[FOM] Finite Set Theory

[Harvey Friedman]

I have had several occasions to explain what set theory is to "ordinary
people" of a variety of age groups.

I have found it very useful and effective to start with finite set theory
only - restriction attention to very small sets.

It works very well, and everybody is quite satisfied. Nobody feels confused
or uneasy.

Then I give a couple of standard examples of infinite sets, and say that the
whole discussion can take place involving infinite sets, instead of those
small finite sets. I leave it at that, for these people are not
mathematically inclined.

It goes something like this. Very simple, very clear, very clean, and very
effective.

1. A set is a "bunch of things" arranged in any order whatsoever, with no
repetition. The only thing that matters about a set is what is in it, and
what is not in it. It doesn't matter how one describes what is in it. There
may be many ways to describe the same set.

2. The simplest set of all is the one which has no elements. It is easy to
describe what is in it: nothing is in it.

3. The next simplest sets are the sets that have exactly one element. Given
any thing x, there is the set whose elements are just x, and it is written
{x}. 

4. Of course, if {x} is the same as {y}, then x is the same as y.

5. Generally speaking, we take {x} if x is a thing that is outside
mathematics. For example, we can form {White House}.

6. Sometimes we want to take {x} where x is in mathematics. In particular,
sometimes we want to take {x} where x is a set.

7. The simplest case of 6 is where x is the empty set. I.e., {empty set}.

QUIZ: How many elements does {empty set} have? ANSWER: One.

QUIZ: How many elements of elements of {empty set} are there? ANSWER: Zero.

8. So far, we have only talked about sets that have either zero or one
element. But sets can have more than one element.

9. If we want to stay in pure set theory, then the simplest example of a set
with two elements is as follows. The elements of this two element set are

i. the empty set.
ii. {empty set}. 

10. I then say that we can mix things up with nonnegative integers and sets.
Again, I stick with small integers - 0,1,2,3 - for all of the examples.

11. Then I talk about pairwise unions of sets, with examples and quizzes.
Then I talk about pairwise intersections of sets, with examples and quizzes.
Then I talk about the set theoretic differences between sets, with examples
and quizzes. Again, only with very small sets - so sets of cardinality at
most 4 arise in the discussion.

12. Then I talk about subsets. This causes no problem, again with quizzes. I
emphasize what we call separation - that given any finite set, say a
particular one with four displayed elements, one can pick out only some of
the elements, and that will form a subset. This is reinforced by examples
involving subsets of a specific 4 element set. Also examples where totally
different ways of picking out the elements arrive at the same set.

13. Then I talk about the power set of a set. I give examples of power sets
of sets with zero, one, two, and three elements.

14. I then say that all of this works the same way with bigger sets. These
relatives always have some informal notion of finite set they are
comfortable with. I say that all of this works the same way with any sized
finite sets. 

15. I encounter no problems. All of this is readily accepted as being
unambiguous and making completely clear sense.

16. Then I give the following example of an infinite set:

{0,1,2,3,...}.

I also give this example from pure set theory:

{emptyset,{emptyset},{{emptyset}},{{{emptyset}}}, ... }.

17. Again, this goes over reasonably well. And then I say that we can rework
all of the above even allowing such infinite sets.

Now, where would FOM subscribers have issues with this development?

Harvey Friedman