Question

What is the point of the Earth farthest from the geometrical center of the planet?

Theseus' ship and Banach-Tarski's paradox

1. I want to speak in this post about notion of "equivalence" between objects, and describe some paradoxes to which it can lead when combined to the notion of "decomposition".

The points a. and b. below are definitions. Point c. states the main problem we will consider.

a. When speaking of "equivalence", we always are in presence of an operation of objects in our universe to the from couples of objects to the truth values (we assume that they are just "true" and "false"): we take 2 objects, say A and B, and the "operation" gives as a result the truth value of the sentence
"A is equivalent to B".

The following basic properties will be (safely) assumed from now on:
- for any object A, "A is equivalent to A"=true
- if A, B are objects and if "A is equivalent to B"=true, then "B is equivalent to A"=true
- if A,B,C are objects and if "A is equivalent to B"=true, and "B is equivalent to C"=true, then "A is equivalent to C"=true

b. When we want to speak of a "decomposition" of an object, we are in presence of :
- a concept of "being a part of"
- a first object A, to which we associate a family F containing some objects B,C,..

The properties required for the family F are:
- A contains as a part of it each one of the elements of the family F
- there is no part of A which is not contained in some element of F (i.e. each object O which is a part of A is also part of some element of F)
- the elements of F are all disjoint (i.e. if we take an object B in F, then there does not exist a part of B which is also part of another object of F)

c. We want now to consider the interplay between decomposition and equivalence. Suppose that the following are true:
- F is a decomposition of A
- F' is a decomposition of A'
- each element of F is equivalent to an element of F'
In other words one breaks A and A' into pieces and observes that the pieces are all equivalent!

Question: Is it true that in such case A is equivalent to A'?
Answer: If you ever played with Lego, you know that this is certainly not the case.
Conclusion: One must introduce the concept of mutual position, at least.

Therefore we will also assume that
- there is a good definition of "relative position" for a family of objects
- the objects forming F are in the same relative position as the ones equivalent to them forming F' (i.e. you positioned the LEGO pieces in the same way as their equivalents)

Question: Is it true NOW that A is equivalent to A'?
Answer: Not yet. One has to be careful with the notion of "part".

This is explained very well through the following example

The Banach–Tarski paradox is a theorem in set theoretic geometry which states that a solid ball in 3-dimensional space can be split into several non-overlapping pieces, which can then be put back together in a different way to yield two identical copies of the original ball. The reassembly process involves only moving the pieces around and rotating them, without changing their shape. Actually the pieces building up the two copies, are also all equal (up to rotation and translation) and they are in the same mutual position!

However they are of a quite complicated shape, in the following sense: From geometry you know the volume of the easily constructable figures (cube, parallelipiped, sphere..). To further extend such notion to other objects, the best way is to state some properties that you want to hold for it, and then look at the greatest class of objects
- which contains the "easy" figures you started with
- on which there is an extension of the "volume" ("volume"=assignment of a number to each figure, such that the wanted properties continue to hold).

The properties for the volume which one wants are basically:
- the volume of a (possibly infinite) union of objects which are each disjoint of the others, is the sum of the volumes of the single elements osf such union
- the volume of an object A is the same as the volume of A' if A' is obtained from A by translation and/or rotation

Then there are so many different shapes in the 3D space, that the "volume" cannot be extended to all subsets of the space. The sets in which the two spheres above are divided are so complicated that a "volume" defined on the sphere cannot be "extended to them".

Conclusion: The definition of "part" should be restricted, otherwise we arrive at apparently contraddictory results.


In everyday life things are getting even more complicated (That's because we usually don't think too much about the concept of "equivalence", until somebody points out that it generates a paradox.) :

2. Theseus is remembered in Greek mythology as the slayer of the Minotaur. The ship in which he returned from Knossos was not in good condition, after all the time Theseus passed in the labyrinth.

As parts of the ship needed repair, it was rebuilt plank by plank. Suppose that, eventually, every plank was replaced; would it still have been the same ship?

Changing a single plank can never turn one ship into another. Even when every plank had been replaced, then, and no part of the original ship remained, it would still have been Theseus’ ship.

Suppose, though, that each of the planks removed from Theseus’ ship was restored, and that these planks were then recombined to once again form a ship. Would this have been Theseus’ ship? Again, the answer is yes: this ship would have had precisely the same parts as Theseus’ ship, arranged in precisely the same way.

If this happened, then, then it would seem that Theseus had returned from Knossos in two ships.

Question: What means "being equivalent to Theseus' ship"?

3. Another example: Also our cells are continuously regenerated. The atoms of which we are made are never the same, still we feel like we have always the same identity. This however causes no paradox, just because nobody can take our dead tissues, "restore them", and make other copies of ourselves.