# Footnotes

The label "second argument against plurality" is from *Encyclopedia of Philosophy*, Edwards ed., s.v. "Zeno of Elea" (written by G. Vlastos).

1. H. Diels and W. Kranz, *Die Fragmente der Vorsokratiker*, 12th ed. (Dublin, 1966), 29a15.

2. *Prm.* 128c-d.

3. G. E. L. Owen ("Zeno and the Mathematicians," *Proceedings of the Aristotelian Society* 58 [1958]: 199) says that "Zeno certainly held . . . that there is only one thing in existence." H. Cherniss (*Aristotle's Criticism of Presocratic Philosophy* [New York, 1971], p. 145), commenting on one of Zeno's arguments, says, "This argument is a refutation of the doctrine that there can be more than a single Real Being; it is in accord with the purpose of Zeno's dialectic as given by Plato."

4. Vlastos's translation ("Zeno," p. 371) of Diels and Kranz, 29b3, and G. S. Kirk and J. E. Raven, *The Presocratic Philosophers* (Cambridge, 1964), no. 366. "Finitely many" translates . "As many as there are" translates .

5. For a discussion of how they are counterexamples, see e.g., B. Russell, *Introduction to Mathematical Philosophy* (London, 1963), chap. 8. Something like premise 3 turns up in Aristotle, *Metaphysics* 1043b37: "When one of the parts of which a number consists has been taken away from or added to the number, it is no longer the same number, but a different one, even if it is the very smallest part that has been taken away or added" (Ross trans.). But "number" in this citation amounts to "finite number." See n. 16 below. In a similar vein is Plato's *Cratylus* 432a7-9, pointed out to me by my colleague Vicki Harper: "Ten, for instance, or any number you like, if you add or subtract anything is immediately another number" (Fowler trans.).

6. Something close to this feature of a collection *x* is sometimes chosen as one way of defining infinite—so-called reflexive—sets. See e.g., A. A. Fraenkel, *Abstract Set Theory*, 3rd ed. (Amsterdam, 1966), p. 29: "Definition VII: A set R is called *infinite* and more strictly, *reflexive*, if R has a proper subset that is equivalent to R."

7. For discussion establishing that some infinite sets are larger than others, see Fraenkel, chaps. 1, 2(especially Sec. 5, theorem 2, Cantor's Theorem).

8. See Fraenkel, chap. 1, for the distinction, which shows (ii) false, between *denumerable* infinite sets, whose members can be indicated by an infinite list, and infinite sets that are not so listable.

9. Vlastos's comment (p. 371) that this part of the argument enunciates "the denseness *of a continuum*" (my emphasis) does not, of course, imply that Zeno recognized hereby the distinction between a continuum and a denumerable infinity.

10. It appears as a Euclidean common notion, although T. L. Heath (*Euclid's Elements* [New York, 1956], 1:232) thinks it is not genuine. Á. Szabó raises the interesting possibility that it had to be included in Euclid because someone, i.e., Zeno, had denied it ("The Transformation of Mathematics into Deductive Science and the Beginning of Its Foundations on Definitions and Axioms," *Scripta Mathematica* 27 [1964]: 131-136). It does not seem to me, however, that Szabó's discussion raises this interesting possibility to the level of much probability.

11. I am indebted to my colleague John Wallace for helping me to shorten this justification.

12. S. C. Kleene evidently credits Galileo with the discovery of this oddity (*Introduction to Metamathematics* [Amsterdam, 1962], p. 3).

13. Szabó (pp. 134-135) suggests that the Zenonian paradox reported by Aristotle as "the half of the time equals its double" (Aristotle, *Ph.* 239b33ff., discussed by Vlastos under the head "The Moving Blocks") is meant to convey that part of an infinite set may be made to correspond one-to-one to the whole set. Here also it seems to me that Szabó's thought-provoking article has raised a very interesting possibility without making it very probable. The relevant point here is that in Szabó's reconstruction of the paradox the sets in question are nondenumerably infinite. The way of generating the correspondence is not, of course, via...