can’t claim myself), you shouldn’t have any trouble with the point of this thread, since it’s far less abstruse. Do you not agree that [sup]3[/sup]/[sub]3[/sub]=1?
(By the way, I currently don’t have internet access at home, and I’m about to leave work for the weekend. So, if you reply to this and I don’t respond, please forgive me. I will respond sometime Monday.)
Correct. It is irrelevant.
Why do you persist in keeping bringing it up?
Read this whole post, please, though it may seem to repeat some things that were said before. As I find myself repeating points, I believe there is some serious miscommunication happening here. You think I’m saying things I’m not saying, and you don’t know that I am saying the things I am actually saying. Here it all is starkly laid out, in paraphrased dialogue form.
You: I don’t actually think that motion is impossible, but nevertheless, here is an argument that motion is impossible. I’m curious what flaw you can find in it. Please don’t refer to limits in your answer.
Me: Okay.
You: Okay, here’s the argument. To move from A to B, you must first move to the point halfway between A and B–call it C. Then, to get to B, you still have to move to a point halfway between C and B. Call that D. And still even then, to get to B, you have to move to a point halfway between D and B. This goes on forever. So, as you can see, you must move through an infinite number of points before you reach B. So, you never reach B.
Me: But in general, just because you must move through an infinite number of points before reaching a point, that does not mean you can never reach that point. It does mean you never reach that point while moving through the prior points. But after you’ve moved through all those points, you’re free to move to that goal point. That is why the conclusion of the argument does not follow from its premises. That is the flaw in the argument.
It seems to depend on the notation! In binary, since 1/3 is an unending repeating numerical expansion, no. In trinary, where 1/3 is just .1, it’s easy!
Understatement of the millennium! (But it’s early; we have lots of time.)
You might say, we move through whole bunches (infinities!) of points all in a “glomp.” We are never obliged to identify, name, or construct the points we pass through. The paradox tricks us by constructing one particular sequence of named points – 1/2, 1/4, 1/8 – through which we pass. But that simply isn’t how real motion works. We pass through points without naming them.
Similarly, in geometry, we accept that a line is a “locus” of points. It isn’t a specified, nominated set of points. We say things like y = 7x + 5 for x > 3 and x < 5. Voila, a line segment, excluding its end-points. No one could ever print out a “roster” of every point in it, but this impossibility doesn’t mean that it is not possible for a car, or a bullet, or a molecule, to follow the path that the function describes.
Correct. So why bother to select some particular set of points?
No it isn’t. An airplane is a physical entity subject to the laws of physics. The number line, and the reals are not physical entities, and are not subject to the laws of physics. They are subject to manipulation by the rules we select for them.
Correct.
And there is your fallacy. You don’t like having an infinite number of points in a finite interval.
My point was that there are an uncountable number of series in the interval, not just yours. Completing the series is hardly something special. Do we have to “complete” every such series? It would seem that if we have to pick one arbitrary series, we must have to complete them all.
But it isn’t different. There are a different number of points, because one is a countable set, and the other uncountable, but that is all. It is the attempt to define infinity in a different manner to that generally accepted (Aleph Null) that causes issues. In either case you pass though an infinitude of points. 0.9999… or 1/2, 1/4, 1/8… is simply a selection from those you passed though.
In the end, any attempt to appeal to motion or Zeno seems to fail on the same problem: a wish not to allow infinity to behave the way it does. To somehow make it behave like a finite number, and thus subject to the ordinary rules. But it isn’t finite, that is by definition. It doesn’t adhere to the intuitive rules of numbers, and trying to make it do so doomed from the outset.
The argument above relies on an unstated assumption that an endless sequence of events cannot be completed. That is true if the events have a (real) non-zero duration. But presumably, you’re allowing for “events” with a duration of zero. In that case, though, I do not know why anyone would think that an endless sequence of such (zero-duration) events could not be completed. What is your argument for that substantive, non-obvious unstated assumption?
Is your argument for this unstated assumption like this?
P3) Completion of a sequence of events implies that the sequence has an end.
P4) An endless sequence of events has no end.
Conclusion) An endless sequence of events can have no end.
If that is your argument, then, assuming “end” means “final member of a sequence,” the flaw is that P3 is false. It is not true that completion of a sequence of events implies that the sequence has a final member. If you think otherwise, you need to justify that substantive and non-obvious premise.
But is that not what you mean by “end?”
Or alternatively, is that not your argument for the unstated assumption I identified in the first argument above?
Okay I think we are finally getting to a meeting of minds here Frylock…
RE: “The argument above relies on an unstated assumption that an endless sequence of events cannot be completed.” - I agree with you 100% that is the flaw.
Let’s go back a few posts to the following post you made as it is also relevant to the statements you are making above I think.
Your post #587:
"If “the end” means “the final member,” then yes I do suggest that we never reach the end.
But we reach something. Just not “the end” in the sense of “the final member.”
There’s also a problem in interpreting your term “reach,” btw. What does “reach” mean to you in this context? "
Fist let’s address what I mean by reach. I don’t understand why this needs clarification, but so be it. By “reach” I mean you are there. In Zeno’s Paradox of motion “reaching” each half way point means you are AT that point. Will further define being at that point by the tip of your nose being inline with the point that marks the halfway point. The tip of your nose would be further defined by the furthest physical point towards the direction of motion that would be considered part of your nose. I hope that clearly enough defines reach for you :dubious:
But the heart of our disagreement may come down to this part:
Frylock: But we reach something. Just not “the end” in the sense of “the final member.”
What do we “reach” as we complete the sequence of an infinite (endless) number of halfway points if not the end as in final member, (yes I would take final member and end to be the same thing… I see no other way to possibly take it).
Let’s take the set of all natural numbers, ok? (Don’t worry we are just borrowing them, we’ll give them back when we’re done)
Your task is to name them off (all natural numbers) 1 by 1 in order, i.e. “1, 2, 3, …”. I am also going to grant you super powers. 
You will be able to name them as fast as you like, in fact infinitely fast. So I don’t expect it should take you long. The only concern I have is how you will know when you are done naming them?
Thanks! I will definitely check it out. I have been currently reading up on hyperreals.
The flaw is the assumption that you cannot add an infinite number of numbers and arrive at a finite number. The “infinite number of numbers” in this case are all the half-way distances. The “finite number” is the total distance traveled. The infinite number of numbers to be added is called an infinite series. An infinite series can sum to a finite number if the terms of the series get progressively smaller fast enough. That’s the crux of the solution. You may want to pick up an elementary calculus book and read the section on infinite series.
First of all no assumption of the sort is being made. The distance is to be traveled is already given earlier on in this thread. Secondly no one is asserting the that the halves do not add up to a finite number (although there is some debate about that being .999… or 1). But at this point we are on a bit of a tangent. If you actually want to be brought up to speed on this I can point you to the relevant issue.
Post #574
Since the scenario is strictly impossible, it shouldn’t be surprising if my answer is strange. It is logically impossible for me, at any point during the counting process, to correctly think “okay, that was the last one–no more numbers to count.” So at no point during the counting process will I know that I am finished. But at some point after the interval within which I was doing the counting, I will find that I am not counting any more. (Even though there will have been no counting number that I missed.) At that point, I will know that I am done.