Pseudo-Surreal numbers are analogous to?

Question

I've been exploring surreal numbers. Real equivalent of the surreal number {0.5|}

I see that pseudo-surreal numbers seem to have an interesting branch of game theory.

Still having a form of {x|y}, where 'x' and 'y' are sets of surreal numbers, this includes integer and real numbers, as they have equivalent surreal numbers.

But pseudo-surreal numbers don't have to follow the other rules, that result in ordinary valid surreal numbers, such as the left term being less than the right term.

How can pseudo-surreal numbers be described?

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Is "The value that is both less than 2 and more than 4." a reasonable description of {4|2}?

Is {2|2} analogous to "Between 2 and 2." or "The value that is both less than two and more than 2.", or 0 as that is the oldest value in that range? With the latter interpretation, does {0|0} behave differently?

Answer 1

As GEdgar said, these "pseudo-surreal numbers" are "games" in the technical sense of Combinatorial Game Theory. Basically, instead of thinking of this as a number system any more, we can think of these as game positions of a game with two players (conventionally called Left and Right, but they might correspond to, say, Black and White in a game of Chess). The "normal play convention" is that the first player to be stuck without a move loses the game, and the two sets tell you the moves.

For example, in $1=\{0|\}$, Left can move to $0=\{|\}$, and right has no moves available. Therefore, whether Left goes first or Right goes first, Left will win for sure. That makes this game "positive": it's better for Left than $0=\{|\}$.

If you want to know how these new values fit with the surreal number line, it's a little complicated: $\{0|0\}$, conventionally and hereafter referred to as $*$ (as columbus8myhw pointed out) is less than every positive surreal (positive surreals are clear benefits for Left compared to this game), and greater than every negative surreal (negatives are benefits for Right) but if you look at the definition of $\le$, this incomparable to $0$. We might say that it's "confused with $0$".

$\{4|2\}$ is a little weirder. It's greater than any surreal less than $2$, and less than any surreal greater than $4$, but confused with everything in the "closed interval" between. As an aside, this is more conveniently thought of as $\{1|-1\}+3$, which might be written as $3\pm1$.

If you would like to learn more about combinatorial game theory, I would recommend getting Lessons in Play (perhaps at a university library), but there are many resources online. One introduction aimed at advanced high school students is at https://simomaths.wordpress.com/2012/08/02/combinatorial-game-theory-i/. I have a slightly drier/more advanced treatment at https://combinatorialgames.wordpress.com/welcome-to-the-blog/. There are also plenty of PDFs and slides floating around the internet with a variety of different takes on the basics.

Answer 2

This isn't an answer

But I have a finite surreal calculator that works for normal surreal number. I entered pseudo-surreal number into the program to get these results:

{4|2} compared equally to {3|} against all other numbers... but was unequal to {3|} or "4"

{2|2} compared equally to {1|} against all other numbers... but was unequal to {1|} or "2"

{0|0} compared equally to {|} against all other numbers... but was unequal to {|} or "0"

This was a simple test comparing only a few numbers, so this may not be true. I don't know if it valid to compare pseudo and non-pseudo numbers into routines originally designed for non-pseudo numbers.

I also wonder what results from adding these numbers?