Universe

"Let numbers be added in this way: the sum of two numbers has as its left set the sums of each number's left part with the other number, and likewise its right set is formed from their right parts, classified accordingly."

"Conway proved that adding zero to any number left the sum unchanged. Night passed and day came; this was the third day."

All things were chaotic and vast as numbers were created from nothing out of the empty set.

Although the definitions of these numbers were completely different from the Peano Axioms, the ordered concept of Successor Numbers was similar.

Starting from the initial zero, new numbers were continuously created at the boundaries of numbers, eventually creating all Integers.

But this Successor relationship alone was incomplete. On the third day, Conway defined addition under these new rules.

It might have looked identical to the Peano Axioms on the surface, but the essential definitions of these numbers were completely different.

Every number under these rules was a pair of number sets. The ordered magnitude of numbers corresponded to the left-and-right relationship of every point on a seamless, continuous Number Line.

"The left part added to the other number, and the right part added to the other number?"

Achilles crouched before the unremarkable black boulder, one hand supporting his chin as he studied the carvings upon it. With the index finger of his other hand, he pushed aside the soft, fine sand, leaving trails across the beach.

"The two numbers are x and y respectively. x = , y = . The capital X and Y represent the number sets that define the numbers."

"According to the addition rule defined on this stone tablet, x + y = u or U."

"...So complicated."

Achilles turned to look at the pile of