(An Initial Segment of) The Ordinals.

Problem

This exercise is intended to familiarize you with the ordinals. The notion of an ordinal is a fundamental notion of set theory. In computing science, ordinals are used to prove program termination. Here is one way of describing the ordinals.

Start with the natural numbers, ordered as usual:

0, 1, 2, ...

Now suppose that there is a "number", w (omega) that is greater than all of the natural numbers. We now have:

0, 1, 2, ..., w

We can continue as follows:

0, 1, 2, ..., w, w + 1, w + 2, ...

Again, suppose that there is a "number" greater than all of the above. We now have:

0, 1, 2, ..., w, w + 1, w + 2, ..., w × 2, (w × 2) + 1, (w × 2) + 2, ...

Let us speed up the process a little:

0, w, w × 2, ..., w × w = w ², w ³ , w ⁴ , ..., w ^w , w ^{(w w )}, w ^{(w (w w ) )} , ..., e ₀ .

The last ordinal shown above, e ₀ , is pronounced "epsilon naught." For technical reasons, e ₀ is defined to be the set of ordinals less than itself (this is how all ordinals are defined). This ordinal plays an important role in logic. In addition, it is closed under ordinal addition, multiplication, and exponentiation, if we start with the set {0, 1, 2, ..., w}.

We can continue adding "numbers" in this way indefinitely. Any book on set theory will have a discussion of the ordinals. Read and understand the basic ideas, including ordinal arithmetic. Here is a list of recommended books that you can consult.

• Keith Devlin. The Joy of Sets: Fundamentals of Contemporary Set Theory. Springer-Verlag, 2nd edition, 1992.
• Paul R. Halmos. Naive Set Theory. Van Nostrand, 1960.
• Kenneth Kunen. Set Theory - An Introduction to Independence Proofs, volume 102 of Studies in Logic and the Foundations of Mathematics. North-Holland, 1980.

Deliverables

Write a report that addresses the following.

1. Prove that e ₀ is countable. (Recall that e ₀ is the set of ordinals up to e ₀ .)
2. Show that for any positive natural number n, the set of n-tuples of natural numbers, ordered lexicographically, can be embedded into e ₀ in an order-preserving way. The lexicographic ordering, <_n, on n-tuples of naturals is defined as follows: <₁ is <, the less than relation on the natural numbers. For n > 1, (x₁, x₂, ..., x_n) <_n ( y₁, y₂, ..., y_n ) iff x₁ <₁ y₁ or [x₁ = y₁ and (x₂, ..., x_n) <_n-1 (y₂, ..., y_n)].
3. Design a way of representing elements of e ₀ in the programming language of your choice (Lisp and Haskell are recommended). Define a function that accepts one argument and returns true if the argument represents an element of e ₀ and false otherwise.
4. Define a function that accepts two arguments. If both arguments are elements of e ₀ and the first is strictly less than the second, it returns true; otherwise, it returns false.

The evaluation will be based on the report. Late reports will not be accepted. Given that students have different backgrounds and levels of mathematical maturity, if it is clear that the project was given serious consideration, I will assign an A. One way that I can determine that the project was considered seriously is if you talk to me.