The Natural Field of Ordinals

Which numbers are transcendental over the ordinals? Which are irrational? Let us introduce the natural field of ordinals and consider the status of √2, √ω, e, and π, among other numbers.

Which numbers are transcendental over the ordinals? To make sense of the question, let us expand our investigation from the natural ring of ordinals ⟨Ord⟩ to the natural field of ordinals, in which we can fully add, multiply, subtract, and divide ordinals and their differences and quotients in the natural arithmetic. In this field, we can consider such numbers as:

\(\newcommand\bminus{\mathbin{\textbf{─}}}\newcommand\bplus{\mathbin{\textbf{+}}} \frac{\omega^{\omega^2}\bminus5\omega^3}{\omega^{\omega}\bplus 3\omega}\qquad\text{ and }\qquad \frac{2\bminus\omega^\omega}{7\omega\bminus\omega^2}.\)

Which numbers arise in this field? Can we represent √2 or √ω this way? Which surreal numbers are algebraic or transcendental over the ordinals? Do the ordinals reveal new algebraic relations concerning e or π? How can we know?

We shall discuss all this and more in today’s installment, part of my series of essays on the ordinal numbers—find them in the ordinals tag.

Let’s get into it!