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Aristotle discusses six dialectical arguments for the non-existence of place in Physics bk. Δ On Place, ch. 1 (209a); Zeno's argument is #5:

if everything that exists has a place, place too will have a place, and so on ad infinitum

Does what I call "idempotence of place" resolve the paradox?:

Place is an idempotent relation because the the place of the place of something is still its place. Thus place(place)=place. Therefore place(place(place(…)))=place.

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    (1) You should be more formal. Your use of symbol is a bit ambiguous. (2) Mathematically speaking it's not transitivity but rather looks like idempotence (3) You should think about what it means to "resolve the paradox" – Boris E. Mar 1 '17 at 19:21
  • Agreed it is idempotence. – samerivertwice Mar 1 '17 at 19:23
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    Aristotle considers the idea that a place is not a part of that whose place it is to be one of our basic intuitions about "place". So this proposal would be a non-starter for him, see Aristotle on Space, Time, and Motion – Conifold Mar 1 '17 at 21:04
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    Why do you think that there is a paradox at all ? A., Phy, IV 4, 212a20-21 "the place of a thing is the innermost motionless boundary of what contains it." Boundary does not mean that it is a thing itself. Thus, if place is not a thing, there is no need of "the place" of place. – Mauro ALLEGRANZA Mar 2 '17 at 14:04
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    If you agree that this is about idempotence and not transitivity, please edit your post to clarify that. Also, I believe your last expression 'place(place(place(...)))=place' should actually read something like 'place(place(...(place)...))=place', to avoid plugging in something else than a place. – user2953 Mar 5 '17 at 12:29
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Aristotle resolves the argument in Physics bk. Δ On Place, ch. 3 (210b):

Zeno's problem—that if Place is something it must be in something—is not difficult to solve. There is nothing to prevent the first place from being 'in' something else—not indeed in that as 'in' place, but as health is 'in' the hot as a positive determination of it or as the hot is 'in' body as an affection. So we escape the infinite regress.


For an in-depth treatment of the "problem" of place, see "Part II: Place" of:

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