Introductory analysis classes make it a point to observe some of the critical properties of the reals that have allowed so much to be done with them.
My takeaways are that the most important properties of $\mathbb{R}$ are that it is a complete, totally-ordered field equipped with a distance metric.
One notable property missing is algebraic closure, which is achieved with $\mathbb{C}$ at the expense of well-ordering.
So this leads me to a couple questions:
1) Are there other "nice" properties that $\mathbb{R}$ is missing?
2) Are there any field extensions of $\mathbb{R}$ which have a total order, and if so, are they algebraically closed? If no such extensions exist, why not?
3) Are there alternatative (even better, non-isomorphic) structures to $\mathbb{R}$ which share most or all of its "nice" properties? Or even other structures we can develop calculus on?
4) In particular (specification of 3), are there other complete fields?
5) What happens to calculus and analysis when we stop requiring some of the properties the reals have?
