Take the set of all countable ordinals $\omega_1$. Define $+$ and $\cdot$ by the Hessenberg sum and product. In a similar fashion to the naturals, construct the integers and the rationals from these by taking equivalence classes of ordered pairs. We'll call these the "$\omega_1$ rationals". I have a few questions about this construction:
- What is the cardinality of the $\omega_1$ rationals? My assumption is that it is $\aleph_1$, as at each step we have only taken pairs of countable ordinals.
- Are the $\omega_1$ rationals a field?
- Assuming they are a field, what are its properties? Is it metrically complete? Algebraically closed? If it is neither, how far is it from being metrically complete or algebraically closed?
You're exactly right about the cardinality being $\aleph_1$. This construction does yield a field. To understand it better, note that by Cantor normal form, the semiring $(\omega_1,+,\cdot)$ can be identified with the set of formal polynomials with coefficients in $\mathbb{N}$ in variables $x_\alpha$ for $\alpha<\omega_1$, where $x_\alpha$ corresponds to the ordinal $\omega^{\omega^\alpha}$. So, taking formal differences of such things will just give polynomials in the variables $x_\alpha$ with coefficients in $\mathbb{Z}$, and then taking formal quotients will give rational functions in the variables with coefficients in $\mathbb{Q}$. So this field is a field of rational functions in $\aleph_1$ variables over $\mathbb{Q}$. In particular, it is very far from being algebraically closed (for instance, it contains no elements algebraic over $\mathbb{Q}$ besides the elements of $\mathbb{Q}$ themselves).
(Of course, there is nothing particularly special about $\omega_1$ here. You could do the same thing for any ordinal $\omega^{\omega^\alpha}$, and you would get a field that is a field of rational functions on variables $x_\beta$ corresponding to $\omega^{\omega^\beta}$ for all $\beta<\alpha$.)