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The paradox in question: If every unit of length is made up of smaller units of length, it seems that you need to have units of length before a unit of length can come into existence. But this is clearly contradictory.

This paradox would seem to imply the following. Say you have two lines lying exactly on top of one another "|", and you want to slide one off the other to create space between the two, so that it now looks like this "||". If we need units of length before a unit of length can exist, then it seems we cannot create a length of distance between these two lines when we started with none, as we would need to have some length of distance already between them before we could have any length of distance between them.

Where does this reasoning go wrong?

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    I do not see why smaller units of length need to "precede in existence" (whatever that means) bigger ones, why can't they all exist eternally, or "come into existence" simultaneously, or bigger ones first? Also, you "existence" is ambiguous between length unit (as a physical aspect) and the notion of length unit. Why can't sliding of the lines happen physically even if there were no notions of length units, or of length and lines for that matter? Or if those came to be distinguished only later after the whole process happened multiple times, which is how it historically was? – Conifold Sep 12 '17 at 17:51
  • I have the same question as Conifold: "Why can't sliding of the lines happen physically even if there were no notions of length units, or of length and lines for that matter?" – Mark Andrews Sep 12 '17 at 19:18
  • The statement "every unit of length is made up of smaller units of length" seems to contradict the way we work with units. To work with units, one fixes one value (one meter, one kilogram are canonical examples), and then one subdivides as desired. Why would a contradiction arise from that? – Martin Argerami Sep 12 '17 at 21:34
  • @Conifold, the same argument can be applied to the concept of distance. If every distance is made up of shorter distances, we need to have distance before we can have distance. While they could come into existence at the same time, the fact that you first need the shorter distances before you can have / comprehend the full distance implies, to me at least, that one must come before the other. – IgnorantCuriosity Sep 13 '17 at 0:22
  • This is not Zeno's argument at all, his argument is that we need to cover half the distance before we cover the whole distance. As for what we need to comprehend, I do not see why we can not comprehend all possible distances at once, they are all of a kind, and even if not why are shorter ones "easier" to comprehend and come first? This seems to conflate order of size with the order of understanding. – Conifold Sep 13 '17 at 0:33
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Most of the Zeno-like paradoxes involve an infinite sequence of events buried in the language used in the problem. Your example appears to go down the path of doubling the distance between the lines enough times to achieve the desired separation, while Zeno's original paradox depends on halving the distance.

In both cases, the tricky part is coming up with a language to describe the problem which is capable of describing this infinite series of steps without contradiction. The problematic portion is typically identifiable with an appeal to reason, such as your "... it seems we cannot create..." It is dependent on the listener agreeing with such a statement. If they agree with that statement, then typically they agree that the problem is paradoxical, and thus there must be something wrong.

If one's listener does not automatically agree with you about this statement, one must defend it. Defending it is where the really precise language often comes into play. Mathematics, for instance, has a extraordinarily precise ways of dealing with the concept of infinity, and prides itself on its self-consistency. If one phrases the problem in the language of mathematics, then one can use the strength of mathematics to argue for their position.

However, the current "preferred" solution to Zeno's paradox is calculus. Calculus handles these infinities in a way that appears to be consistent with the world we live in without causing paradoxes from self-consistency issues. If your efforts to phrase the question in mathematical terms leads you to use notations from calculus, one will find that the issue is dealt with by the handling of limits which elide away issues that might come up regarding infinitesimals. These methods have been heavily analyzed over the years, so lead people to have a great deal of confidence in answers derived from them.

In my opinion, your Zeno-like argument is going down a direction which would be proven using set theory. There are systems for handling sets like ZFC which one would be tempted to use, but your particular construction is likely to go down the path of having an infinite descending set, which is forbidden in ZFC by the axiom of regularity. One would need to look at different solutions, such as Quine atoms which are used in some non-well-founded set theories. Such set theories are much less popular than their well-founded brethren, so they do not encourage the same level of confidence.

  • Currently, this is nothing more than a statement of your opinion without argumentation or references. Please edit this to make it objective (not with the aim to convince someone) and add proper references. keelan◆ [hehe kidding!] – Ron Royston Sep 12 '17 at 20:18

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