Zeno is said to have composed forty arguments, but only a few are preserved in the form of discussion of them by later writers. Some of these arguments may be and have been attributed to Zeno's teacher, Parmenides. Arguments #6-9 below are especially famous due to their having been discussed by Aristotle. Argument #9, the Stadium, speaks especially to modern thinkers because it may represent a confusion between motion and relative motion. Zeno's reconstructible arguments are:

1. Infinity and Betweenness
Argument #1: If there are many things then they must be finite in number, for there must be as many as there are and neither more nor fewer. On the other hand, if there are many things then there must be infinitely many things, for there are always other things between any two things that exist, and between these yet others. So, if there are many things, there must be both finitely many of them and infinitely many of them, which is impossible. Thus there cannot be many things, not even as many as two.

Argument #1 must make us wonder whether Zeno has anticipated Cantor in understanding that an infinite set may be put into one-to-one correspondence with some of its proper subsets and supersets. What would lead Zeno to imply that an infinite set contains more or fewer elements than it contains?

The argument, however, is fallacious. An infinite set, like a finite set, has neither more nor fewer elements than it has.

Concerning the betweenness argument, one needs to distinguish discrete sets from dense sets. There are points between any two points and further points between any two of these, and so forth, and thus infinitely many points. With respect to atoms, it is not the case that between any two atoms there are other atoms. In all probability, there does not exist, at any instant, any set of even so many as three collinear atoms. Given any two atoms, it is very improbable, though not impossible, that there is any atom in between the two given atoms. If two given atoms are far enough apart, then there will be atoms that are between the two given atoms in the sense, say, that they are closer to both of the two given atoms than the two given atoms are to each other. On the other hand, for each atom, there is some atom that is closer to it than is any other atoms. Between two such atoms there are no other atoms, even with such a loose meaning to the word "between."

The proof of the existence of infinitely many atoms that parallels the proof of the existence of infinitely many points thus fails. There is, however, no reason to assume that there are fewer than infinitely many atoms. However many atoms there are, they are simply not densely arranged in space.

2. Confusion Between Space and Objects
Argument # 2: If everything that exists must be in some place, then, if places exist, they must be in some place, and these places must be in other places, ad infinitum, so that there is finally no place in which anything can be.

Argument #2 confuses two different kinds of existence, the existence of space and the existence of objects occupying space. It is only the latter that entails being located somewhere. Space does not exist in the same sense as that in which matter exists. Space exists in the sense that pieces of matter are at various (and varying) distances from each other in various (and varying) directions. (The variation, both of the distances and directions of physical objects from each other and of other physical properties that vary, is what is meant by the existence of time.)

3. The Distance Between Endpoints
Argument #3: No positive volume, area or length is the smallest, since one can always have half as much. So if there were a smallest volume, area or length, these would have to have magnitudes of zero cubic meters, zero square meters and zero meters, respectively. However, the sum of any number of zeroes is zero also. Thus a positive volume, area or length can not be the union of things of zero volume, area or length. Therefore, things of positive volume, area or length do not have smallest parts.

Argument #3 supposes that the magnitude of a set is the product of the number of the elements in the set and the magnitude of the elements of the set. This is not the case. The length of an interval of points is not the product of the number zero and the cardinal of the interval but is the distance between the endpoints of the interval.

The length of the intervals (a,b), [a,b), (a,b] or [a,b] is the integral from a to b of 1 dx, which is b - a meters, assuming we are using a co-ordinate system with the meter as its unit of distance. Similarly, the area of the plane region bounded by the lines x = a, x = b, y = c and y = d is the integral from c to d of the integral from a to b of 1 dx dy, which is (a - b)(c - d) square meters. The volume of the region bounded by the planes x = a, x = b, y = c, y = d, z = e and z = f is the integral from e to f of the integral from c to d of the integral from a to b of 1 dx dy dz, which is (a - b)(c - d)(e - f) cubic meters. The length, area and volume of the point (a,b,c) are, respectively, the integral from a to a of 1 dx, which is 0 meters, the integral from b to b of the integral from a to a of 1 dx dy, which is 0 square meters, and the integral from c to c of the integral from b to b of the integral from a to a of 1 dx dy dz, which is 0 cubic meters.

Similarly, the area and the volume of a line segment is zero and the volume of a plane region is zero.

4. Confusion Between Size and Elements
Argument #4: Again, if to a subset of space of positive volume, area or length a subset of space of zero volume, area or length is added, then the resulting subset of space would not be any larger. Similarly, if from a subset of space of positive volume, area or length a subset of space of zero volume, area or length is subtracted, then the resulting subset of space would not be any smaller. However, that which, when added to or subtracted from something else, makes that something else no larger or no smaller, is nothing at all. A subset of space of zero volume, area or length is thus nothing at all and such a thing does not exist.

Argument #4 confuses size with elementhood. What makes two sets be the same set is that they have the same elements, not that they have the same size. If A is a set of points, or of instants, or of real numbers, and P is an element of A and Q is not an element of A, then A \ {P}, A and A È {Q} are three different sets. A \ {P} consists of all the elements of A except P. A È {Q} consists of all the elements of A and also Q. P is an element of A and of A È {Q}, but not of A \ {P}. Q is an element of A È {Q}, but not of A or of A \ {P}. Whether the size of P and Q is zero or something else or not defined is an irrelevancy. If A is a subset of the underlying set of a metric space, then A is a set each element of which has a size of zero, yet A may have a non-zero diameter, which will be the least upper bound of the distances between its elements.

In particular, if P is a point then the empty set and the set {P} the only element of which is the point P are the same size but are different sets. The length of both sets is 0 meters. The empty set has cardinal zero. It has no elements. Its elements do not exist. The set {P} has cardinal one. It has one element. Its element does exist.

5. Motion in Space
Argument #5: If something is moving, then it must move either in the place in which it is or in some other place. However, it cannot move, nor do anything else, in a place in which it is not. Also, it cannot move in the place in which it is, for that place is the same size as the moving object and hence allows no room in which to move.

Argument #5 disappears when one understands that motion consists of being first in one part of space and later in a different part of space, passing meanwhile through a continuum of other parts of space. All of these parts of space have the same shape as that of the moving object and the same size as that of the moving object, assuming that the moving object does not change its shape or its size as it moves.

6. The Dichotomy
Argument #6: You cannot move from A to B, for in order to get to B, you must first get halfway there. Supposing that you had done so, you would still have half the remaining distance to cover, and so forth ad infinitum. Thus in order to get to B you must do infinitely many other things first, which cannot be done in a finite amount of time. You cannot, in fact, even get halfway to B, for the same reason, nor a fourth of the way there, and so forth. It follows that you can never leave A. Thus motion is impossible. For Argument #6, one must note that since an interval of either space or time has a finite length while consisting of an infinitude of points or instants, one can indeed do infinitely many things, e.g., pass through infinitely many points, during an interval of time of finite duration.

7. Achilles
Argument # 7: If Achilles undertakes to race the tortoise and gives the tortoise a head start, then Achilles can never overtake the tortoise. For when Achilles starts at A, the tortoise is at some point B ahead of Achilles. By the time Achilles gets to B, the tortoise has moved ahead to some point B’, but by the time Achilles gets to B’, the tortoise has moved ahead to some point B," and so forth ad infinitum. Thus the tortoise is forever ahead of Achilles, or Achilles never overtakes the tortoise.
For Argument #7, suppose that Achilles runs 10 miles an hour and the tortoise runs 1/10 of a mile an hour and Achilles gives the tortoise a head start of one hour. Then an hour after the tortoise starts, the tortoise has run 1/10 of a mile. It takes Achilles 1/100 hours to cover this mile, during which time the tortoise has moved ahead 1/1000 of a mile. It takes Achilles 1/10000 if an hour to cover this distance, during which time the tortoise has moved ahead 1/1000000 of a mile. This process continues until Achilles catches up to the tortoise. This happens 1/100 + 1/10000 + . . . hours = .010101 . . . hours = 1/99 hours after Achilles starts running, and at a point 1/10 + 1/1000 + . . . miles = .101010 . . . miles = 10/99 miles ahead of the starting point. The fact that 1/99 or 10/99 can be written as the sum of infinitely many increasingly smaller numbers does not imply that these quantities are themselves infinite.

8. The Arrow
Argument #8: At each instant, an arrow occupies a space equal to its own length and therefore the arrow does not move at that instant. However, if it moves at no instant, then it also does not move during any interval of instants.

Argument #8 arises from a confusion between the related but distinct notions of the net speed over an interval of time and the instantaneous speed at an instant. A moving object covers no net distance during an instant, but this does not imply that it is not moving. Both a moving arrow and an arrow at rest occupy, at any instant, spaces equal in length to the lengths of the arrows. The relevant question is not whether the arrow occupies a space equal to its own length at each instant but whether the arrow occupies the same interval of space during an interval of instants. If it does, then it is not moving. If it does not, then it is moving.

9. The Stadium
Argument #9: This argument, as nearly as I can see, is impossible to reconstruct in a way that makes it comprehensible. The Stadium argument, as Aristotle presents it, imagines two equally spaced columns of men marching in opposite directions at equal speeds and passing by an equally spaced column of stationary men. On the ground that the men in the two moving columns pass each other at twice the rate or in half the time that the men in either moving column pass the men in the stationary column, it is concluded that the half is equal to the whole or that the whole is equal to the double.

As for Argument #9, it may suffice that we rest content with Aristotle's analysis. Aristotle remarks (Physica 239b) that the fallacy lies in failing to distinguish between a comparison with something moving and a comparison with something at rest. This, however, does not explain why the argument itself is incoherent or how it is possible that Zeno could have been so confused as to give this argument and believe himself or expect others to believe that the stated conclusions follow from it.

It seems likely to me that Zeno actually said something at least a bit different from what Aristotle reports and that Aristotle did not understand what Zeno had said. In order to understand Argument #9, I suggest that it may be helpful to remember that all the atoms move at the same constant speed. This you may read in the works of the ancient atomists whose writings are extant, Lucretius (De Rerum Natura) and Epicurus (Letter to Herodotos). It is a doctrine that probably goes back to Pythagoras, since Zeno here seems to argue against it, although this is difficult to tell from Aristotle's probably garbled account of Zeno's argument. Still, Zeno's other three arguments against motion argue that motion is impossible, while the Stadium is an argument against the proposition that all individuals move at the same speed.

The Stadium is also the first recorded confusion of motion with relative motion. Something that should be expected of the one is attributed to the other with an absurd conclusion being deduced from the confusion.

Zeno's argument admits of another interpretation also. It is possible that the purpose of Zeno's argument is to refute the premise that space and time exist each in the form of discrete units of positive length.