What exactly is surreal star? What does it mean that it is incomparable to zero

combinatorial-game-theoryorder-theoryphilosophysurreal-numbers

I have been thinking about this for a while & I am perplexed.

What exactly is surreal star? I am aware that it is a fuzzy game. What I don't understand is what exactly that means. Wikipedia says a fuzzy game is:

incomparable with the zero game; it is not greater than 0, which would be a win for Left; nor less than 0 which would be a win for Right; nor equal to 0 which would be a win for the second player to move. It is therefore a first-player win.

This brings me to the first thing I don't understand: what does it mean for a value to be incomparable to a number? I have read the answer to Distinction between “fuzzy” and “confused with.”, however I still do not really comprehend why star & zero are incomparable.

A part of my confusion is that in the surreal numbers there is the concept of $\uparrow$, which is defined as $\{0|*\}$ (greater than zero & less than star) & $\downarrow$, which is defined as $\{*|0\}$ (greater than star & less than zero). If star & zero are incomparable, how can this comparison be made?

Best Answer

Numbers and Games

The $\{a,b\mid c,d,e\}$ notation used for surreal numbers is also used for representing certain games more generally. Basically, a "game" lets you put any sets of games as the left and right set. But a game is only a (surreal) number if all elements of those sets are numbers and no right element is less than or equal to any left element. For clarity, $*$ and $\uparrow$ are not (surreal) numbers, just games.

It turns out that numbers have nice properties: If $x=\{a\mid b\}$ is a number, then $a<x<b$ is true. However, that does not hold for games in general. So "greater than zero & less than star" is not a correct way of thinking of "$\{0\mid*\}$".

Definition of Inequality

To understand the meaning of inequalities and what incomparable would mean, we need a definition of inequality for games. There are a few equivalent definitions, but one that takes the least work to set up is given in Claus Tøndering's Surreal Numbers -- An Introduction. Paraphrased, Definition 2 says:

$x\le y$ if and only if $y$ is less than or equal to no member of $x$'s left set, and no member of $y$'s right set is less than or equal to $x$.

Now that we have this recursive definition of $x\le y$, we can define other (in)equality symbols:

  • $x=y$ when $x\le y$ and $y\le x$ both hold.
  • $x<y$ when $x\le y$ holds but $y\le x$ does not.
  • $x\not\gtrless y$ ($x$ is "incomparable to" $y$) when neither of $x\le y$ and $y\le x$ hold.

You can see a notation-heavy use of this definition of $\le$ in this answer of mine explaining in detail how to check that $\{0\mid1\}$ is a number.

How can things be incomparable?

For example, consider the game (not a number) $s=\{1\mid-1\}$. If you check the definition of inequality above (or any equivalent one), you'll find that it's greater than $-2$ and less than $2$. But $s\le1$ and $1\le s$ are both false, so that $s$ is "incomparable to"/"confused with" $1$ (we might write $s\not\gtrless 1$). Similarly, $s$ is confused with $0$ (so "fuzzy") and confused with $-1$ as well. It is simply not true that "$s$ is somehow greater than $1$ and less than $-1$".

Your examples of $*=\{0\mid0\}$ and $\uparrow=\{0\mid*\}$ are similar. $*<1$ is true but $*\le0$ and $0\le*$ are not true (so $*\not\gtrless 0$). $0<\uparrow$ happens to be true, but $\uparrow\le*$ and $*\le\uparrow$ are not true (so $\uparrow\not\gtrless *$).

Related Question