I ve been busy with this: If you have to go from one point that is ten feet from the other you have to pass first by the 5 feet mark to the goal point,then 2.5,then 1.25,...if you always must cover the middle point of what is left,you always advance less than the remaining distance... If such a thing as Planck s lenght is kind of a minimum space,how can an object move faster than the other?... If they cross the Planck s unit in different times,that would mean that when the faster finalizes the track,the slower is in some point less extent,but different from the start because if not it wouldn t be moving at all...

All right,I will try my own attempt at an explanation,tell me if it makes any sense... The concept of infinite divisibility would challenge any valuation of the world, because every event must be measured in spatial-temporal coordinates, and these imply quantities:if any finite number representing a magnitude contains an infinity then its mere enunciation would be a contradiction... From the strictly abstract,if we consider,for example the number 2,111...we see that is an infinitely extense number altough we say that is less than 3;in this case we can solve the problem just by establishing conventionally the indivisibility in some point of the scale:we can see here that any number has no value else in itself more than which we give to it by convention: If you go from 1 to 10 in the abstract realm of mathematics,and you are in 9,but then you divide by ten the remaining distance(1/10); you would have to cover ten units to the goal,just as in the start point,and the difference between this units(0.1 of the initial unit each) and the initial ones only arises if you consider the initial ones as a comparative frame of reference,with no other "autonomous" value as being of different size as units than this comparison itself,and you only reach the "goal line" 10 when you stablish conventinally the indivisibility of the units in some point(let s say 0.1,from 9=0.1 x 10,and you reach the goal)... Now,in the physical world,the role played by conventions in maths is accomplished by the concrete frame of reference that the perceptive limitations gives to our senses(frame of reference): As Galileo pointed out,in a good example by physicist Mach(is the spelling right?): Lets consider an apple on a table in apparent state of rest;an observer within the same room has the right to claim that the fruit is not moving,but as the Earth spins on its axis the apple is moving as well,what an observer in orbit can affirm;but the Earth revolutes around the sun(observer 3),the sun around the galactic core(observer 4 ),the galaxy with the local group of galaxies(observer 5),and so on: Another example,if we show to a 2 mts man a room,and then to a 1 mt tall man the same room independently. Which is the dinamic state of the apple? Is the room big or small? Well, all of the apple observers are right and both men can claim either that the room is big or is small because there is no absolute parameter to measure space(and therefore movement),and the value to it will be given by the frame of reference provided by the point of view of the observer,which in turn depends on his/hers senses of perception,natural or artificial;but as any number different from zero divided by other distinct as well from zero will be always different from zero,our point of view will always deliver a finite quantity of units to measure the object(notwithstanding the fact that a physical limit as the size of photons may prevent us further sharpening of the observation,this limit has a value different from zero for each unit) ,and thus,in the case of covering a given distance we ll observe the object achieving the goal line after passing the amount of minimal units perceivable by us(which constitutes our frame of reference)AS LONG AS WE DON T CHANGE OUR RESOLUTION TO COUNTERMEASURE THE ADVANCE;for as we noticed in the abstract example,the numbers(or magnitudes)only depend of the value given by the basal units that constitute the frame of reference;this means that to change the resolution,or the extension of the minimal perceivable unit is to change the value of space itself: Then if we imagine a distance to cover in the example of the ten feet of the initial post,independently of our capacity of resolution,this would mean to change arbitrarily our resolution(unit) beyond our minimal point(or frame of reference),or our valuation of space itself: If we double it as we double the advanced space(1/2+1/2+...),we won t move:but if we aumengt the resolution by a factor of ten while doubling the advance:we ll be getting away from the goal¡¡¡¡ But as long as we remain stationary from the perceptive point of view,the object achieves the goal with no problem. So we don t need an "absolute base" to contain magnitudes as we don t need it to claim the state of rest of the PC while the Earth is moving... But if we have it,as Planck s lenght,then what happens with the faster/slower relation of the first post...? Has this any sense?Help... Javier

I heard about this from mr. Brian greene as he beautifully describes solving this paradox. Since space and time are interwoven together it gave birth to concept such as velocity and acceleration. To cover 10ft u must have definite velocity which allows it. This paradox only stands when rate of acceleration is decreasing and the limit is infinity.hope i am not making fool out of myself.

If you have to go ten feet and you do it in 10 seconds, then you'll reach the 5 foot mark in 5 seconds. With only 5 feet left to travel, you'll get to the 9 foot mark in another 4 seconds. Now you only have 1 foot left to travel, but before you can do that you have to travel half a foot. But that half-foot will only take half a second. Now you have half a foot left, but to cover it you first have to cover one-quarter of a foot (to bring you to the 9.75 foot mark). But that will only take you 0.25 seconds. To cover the next 0.1 feet, will take 0.1 seconds. And so on and so forth, until you have 0.00001 feet left to travel, which will take you 0.00001 seconds. The point is, you cover smaller distances in less time. You can divide up the total distance you have to cover as much as you like, but as the distance divisions get smaller and smaller, so will the times you require to cross those divisions. The ultimate limit would be if you tried to divide up the distance into infinitesimally small divisions, but each one of those could be crossed in an infinitesimally small amount of time. It may seem that an infinite series of steps, like 1/2 + 1/4 + 1/8 + 1/16 + 1/32 +... can't ever add up to a finite number, but mathematically it can and it does.