Set Theory 7:
Containers

Page last updated 10 Jun 2024

A criticism of the natural set theory set forth of the previous pages might be that it appears to not provide anything that could be considered analogous to the real world situation where things are placed into different containers, and where those containers themselves could be inside other (physically bigger) containers.

With respect to this, the crucial point to be realized is that since real world containers are actual physical things that means that if a thing is inside one container, it cannot at the same time be inside any other container, except for the case where the container is nested within a bigger container. Hence a mathematical analogy of such physical situations would be that one can have mathematical objects that we may call “containers” with the proviso that any object/element that is within one container cannot at the same time be within any other container, other than by way of nested containers.

It can be seen that this is very different to any set theory, where a set can be an element of multiple sets at the same time. Obviously, in the real physical world an item can only be in one place at any given time, and mathematical “containers” such as are envisaged here offer a nice reflection of real world situations. For example, in a library, a bookshelf may be divided into sections, and that bookshelf is part of a block of shelves that constitute a section of the library. This gives us different levels of nesting:

• Section of a bookshelf
• The bookshelf
• The block of bookshelves
• The section of the library
• The library

Hence a book in this library could be represented by mathematical objects that we can call containers. Obviously, some rules would be required for such containers. Possibilities are:

1. Uniqueness of location: Any object/element that is within one container cannot at the same time be within any other container, other than by way of nested containers.
2. A non-circular hierarchy of levels: A container can only be in another container if that container is a higher level container.
3. There are no empty containers: This is in accordance with the observation that in a real world scenario a thing that contains nothing is not a container at that point in time.

With reference to our previous example using vehicles (see Sets as elements in part 3), we could divide our vehicles into two containers, Either:

1. a container that contains vehicles made prior to 1970, and a container that contains vehicles made in 1970 or later,
   or
2. a container that contains vehicles whose rear wheels are driven, and a container that contains vehicles whose rear wheels are not driven,

but we could not have both scenarios at the same time.