I am five feet away from the door right now. So far as I know, there is an infinite amount of feet between me and the door (4.9 ft, 4.99 ft, 4.999 ft, etc) which means that there is an infinite amount of space separating me from the door. If that is the case, how is it possible for me to get to the door?
It’s not an infinite amount of feet, it’s five feet.
It’s true that you could subdivide five feet into infinitely many, infinitely small intervals, but that’s just a mathematical exercise. There’s no connection to reality.
Apart from your incorrect assumption that there are an infinite number of feet involved, you are simply restating Zeno’s “Paradox.” The “solution” to the supposed paradox resolves your issue as well.
Zeno’s paradox explains all that I was wondering, thanks.
Basically, the answer hinges on the fact that an infinite series can sum to a finite value. For example, 0.5 + 0.25 + 0.125 + … sums to 1. There’s a nice pictorial “proof” of this where one half of a square is recursively halved.
ETA: my post may be ambiguous—not all infinite series converge, of course.
And if you are looking at it that way, if you take a step 2.5 feet in length, you can divide it into an infinite length as well, so the question is not “how can I traverse the infinite distance between me and the door?” but “how can I keep from overshooting it?”
Thing is:
Its not a trick, they’re really exactly equal. Although, to quote from the above article:
"Interviewing his students to determine why the vast majority initially rejected the equality, he found that “students continued to conceive of 0.999… as a sequence of numbers getting closer and closer to 1 and not a fixed value, because ‘you haven’t specified how many places there are’ or ‘it is the nearest possible decimal below 1’”
Hard to not think of it like that!
I don’t know what your last sentence means, exactly, but the rest of it is exactly what I keep trying to emphasize in discussions of Zeno’s paradox. To wit, you don’t need to bring up anything about sums of infinite series to see the fallacy here; don’t think, but look! Just stop and look at the very situation under discussion.
Just because an interval contains infinitely many points, it doesn’t follow that the length of the interval is itself infinite; this is a completely fallacious inference. There’s no valid way to conclude the latter from the former. This is where the supposedly problematic reasoning breaks down. Calculus, schmalculus; it’s just a simple unsupported leap. You don’t need to recast it in terms of convergence of series to see that (indeed, you could easily recast Zeno’s paradox into a form where analysis in terms of convergence of series would be difficult).
It is not wrong to think that way if the infinite segments (points) are of finite length. The assumption that the segments are of finite length is what is wrong. The segments are infinitessimally small.
What you are doing is taking a finite length, dividing it by infinite and then multiplying it by infinite and pretending it makes sense. It doesn’t.
I don’t understand why you guys refer to Wikipedia for Zeno’s paradox, or for why .9999… = 1, when you could refer to a wonderful website called The Straight Dope, whose archives include: An infinite question: Why doesn’t .9999… = 1 which covers both Zeno’s paradox and the infinite decimal approximation in one swell foop.
No. There are no such things as infinitesimals in the standard real numbers. Any interval is either of finite length greater than 0, a single point, or the empty set.
There is neither division by infinity then so there is no problem to be solved in the first place.
I bet you build up quite a hunger fighting ignorance like that, Dex!
I came in here to see if anyone had provided that link - shoulda known a mod was on the case!
I’m not sure what you think division by infinity has to do with it. There are infinitely many points between 0 and 1. If one likes, one can break the interval between 0 and 1 into infinitely many intervals of positive length (though not all of the same length). All the same, the distance between 0 and 1 is, obviously, 1 - 0 = 1 which is finite. This is supposedly paradoxical, but, clearly, it is it not.
For some people, the problem is “How can I move from A to B if, for every point along the way, I have to first reach another point before I can reach it?”. To which the answer is “Sure, what’s wrong with that? Why would you think there should be a first point after A to reach? Is there a first moment in time after the time you start at A?”.
People want to accept three premises (of some form or another):
A) There exist nontrivial intervals of finite length (or, which “can be traversed”)
B) Every nontrivial interval contains infinitely many points (or, contains infinitely many intervals; or, is such that there is some point between the first endpoint and every other point of the interval)
C) Every interval which contains infinitely many points (or, contains infinitely many disjoint intervals; or, is such that there is some point between the first endpoint and every other point of the interval) is of infinite length (or, “cannot be traversed”)
The three are of course directly contradictory. Any two could be consistently maintained for some system, but for most applications, the reasons to maintain A) and B) are very, very strong, while the reasons to maintain C) are just about nonexistent. C) is just pulled out of nowhere, with no support. It’s the flaw in Zeno-style arguing. C) simply isn’t true in such situations, and there’s no good reason to have ever believed it was
(To toss in some jargon, the paradoxes (both the alleged impossibility of motion and Achilles’ alleged inability to catch up to the tortoise) operate on irrational paranoia regarding supertasks)
Sort of along the lines of what ultrafilter said, but every one of those segments has a nonzero positive length. Every one of them.
Either the segments have zero length, or there are a finite number of them. Pick one, but not both.
Why do you say that? What reason at all is there to believe that we can’t have both?
Consider the OP’s example. Look at the disjoint segments [4, 4.9), [4.9, 4.99), [4.99, 4.999), …, contained in the interval [4, 5), which has finite length 5 - 4 = 1. There are infinitely many of them, and each has positive length (the first has length 4.9 - 4 = 0.9, the second has length 4.99 - 4.9 = 0.09, the third has length 4.999 - 4.99 = 0.009, …)
They don’t all have the same length, but there are infinitely many of them, and they all have positive (i.e., nonzero) length. The two conditions are not incompatible. There’s no reason to think they are.
Er, where by “have both”, I mean “all the segments have non-zero length and there are an infinite number of them”.