I've been reading the absolutely great "Surreal Numbers" by Knuth, but I've been stuck on the $\aleph$ day problem (apologies for not using the right symbol: the right symbol is \aleph
). Alice and Bob (pretty great mathematicians) are discussing the $\aleph$ day on page 94 of the book.
What I do understand is that on the $\aleph$ day, all real numbers are been created: Bill says
… when it says the universe was created on $\aleph$ day: The real numbers are the universe.
Now, from previous chapters and how numbers like $1$, $-1$, etc. have been created, they've always needed numbers from the previous days in $X_L$ or $X_R$.
What I'm struggling to understand, however, is how $\omega$, (or $\epsilon$, for that matter), gets created on this day, when $\omega$ is also not in the real number system, and is greater than all real numbers. Wouldn't it require all the reals in $X_L$ to be greater than the reals themselves? (And hence require the reals to have been created on a previous day?)
Best Answer
As @jjagmath commented,
Each surreal number in $X_L$ is less than each surreal number in $X_R$, trivially in this case since $X_R$ is empty here.
Note that what Knuth wrote as $\aleph$ would be written as $\omega$ by Conway in On Numbers and Games, since it fits with the class of surreal numbers including all ordinals. We can say a little more: