Hilbert's First Problem

"Discrete and continuous, Countable Infinity and Uncountable Infinity—Cantor's research lit a glimmer of dawn in the Infinite Domain."

"Actual Infinity, which had remained in chaos since Ancient Greece and was regarded by humanity as vague and incomprehensible, thus acquired a concrete form."

"In mathematics, it would not be an exaggeration to call Cantor's Set Theory and Transfinite Number Theory pioneering."

"The most crucial point was that Cantor discovered that Infinity also had different levels."

"Since there was an infinity distinct from Countable Infinity, were there a second, a third, or even infinitely many beyond it?"

"Cantor's research into lines, planes, and solids was a product of this line of thought."

Li Heng leaned against that bare jujube tree and folded a leaf in his hand into a three-dimensional shape.

"More dimensions require more coordinates to determine them. There should be more points in two-dimensional space than on a one-dimensional line, so the set of points in space should be of a higher order than an uncountable set."

"This was especially obvious in calculus. The area enclosed by an infinitely long line and the number axis could be finite."

"Cut apart and unfold a disk with finite area, and it could form a chain of infinite length."

"An object finite in higher-dimensional space could be infinite in lower-dimensional space. This seemed to show that higher-dimensional space was a greater infinity."

"Unfortunately, as mentioned before, Cantor instead proved that the number of dimensions had no effect whatsoever on the size of point sets in continuous space."

"In the world of infinity, humanity's intuitive geometric concepts were clearly useless."

"To seek larger cardinalities, Cantor started again from Set Theory, beginning with the relationship between every set and its own subsets."

A set contained three elements.

Its nonempty subsets were {a}, {b}, {c}, {a,