Using this definition, the twenty numbers from generations 0-2 break down into
ten equivalence classes based on similarity, as shown below.
-$$\surreal{0}{} = \surreal{-1,0}{}$$
+$$\surreal{0}{} \similar \surreal{-1,0}{}$$
$$\surreal{}{}$$
-$$\surreal{}{0}, \surreal{}{0,1}$$
+$$\surreal{}{0} \similar \surreal{}{0,1}$$
$$\surreal{-1}{}$$
-$$\surreal{1}{}, \surreal{0,1}{}, \surreal{-1,1}{}, \surreal{-1,0,1}{}$$
+$$\surreal{1}{} \similar \surreal{0,1}{} \similar \surreal{-1,1}{} \similar \surreal{-1,0,1}{}$$
$$\surreal{-1}{1}$$
-$$\surreal{0}{1}, \surreal{-1,0}{1}$$
+$$\surreal{0}{1} \similar \surreal{-1,0}{1}$$
$$\surreal{}{1}$$
-$$\surreal{-1}{0}, \surreal{-1}{0,1}$$
-$$\surreal{}{-1}, \surreal{}{-1,0}, \surreal{}{-1,1}, \surreal{}{-1,0,1}$$
+$$\surreal{-1}{0} \similar \surreal{-1}{0,1}$$
+$$\surreal{}{-1} \similar \surreal{}{-1,0} \similar \surreal{}{-1,1} \similar \surreal{}{-1,0,1}$$
From this we see that, since Axiom \autoref{ax:leq-comparison} makes its
comparison element-wise, every surreal number generated by our current methods