Ordinal arithmetic
Let's review the basics of ordinal arithmetic, addition, multiplication, and exponentiation, providing both the order-theoretic semantic definitions as well as the recursive definitions.
Ordinal arithmetic!
In this series of essays—you can find them in the ordinals tag—I shall cover all the basics of ordinal arithmetic, starting with the standard addition, multiplication, and exponentiation operations, but eventually getting to indecomposable ordinals, irreducible ordinals, Cantor normal form, binary ordinal representation, and more. Those who are new to the ordinals might want to start with How to Count.
A foundation for what is coming
I plan to lay down a solid foundation on these ordinal matters, with the aim in subsequent posts to grow the discussion into several deeper matters beyond, which require mastery over these concepts.
In particular, we shall continue on with my essays on the surreal numbers (in the surreal numbers tag), since our further work with that requires grounding in the ordinals. To foreshadow the coming topics, I shall subsequently introduce and investigate the so-called natural ordinal arithmetic (also known as the Hessenberg operations), which unlike the standard classical ordinal arithmetic are commutative operations. With the natural operations, the ordinals form a commutative semiring, the natural semiring of ordinals. The relevance of the natural ordinal arithmetic for the larger project of this book is that these are the same operations the ordinals experience in the surreal number field. After this, I shall introduce and develop the theory of what I call the natural ring of ordinals ⟨Ord⟩, which is the commutative ring generated by the ordinals under these operations, in which you can form such numbers as ω3 · 5 - ω2 + ω - 7. This is precisely the subring generated by the ordinal numbers in the surreal field. How does the natural ring of ordinals compare with the Omnific integers? We saw that √2 is rational in Oz, for example, but what is the situation in the natural ring of ordinals ⟨Ord⟩? Do we have unique factorization in ⟨Ord⟩? Those are the questions at which we shall aim in the coming posts.
But first, in this essay, we shall put down a proper reliable foundation for the standard order-theoretic operations and basic theory of the ordinals. Everything coming later will build upon this.
