pseudo numbers and surreal numbers

Question

A surreal number $\{x_L\|x_R\} \in No$ is a number when for all $\xi\in x_L$ and all $\eta \in x_R$ we have $\eta > \xi$. All the things $\{x_L\|x_R\}$ which are not of that form are called "pseudo-numbers" and usually ignored (except in certain game-theoretic applications).

Questions:

• Is it possible (in an algebraic, analytic or topological sense) to make sense of these pseudo-numbers ?
• Can one obtain a "filling of the gaps" of $No$ by extending $No$ with pseudo-numbers ?
• Is there research in which the pseudo numbers are studied ?
• When you study $\mathbb{A}^{1,an}_{No}$ as the set of (bounded, non-archimedean) seminorms on $No[T]]$, is there a connection between pseudonumbers and the set $\mathbb{A}^{1,an}_{No} - No$, also known as the "hyperbolic space" $\mathbb{H}^{1,an}_{No}$ among those who a acquainted with the Berkovich terminology ?

Answer 1

Fleshed out to an answer since it got too big for a comment:

• 'things which are not of that form are called "pseudo-numbers" and usually ignored.'
• 'Is there research in which the pseudo-numbers are studied?'

These two statements seem rather at odds with each other, but yes — as Ted's answer notes, the subject you're looking for is Combinatorial Game Theory and it devotes quite a bit of study to these objects. While the order on games is only a partial order, there are games that can be compared with all numbers and do fall into the gaps you're speaking about - for instance, the game $\uparrow = \{0|\{0|0\}\}$ can be shown to be larger than zero but smaller than any positive number, even infinitesimal ones (this is a nice exercise!).

Note that even after plugging in games that can be ordered with respect to all numbers, there are still gaps that can be filled; this is usually done by taking $X_L$ and $X_R$ as proper classes rather than sets, although another tricky appraoch that shows up in Winning Ways is to give up the axiom of foundation (in their notation, go beyond the class of enders) and consider, e.g., games like $\mathbf{On}=\{\mathbf{On}|\}$ (which can easily be seen to be larger than any 'proper' game in a suitable sense). I believe this is equivalent for all practical purposes to defining $\mathbf{On}=\{V|\}$, where $V$ denotes 'the universe', but abandoning foundation allows games to be considered that couldn't be expressed otherwise.

Unfortunately, I don't know anything about the topological properties of these objects - I wasn't even aware that the Surreals had gotten much study in a topological sense! That's a subject that I'd like to know quite a bit more about myself.

Answer 2

A game is an object $\{X_L | X_R \}$ where $X_L$ and $X_R$ are sets of games. So this includes both the surreal numbers and the "pseudo-numbers".

Equality, comparison, and addition of games are defined in the same way as for surreal numbers. Just as for surreal numbers, the games form an abelian group, and the group structure is compatible with the ordering.

However, unlike surreal numbers, 2 games cannot always be compared with each other (there exist games $x$ and $y$ for which neither $x \le y$ nor $y \le x$ holds), and multiplication of two games is not in general defined (the definition for surreal numbers does not carry over to games because it's not well-defined with respect to the notion of equality). But some products can be defined on limited subsets of games, which have reasonable properties.

Combinatorial game theory is the subject which studies all of this.