The Turtle He Could Never Catch

The cardinality of the Continuum was not Aleph-One.

Achilles recalled the Turing Degree Hierarchy diagram on the previous sheet of paper, which had listed the power levels of Hyper-Turing Machines. He asked with a puzzled expression:

"But on that chart, didn't you draw a line beneath the Infinite Time Turing Machine and write down the Real Continuum?"

On that Turing Degree Hierarchy chart, every Hyper-Turing Machine belonged to the level of Countable Infinity. Only the Real Continuum at the very end was Uncountable Infinity.

By that reasoning, Cantor's Continuum Hypothesis should hold true in this world.

The Power Set of Natural Numbers, the set composed of all real numbers, and the set composed of all countable ordinal numbers all had the cardinality of Uncountable Infinity 1.

"No, wait, I think I understand!"

Achilles glanced at the White Light radiating from the sunlike halo above his head and suddenly realized.

"What you wrote on that paper was the Real Continuum, not the Continuum."

"You mean that, in this world, even all real numbers still cannot fill the entire Number Line?"

Li Heng nodded.

"Correct."

"It actually isn't hard to imagine. The Second Mathematical Crisis was a crisis caused by the confusion between Actual Infinity and Potential Infinity—or, more precisely, the contradiction between infinitesimals and zero."

"Leibniz used Actual Infinity in his calculus, and this was also the main angle from which Bishop Berkeley attacked the theoretical foundations of calculus."

"From an ontological perspective, Leibniz regarded infinitesimals as the smallest indivisible atoms from which all things were composed. Their absolute values were Actual Infinity smaller than any real number."

"Cauchy and Weierstrass's concept of limits, Dedekind Cuts using Rational Numbers to divide a continuous straight line, and Cantor's method of representing decimal infinite numbers with sequences of Rational Numbers—these three were equivalent