Logic and Language

Copyright   James R Meyer    2012 - 2023 https://www.jamesrmeyer.com

Page last updated 29 Aug 2022

The formulation of this paradox was inspired by the Platonist interpretations of the Balls in the Urn paradox. (Footnote: Unlike the case of Balls in the Urn paradox, in this paradox no piece that is added is ever subsequently removed.)

Consider the following:

We start with a rod of exactly 1 unit length in an infinitely large container

At 60 seconds to noon 9 rods of length 1 are added, and then 9/10ths is cut off each rod, and the remaining pieces are set end to end, so the total length is 1 unit, as before.

At 30 seconds to noon 99 rods of length 1 are added, and then 99/100ths is cut off each rod, and the remaining pieces are set end to end, so the total length is 1 unit, as before.

At 15 seconds to noon 999 rods of length 1 are added, and then 999/1000ths is cut off each rod, and the remaining pieces are set end to end, so the total length is 1 unit, as before.

And so on…

Each step takes zero time to accomplish.

Now consider the following questions:

1. At noon what is the length of the composite rod?
2. At noon how many pieces are there?
3. At noon what is the length of each piece?

A typical response is that:

1. The length of the composite rod is 1 (since its length is always 1 after each step).
2. There are infinitely many pieces (since their number increases at every step and there is no limit to the number of steps).
3. The length of each piece is zero (since the size decreases at every step and there is no limit to the number of steps).

But this is not the result that the Nested Intervals Theorem of set theory gives, (Footnote: For example, as described in Ethan D Bloch’s book, The real numbers and real analysis, Springer Science & Business Media, 2011, Bloch: The real numbers and real analysis: Details.) and which states that:

1. the intersection of infinitely many nested intervals is not empty, and
2. if the limiting value of the left ends of the intervals and the limiting value of the right ends of the intervals are identical, then the intersection is a single point which is that limiting value.

But the limiting value of all the left ends are all zero, and the limiting value of all the right ends are also all zero, which gives the result that at noon the rod is a single point rather than a rod of length 1. This is in direct disagreement with (1) and (2) above, hence the paradox.

So how does the paradox arise?

It arises because the process never actually completes. As in the case of the Balls in the Urn paradox, the reference to time in the description is a red herring, see the section Time as a red herring on the page Balls in the Urn paradox. Since the process cannot complete, the pieces never actually become of zero length, and must always have some finite length, regardless of how small that might be. And there is no step where the left end of any piece becomes coincident with the right end of another piece.

So the correct answer to (ii) is that there is no limiting value to the quantity of pieces, noting that there is no step at which the value actually becomes infinite (note that the term “infinite” is frequently treated as though it refers to a particular quantity, whereas in fact, it refers to a situation where there is no upper limit to a quantity).

And the correct answer to (iii) is that the limiting value of the length of the pieces is zero. Since the process never terminates, the pieces must always have some length greater than zero, regardless of how small that length can be.

In this way, the apparent paradox is resolved.

However, some people seem to think that, even with the application of limit states, there is still a contradiction, since there seems to be an enumerable infinity of pieces of zero length - with the claim that all the added lengths are enumerable, so that we have a definitive length of 1 that is composed entirely of an enumerable quantity of zero width points, where the first point is the leftmost point, the second is intermediately adjacent to the right of it, the third immediately adjacent to the right of the second, and so on. But this is easily shown to be a fallacy by considering the following:

Limit of the length of each piece = Limit 1n = 0.

Limit of the quantity of pieces = “Limit n ”, but such a limit does not exist.

Limit of total length = Limit of sum of lengths of pieces = Limit (n · 1n) = 1.

The fallacy corresponds to the supposition that Limit (n · 1n) = Limit n  · Limit 1n but this is not the case and has no logical foundation, since Limit n does not exist. The Platonist notion that the infinitely recursive iterative process can actually “complete” leads to the paradox. But the application of a limit does not imply that the limiting state is somehow the “completion” of the infinite process. It isn’t, and the failure to understand this engenders the false appearance of a paradox. For more on this, see also Understanding Limits and Infinity and Lebesgue measure.

The Platonist notion of a “completion” of an infinite process goes hand-in-hand with the Platonist notion that mathematical things actually “exist” independently of any definition. According to Platonist beliefs, every mathematical entity (in the above definition, every interval that can be defined by that definition) already somehow “exists” prior to the creation of the definition, and there is a non-physical mathematical “reality” where every step in the process already “exists” prior to the creation of the definition.

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Copyright   James R Meyer   2012 - 2023
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