But eventually you reach a point where the ‘half’ is a single pixel on the screen, and I don’t think you can really divide it in half any further. Either you ‘read’ that pixel or ya don’t. 
Which reminds me of a method I read about in one of Martin Gardner’s books of encoding a message of any length with just a single cut of a one-inch-long bar: Convert the message to numbers (using ASCII code; or A = 01, B = 02, etc.; or whatever), write all those numbers run together to the right of a decimal point, and you have a unique real number between 0 and 1. Then cut the bar at the point corresponding to that number. Of course, this only works in the world of theoretical math, not in the real world, because you’d have to be able to cut and measure with infinite precision.
You’d need to make sure that you have a code that cannot be ambiguous… three-digit ascii numbers for instance… otherwise you might not be sure how to convert a number back into a message.
Reminds me of a scene in an early ‘reboot’ episode where somebody tells a joke in ‘binary’ – not in ASCII code or anything sensible like that, but an encoding of 1 for A, 10 for B, 11 for C, 100 for D, and so on. But with the digits in a mostly unbroken string, you can’t tell if 101 meant to stand for E or BA, or the start of some other letter. 
Such a code is known as a prefix code. Huffman encoding is the best-known version of this.
[pedantic nitpickery]Actually, such a code is called “uniquely decodable.” Not all UD codes are prefix codes, but all prefix codes are UD. Prefix codes (and fixed-length codes, which are a subset of prefix codes) are the easiest UD codes to deal with, though, since they can be efficiently decoded inline.[/pedantic nitpickery]
Yes, that’s exactly what I said (trying to be clear, not attacking you ultrafilter.
As for .5000… or .4999…, either works. Better yet, let’s get very specific:
Given a sequence a_i of numbers between 0 and 9, the series
\Sum_{i=1}^\infty \frac{a_i}{10^{-n}}
converges. Its limit is the ratio of two natural numbers iff the sequence is eventually periodic.
Really, granularity doesn’t enter into the solution to Zeno’s paradox. It’s a little subtler than that and involves showing that the paradox itself is ill-posed.
I guess it depends upon which version you’re addressing, or maybe which paradox. I don’t remember this one being ill-posed.
The classic one about firing an arrow is based on the assumption that an object is stopped or in motion, and further that positional information is everything. When you freeze-frame as Zeno wants you to, you lose momentum information.
If mechanics took place solely in configuration space, there would indeed be a problem, but at each moment the arrow not only is somewhere, but has an instantaneous velocity. When Zeno tells you “everything is either moving or it’s not”, it’s false.
Hmm… I always thought that Zeno’s paradox was the thing with Achilles and the tortoise, but evidently there are several paradoxes
Well, the problem with that is that the very act of reading the pixel may or may not kill a cat.
The arrow paradox is over at the “Fruit flies like a banana” thread
Specifically, those are the paradoxes relating to motion. There are other surviving paradoxes aimed at showing that reality is indivisible, but those are less well-known. IIRC, one of them is a little trickier to resolve than the motion paradoxes.
“Tell Zeno I’m Willing to meet him Halfway.”
– CalMeacham’s first sig line.
For anyone who’s interested, it looks like this site has a more complete listing, although I haven’t had time to read all the way through it myself yet:
http://plato.stanford.edu/entries/paradox-zeno/
Sorry, Mathochist, but I can’t resist a straight line like that. At any given moment, the arrow has a distribution of positions and of velocities, such that the width of the position distribution is at least inversely proportional to the width of the position distribution, and vice versa.
Not, of course, that this has anything at all to do with the OP.
I think some of you guys should be making it clearer to the non-experts when you are talking about decimal fractions (which are mathematical abstractions with infinite precision), and when you are talking about physical objects, like bars and arrows. Newton apparently thought that those bars and arrows could be measured with indefinite precision, but Heisenberg was certainly uncertain about where arrows are and what they are doing.
Well, this whole discussion has left the original one about decimals. Infinite divisibility is a tangent, which of course is derived from the first thread. As such I can tell you exactly where we are but I have no idea where it’s going.
