Creation from Nothing

"The world of Nonstandard Analysis is far more complex than the world of Standard Analysis."

"Although they are collectively called Nonstandard Analysis, they actually refer to a class of mathematical systems different from the axioms of standard Set Theory."

"Some are merely conservative extensions of ordinary set-theoretic axioms. Although they introduce infinitely large and infinitely small numbers, their conclusions can still be translated into the mathematical framework of Standard Analysis."

"Others describe worlds larger than those covered by conventional set-theoretic axioms."

"It is not as simple as adding a few Hyperreal Numbers to the original foundation. Even the basic definition of numbers changes."

"In short, just as before, we will start with the most fundamental part."

Li Heng reached out, took the pink-and-white Spiral Key hanging from Achilles's chest, and aimed its tip at the white Number Line.

Under Achilles's slightly puzzled gaze, he murmured as though chanting a spell:

"Upgrade, Two-Star Zeno Machine."

A red pattern immediately appeared across the surface of the pink-and-white spiral key, marking this Hyper-Turing Machine as different from the previous one.

"All right. I have now granted it the power of Uncountable Infinity. You can use it to cut through the gaps between Real Numbers."

Hey, isn't this way too easy? What kind of card game are you playing?

Achilles grumbled inwardly.

During their previous journey, he had made every property of Uncountable Infinity sound terrifying.

He had proven that the main constituents of Real Numbers were Undefinable Numbers that could not be precisely defined with finite symbols, incomprehensible and indescribable beings to finite mortals.

Yet now, with a casual sentence, he had created an upgraded Two-Star Zeno Machine, instantly making Uncountable Infinity seem far less impressive.

"In mathematics, you do not understand anything. You merely become accustomed to it."

"There is no singular