Is the argument valid? Is it sound (i. e., are all the premises true)?
Aristotle rejects premise (2) on the grounds that Zeno has failed to distinguish between infinite divisibility and infinite extension. How might one justify (2)? Aristotle suggests the following argument:
Aristotle rejects this argument in terms of the distinction between infinite divisibility and infinite extension. What then, he asks, is the justification for (2)?
But, Aristotle's argument is not quite right, since the infinite divisibility of the motion is reduced to the infinite divisibility of the temporal interval of the motion. If this interval is infinitely divisible then it has an infinite number of parts and to get from one moment to the next requires an infinite series of steps. Aristotle introduces a new distinction between potential and actual infinities. Infinitely divisible intervals are only potentially infinite.
But, the fact remains that the interval (either of motion or time or space) contains an infinite sequence of non-zero intervals.
The following Zenoesque argument for (2) can be constructed:
(Z1) Every interval can be subdivided into an infinite number of non-zero parts.
(Z2) To reconstruct the interval we would have to "stitch together" all the parts.
(Z3) The sum of an infinite number of non-zero parts is infinite, i. e., can never be completed.
(Z4)No intervals can be constructed.
Mathematical techniques developed in the 19th century (2500 years or so after Zeno first put forth the paradoxes) suggest that (Z3) is false. This argument for (2) fails. But, we have no argument that (2) is false. What, then, if anything is wrong with the argument?
(1) For every i, if Achilles is at T(i), then the tortoise is at T(i+1).
(C) For every point P on the race course, if Achilles is at P, the tortoise is at some point P" such that P" is farther from A(0) than P.
In order to derive (C) from (1), Zeno needs another premise
1.At every moment during its flight, the arrow occupies a space equal to itself.
2.When the arrow occupies a space equal to itself, it is not moving.
3.If the arrow is not moving when it occupies a space equal to itself, it must be at rest.
4.At every moment in its flight, the arrow is at rest.
5.The arrow never moves.
This is a modern restatement of the paradox as it has been passed down to us from Aristotle.
Aristotle challenges premise (3) and what he takes to be the atomistic assumption underlying the argument. For Aristotle, an object, to be at rest, must remain in the same position for a certain non -zero interval of time. For Aristotle, there is no "rest at an instant." Aristotle's objection to premise 3 then, is that, at an instant, no object is either at rest or moving. Both these predicates are only applicable to objects considered over finite intervals of elapsed time.
Even if you do not accept Aristotle's argument (from "ordinary Greek," as it were) the argument, as formulated is invalid. The conclusion (5) does not follow from (4). This inferential step commits what is called the fallacy of composition (whatever is true of all the parts of a thing is true of the whole).
But Zeno is not down and out yet. The fact that one way of formulating a line of reasoning is invalid does not mean that all formulations of the line of reasoning are invalid. Zeno, were he alive today, might well respond by offering the following reformulation of his view.
A Valid Reconstruction of the Arrow.
6.If the arrow is moving at t, then (at t) the arrow occupies a space equal to its own volume. [If P then Q]
7.If the arrow occupies (at t) a space equal to its own volume, then it is not moving at t. [If Q then not-P]
8.If the arrow is moving at t, then the arrow is not moving at t. [If P then not P: Hypothetical Syllogism from 6 and 7]
9.The arrow is not moving at t. [not-P: this follows from 8 (check this by showing that there is no way to make the premise (=8) come out TRUE when the conclusion (=9) is FALSE]
This reconstruction of the argument IS VALID. Since the conclusion, presumably, is unacceptable either premise 6 or premise 7 must be false. But, which one and why?
Note that this formulation implicitly rejects Aristotle's argument that one cannot meaningfully talk of rest (or motion) at an instant.
SOLUTION(Uncover only after you have given up or think you have the answer)
The fourth is an argument concerning two rows with an equal number of bodies all of equal length, the rows extending from the opposite ends of the stadium to the midpoint and moving in opposite directions with the same speed; and the conclusion in this argument, so Zeno thinks, is that the half of an interval of time is equal to its double . . . For example, let A1A2A3A4 be a set of stationary bodies all of equal length, B1B2B3B4 another equal set of moving bodies starting on the right from the middle of the A's and having lengths equal to the A's, and C1C2C3C4 a third equal set with speed equal to and contrary to that of the B's, also of lengths equal to those of the A's and ending on the right with the end of the stadium
Now as the B's and the C's pass over one another, B1 will be over C4 at the same time that C1 will be over B4
Thus, (1) C1 will have passed all the B's but only half of the A's, and, as C1 takes an equal time to go through each B as through each A, its time in covering half the A's will be half that in covering all the B's. Also (2) during this same time the B's will have passed all the C's; for since C1 takes an equal time to pass each A as each B, C1 and B1 will reach the contrary ends of the courses at the same time because each of them takes an equal time to pass each A. (240a, 5-15)
The argument can be more formally set out as follows:
Aristotle rightly focusses on (8) as suspect and he argues that it is false. But, he does not pick up on the significance of relative motion. No one does until the 17th century AD.
Nonetheless, the argument can be reconstrued as an argument against the view that spatial and temporal magnitudes can be decomposed into atoms. The first seven premises are the same. The argument proceeds
2)The interval can be infinitely subdivided.
3)The ultimate parts produced by this division, once completed, will
4)If 3a, then the sum of the parts will be an interval of infinite magnitude.
5)If 3b, then the sum of the parts will be an interval of zero magnitude.
6)The interval has either zero magnitude or is infinitely long.
Measure Theory and the Plurality Paradox
Solving the Plurality Paradox
The bottom line is that
The measure of a set
(1)The measure of a set is a generalization of the ordinary concept of length. We can assign measures to finite sets of points as well as continuous intervals.
A set of cardinality = c with measure = 0.
1. Consider the interval 0 ¾ x ¾ 1.
2. Remove open middle thirds in successive steps.
After the first cut:
0 1/3 2/3 1
After the second cut:
0 1/9 2/9 1/3 2/3 7/9 8/9 1
[______] [______] [______] [______]
3. The removed parts sum up to 1 (i. e., has measure = 1).
4. What remains is the Cantor set.
5.Measure(of the interval from 0 to 1) =
i. e., Measure(of the Cantor set) = 0.
6. Cardinality of the Cantor set = c (the cardinality of reals)
3.The (n+1)th decimal place of each number in the nth cut follows the same pattern.
4.The removed parts contain all numbers with 1's anywhere in their decimal expansion.
5.Therefore, the Cantor set (= all the remaining parts) contains all combinations of infinite sequences of 0's and 2's.
6.Change all the 2's to 1's (relabel the numbers).
7.Therefore, the Cantor set contains all combinations of infinite sequences of 0's and 1's.
8.Every real number between 0 and 1 can be represented as an infinite sequence of 0's and 1's. Similarly, each such sequence corresponds to a real number in the interval from 0 to 1.
9.Therefore, the cardinality of the Cantor set = the cardinality of the reals between 0 and 1.
10.But, there is a 1-1 correspondence between the reals lying in the interval from 0 to 1 and the total set of all the reals.
1.For numbers greater than 0, divide through by new base number b until you reach 0. The remainders, in reverse order, are the digits in the number in the base b.
2.Converting Fractions: The following theorem holds:
Thm.If k/l is a rational number (k, l are relatively prime) then (k/l)b has a finite representation IFF every prime factor of l divides b.
1/2 = 1x2-1 = .12
2. Represent 1/9(base 10) in base 3.
1/9 = 9-1 = (32)-1 = 1x3-2 = (.01)3
3. Represent (1/3) and (2/3) (base 10 in base 3.
(1/3)10 = 1x3-1 = 1x(.1)3 = (.1)3
(2/3)10 = 2x3-1 = 2x(.1)3 = (.2)3