Is$\ +\infty$ greater than any other number (surreal, superreal, hyperreal, ...)?

Question

Let$\ \mathbb{A}$ be an arbitrary totally ordered set and consider the largest element of the set of extended real numbers,$\ +\infty$. Can we say that$\ +\infty > \chi $, for *any*$\ \chi \in \mathbb{A}$?

Answer 1

We can make it a convention that we would only use $+\infty$ in that way. For example, the natural interpretation (via transfer) of that symbol in non-standard analysis is, in fact, the largest extended hyperreal.

This isn't too different from the idea that we often use symbols $1$ and $0$ to denote the largest and smallest elements in a bounded lattice.

It would even be reasonable, in the construction where we take an ordered set and adjoin a new element and extend the ordering so the new element is the largest element, to adopt the convention that we use $+\infty$ as the name for this new element.

However, if we are using $+\infty$ to refer to a specific object with a prior meaning, then we can't always assume it is the largest element of an ordered set. For example, we can construct an ordered set $(S, <)$ defined by

• $S = \overline{\mathbf{R}} \cup \{ \star \}$
• $x < y$ is true if and only if ($x,y \in \overline{\mathbf{R}}$ and $x<y$ as extended real numbers) or ($y = \star$)

In this example, we would have $+\infty < \star$.

Or for a more amusing example, we could use the ordered set $(\overline{\mathbf{R}}, >)$; that is, we take the extended real numbers and reverse the ordering. This is still an ordered set, and $+\infty$ is its smallest element.