Surreal numbers: the day $\omega$ was created.

surreal-numbers

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,

$\omega = \{0,1,2,3,\ldots| \}$. And all those elements are created in a finite day, so by the day $\omega$ all its left members are already created.

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:

  • each integer is created on a finite day
  • each dyadic rational (with denominator a power of $2$) is created on a finite day
  • all the other rationals and reals are created on day $\omega$,
    • for example $\frac13 = \{0,\frac14, \frac5{16}, \frac{21}{64}, \ldots | 1, \frac12, \frac38, \frac{11}{32}, \ldots\}$
  • some other numbers created on day $\omega$ are:
    • $\omega = \{0,1,2,3,\ldots| \}$
    • $-\omega = \{|0,-1,-2,-3,\ldots \}$
    • $\epsilon = \{0 | 1,\frac12, \frac14, \frac18, \ldots \}$
  • more surreal numbers are created the next day $\omega +1$, including
    • $\omega +1 = \{\omega| \}$
    • $\omega - 1 = \{0,1,2,3,\ldots| \omega\}$
    • $\frac\epsilon2 = \{0 | \epsilon \}$
  • and many more in later days, so many that the surreal numbers make up a proper class rather than a set.
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