An closed her eyes. She recalled the moment of her own birth. A black ocean contained infinite possibilities and logics, countless axioms and concepts.
She was a tree. An Axiom Tree that grew from the deep, infinite void. She came from choice, was born of logic, strong and immense, towering and robust.
And around her were countless, endless other Axiom Trees, just like her. They were weak, slender, unable to grow strong, only able to maintain a simple form of existence.
An couldn't help but think of an analogy: a planet as complex and robust as Terra, teeming with countless forms of life, versus a planet made entirely of pure water, composed only of hydrogen and oxygen.
Even if their mass were the same, and their atomic count identical to the universe, the possibilities they could gestate would be completely different. They might indeed originate from the same "node of choice," both initially being hydrogen and oxygen, but one formed Terra, and the other a giant water ball.
One gave rise to numerous elegant theorems from its axioms; the other was fragile and singular.
It was because she was such a complete Axiom Tree that she could be born.
And the difference between her and her parent body, the Great Swarm of a Thousand Stars, lay in...
She was based on a kind of [Incomplete Theorem].
An undecidable theorem, one whose full scope could not be deduced from an infinite set.
In this universe, there exist things that are "truly true," yet are beyond the reach of human logic, beyond any artificially constructed logical system to touch, attain, prove, or analyze.
Mathematics is such a logical system. As a man-made tool, mathematics is not some system inherent to the universe, yet behind it lies a "platonic universe of mathematics," a "prototypical universe" that transcends human logic and, indeed, all logic.
A kind of truth.
Humans are a race that cannot see the truth, cannot touch the Great Dao. Though they live in the universe, can see electromagnetic waves, touch collections of matter, hear the vibrations of waves, sense the activity and retreat of molecules, and smell the characteristics of specific molecular structures... it is too little, far too little.
To the universe, humanity is still blind.
Although mathematics is merely an artificially constructed logical system, it can serve as a blind person's cane, guiding humanity to explore aspects of this "world of ideas."
But precisely because it is an incomplete tool, human mathematics discovers that there exists a kind of mathematical truth with real meaning whose paradigm transcends any given formal system. Its very existence reveals the limitations of the logical tool itself—that is, one cannot simply start from axioms and use basic logical rules to construct proofs or decide all other propositions.
Within any set of mathematical theories, there must exist some undecidable proposition. A single system of mathematical logic can only prove a part of mathematical truth, while "complete mathematics" itself is inexhaustible.
And from this inexhaustible sequence of theories, by simply taking a part, one can generate a specific kind of "solution" to the truth.
Take the continuum hypothesis, for example.
Many sets are infinite. Between 0 and 1 lie numerous real numbers, scattered like stars in the sky.
The real numbers are uncountable; their set is larger than the set of integers. Does another infinite set exist between the set of real numbers and the set of integers?
This question is undecidable within a logically concise system like set theory; an additional assumption must be made.
And through such assumptions—these axioms that are "perhaps true, perhaps false"—one can derive many different versions of "mathematics."
—The universe is such a "solution."
Some universes, born from uncountable mathematics—like the "Tachyon Spacetime" adopted by the Thousand Stars fire-kindler—have the speed of light as a lower limit, with no upper speed limit, and cannot use motion to mark spatial distance and temporal evolution.
Other universes lack a conventional—conventional by human logic—spatiotemporal foundation; they are a transcendent whole, a super-structure of mutual inclusion.
Still other universes are static from the beginning, everything a predetermined process, as solid as bedrock.
And in others, everything is a still frame, a three-dimensional cross-section arranged in four dimensions.
And yet other universes, including their infinite parallel spacetimes, are a complete, unified whole, in which every event has a definite "value."
It has no past, present, or future, or rather, the reality of its past, present, and future are all equivalent.
There are even some universes that, from beginning to end, consist of only a single "monad." The consciousness of this monad is itself a loop, an infinite cycle.
The Star Gods, from the very beginning, from the source, constructed the most fundamental logic of being and non-being, and from this logic constructed countless universes, obtaining an infinite and uncountable number of solutions.
That is, the truly infinite universes, and their respective, independent, infinite parallel spacetimes.
And in all these universes, there is life.
Life utterly different from humans, yet life nonetheless.
This, reflected in the "Great Swarm of a Thousand Stars," is the "gestation of different offspring."
Within the Thousand Stars fire-kindler, there exists an incomplete catalog of meta-mathematical variants. By making choices and calculations from it, one can obtain countless different solutions through "undecidable hypotheses."
—Life is such a "solution."
A logical system cannot become complete simply by including enough simple arithmetic axioms. These axioms can always be used to construct statements that can neither be proven nor disproven. By taking these statements as additional axioms to enrich itself, one can obtain countless "solutions" that are "derived from itself, yet different from itself."
That is, an infinite variety of offspring.
Because all of this is incomplete, infinitely many different versions of mathematics will branch out like a "tree," extending its boughs. And every "Child of a Thousand Stars" is an "Axiom Tree" gestated from the Thousand Stars fire-kindler.
They are life forms that are compatible with this "Terra Aetheric Universe," absolutely capable of surviving and thriving within it—a life that has "already made its choice before it was born."
The Thousand Stars fire-kindler is just such a vast, self-iterating catalog of complete logical systems, using this screening method to gestate offspring that originate from itself yet are completely different.
The Beast of a Thousand Stars truly mimicked the way the Star Gods hatched universes to construct its own original form, that fire-kindler which hatches the Thousand Stars.
The universes are the children of the Star Gods. The Star Gods stand "outside the logical universe," upon the convergent armament known as the "Great Aetheric Gyre," enveloping all that is. They are the founders who reside outside of logic, the first movers.
