Mysterious Extremity

On the Day, all real numbers were born simultaneously, forming a universe of uncountable infinity.

The method used in standard analysis to define real numbers was the Dedekind Cut, defining real numbers through the gap between two finite rational numbers.

According to the rules created by Conway, the two sets used to define 1/3 were respectively the left set and the right set.

Among them, 0.01 was the fraction 1/4, while 0.1 was the fraction 1/2.

Every number in the left and right sets was a binary fraction, like cutting an inexhaustible stick through the ages, continuously dividing the number line in half.

As the cutting went on without end, the numbers in both sets grew more numerous, while the remaining length of the number line grew shorter.

In the end, after infinitely many steps, both the left and right sets contained infinitely many numbers, creating 1/3.

Similarly, irrational numbers such as the square root of 2 and Pi could also be defined this way.

This approach was exactly the same as Cantor's method of handling real numbers.

When studying the set of real numbers, in order to ensure that every number had a unique representation, Cantor represented numbers such as 1 and 0.5 as 0.999... and 0.4999..., forms consistent with irrational numbers.

"This feels so familiar."

Achilles lowered his head and looked at the pink-and-white Spiral Key hanging from his chest.

The process by which Conway created the universe of real numbers resembled a Zeno Machine.

On the first day, it spent 1 second and created 2 numbers.

On the second day, it spent 1/2 second and created 4 numbers.

On the third day, it spent 1/4 second and created 8 numbers.

But an ordinary Zeno Machine could only process a single infinite sequence.

The universe of real numbers