Is there a strict superset of the reals with total order?

Question

The ordinals immediately come to mind, but I am mainly interested if there are new elements bounded between two real numbers $[a,b]\subseteq\mathbb{R}$ that can be added to the reals while preserving total order.

Answer 1

Consider $\Bbb{R\times R}$ with the lexicographic ordering, and identify $\Bbb R$ with the diagonal, namely the points $(r, r)$.

More generally lexicographic products with linear orders and identifying the reals in the product will work pretty much always and will allow you some nice degree of control what sort of new elements you add between real numbers.

The ordinals are not really suitable since they are well ordered, and the reals are pretty far from being well ordered.

Answer 2

Yes. You can add infinitesimal elements to the reals which satisfy types of the form:

$$\{0<x\}\wedge \{x<a: a \in \mathbb{R}^{+} \} $$

We can do this for every element of $\mathbb{R}$ (we will also have to add a copy of $\mathbb{Q}$ to maintain density criterion). This new structure $\mathbb{H}$, will contain the reals and $\mathbb{R}$ will be a proper subset of $\mathbb{H}$. However, $\mathbb{R}$ will not be dense in $\mathbb{H}$. $\mathbb{H}$ will, however, be dense in itself.

Also, see here.