I'm reading Surreal Numbers, by Knuth, and I don't understand well the formalism at the very beginning. Here is what it is unclear:

We define that a number $x $ corresponds to a couple of set (of numbers), namely $x=(X_L,X_R) $, where $X_L\ngeq X_R$. However Knuth just defines the relation $x\le y $, that corresponds equivalently to $$(*)\ \ \ x\ngeq Y_R, \ X_L\ngeq y.$$ So for example, once we obtained $0$, $-1$ and $1$, how can we say if $(0,1)$ is a number? (Namely that $0\ngeq1$). We only known that $0\le 1$, not that $0\ngeq1$; and we know that $0,1,-1$ are numbers and that $-1\le0\le 1$ just because the conditions ($*$) are easily verified with empty sets. Or does it suffice that $x $ and $y $ are different numbers (in the sense that $X_L\ne Y_L$ or $X_R\ne Y_R $), to have $x\ngeq y? $ (Obviously assuming that $x\le y $ is verified). Thank you in advance

  • $\begingroup$ Having defined $\leq$, $a=b$ is defined by $a\leq b\wedge b \leq a$ and $a<b$ is defined by $a \leq b \wedge a\neq b$. Since $0>0$ is false (because $0=0$) and $1=\{0\ | \ \varnothing \}$, it follows that $0\geq 1$ is false. $\endgroup$
    – nombre
    Mar 1 '20 at 17:20

Notation for sets and numbers:

This is not a big problem now, but will be when you do a bit more with the surreal numbers: We must be very careful to distinguish numbers from sets of numbers.

As you said, a number is a pair of sets of numbers. That means that $(0,1)$ should definitely not be a number, since $0$ and $1$ are not sets of numbers, they're numbers. When you wrote $(0,1)$, I assume you intended $(\{0\},\{1\})$.

Similarly, When you see capital letters as in $X_L$, we are discussing sets of surreal numbers, not individual numbers. When we are clear about this, that gives us a chance to use convenient shorthands. (Here I assume you are comfortable with $\forall$ and $\in$ for "for all".) Make sure you are clear that $x\ngeq Y_R$ is a shorthand for something like $\forall y_2\in Y_R,\,x\ngeq y_2$. And $X_L\ngeq X_R$ is a shorthand for something like $\forall x_1\in X_L,\,\forall x_2\in X_R,\,x_1\ngeq x_2$.

Because of these shorthands, writing something like $X_L\ne Y_L$ would almost never be done because of the ambiguity. The reader would have to guess whether it means they are not literally the same sets (as was intended in the question post) or if it means that $\forall x_1\in X_L,\,\forall y_1\in Y_L,\,x_1\ne y_2$? etc.

Main answer:

To make this more self-contained and to clear up any confusion about the basic facts, I'll build up to $(\{0\},\{1\})$ being a number from the beginning.

Consider $(\varnothing,\varnothing)$. If we want to check if it's a number, we must check the condition $\varnothing\ngeq\varnothing$. That shorthand expands to something like $\forall x_1,x_2\in\varnothing,\ldots$, which is vacuously true. So $(\varnothing,\varnothing)$ is a number, and we call it "$0$".

Now consider $(\{0\},\varnothing)$. We must check $\{0\}\ngeq\varnothing$. But this expands to something of the form $\forall x_1\in\{0\},\forall x_2\in \varnothing,\ldots$, and is still vacuously true. So $(\{0\},\varnothing)$ is a number, and we call it "$1$".

Now consider $(\{0\},\{1\})$. Let us check whether it is a number, without knowing what background facts might help. The condition is $\{0\}\ngeq\{1\}$. That is shorthand for $\forall x_1\in\{0\},\,\forall x_2\in\{1\},\,x_1\ngeq x_2$. Since those sets only have one element each, the only statement we need to check is $0\ngeq1$. By standard convention for the slash $/$, this means we must check that $0\ge1$ is false. By standard convention for symbols with left-right orientation like this, we must check that $1\le0$ is false.

By definition, $1\le0$ (i.e. $(\{0\},\varnothing)\le(\varnothing,\varnothing)$ would mean both $1\ngeq\varnothing$ and $\{0\}\ngeq0$ hold. So we can show it is false by showing $\{0\}\ngeq0$ is false (incidentally, $1\ngeq\varnothing$ is vacuously true). And $\{0\}\ngeq0$ is shorthand for $\forall x_1\in\{0\},\,x_1\ngeq 0$. Since the set has only one element, this reduces to $0\ngeq 0 $. And to show $0\ngeq 0$ is false, we just need to show that $0\geq 0$, i.e. $0\leq 0$.

Why is $0\leq 0$ true? Well it means $0\ngeq\varnothing$ and $\varnothing\ngeq 0$. But both of those are vacuously true!

This was a lot of drilling down into definitions, so I'll reverse the order of the deductions to summarize:

  1. $0\ngeq\varnothing$ and $\varnothing\ngeq0$ are vacuously true.
  2. $0\le0$ (is true).
  3. $0\ngeq0$ is false.
  4. $\{0\}\ngeq0$ is false.
  5. $1\le0$ is false.
  6. $0\ngeq1$ is true.
  7. $\{0\}\ngeq\{1\}$ (is true).
  8. $(\{0\},\{1\})$ is a number.

On Notation:

Many sources discussing surreal numbers will use a more compact notation. $(\{a,b\},\{c,d,e\})$ would instead be written $\{a,b\mid c,d,e\}$, and something like $1=(\{0\},\varnothing)$ would be written $\{0\mid \,\}$.


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