The remnant of matter after annihilation was inevitable.
Thinking from a Gödel's Incompleteness Theorem Perspective:
Incompleteness is Inevitable; a Remnant After Annihilation
■ Introduction
"Why does matter exist in this universe?"
One of the great mysteries of physics is the asymmetry between matter and antimatter. Immediately after the Big Bang, matter and antimatter should have been created in nearly equal amounts. If the amounts were perfectly equal, everything would have annihilated, leaving a universe of only light. So why was there a "slight" excess of matter?
Traditional explanations have pointed to symmetry breaking (CP violation) and subtle asymmetries in physical laws. However, this article re-examines the problem from the perspective of a giant in mathematical philosophy: Gödel's Incompleteness Theorems.
■ What is Gödel's Incompleteness Theorem?
In simple terms, the Incompleteness Theorem states:
Any sufficiently complex system will contain truths that cannot be proven within that system.
A completely perfect, self-contained system cannot exist.
This means that "completeness" is logically impossible. Any system is inherently incomplete.
■ Incompleteness is the Destiny of the Universe
Let's apply this theorem to the universe.
The universe is a vast machine, governed by the "formal system" of physical laws. What Gödel demonstrated in mathematics applies equally to the natural world. A state of perfect symmetry and perfect uniformity is logically unsustainable. An "unprovable deviation" must inevitably arise somewhere.
This deviation is what likely caused the subtle difference in the amounts of matter and antimatter after the Big Bang. The probability of them being in perfectly equal amounts is mathematically and physically infinitesimally close to zero.
■ We Just Call the Remnant "Matter"
From this perspective, there's no need for a complex explanation of why matter survived.
The survival of a small number of particles during the annihilation process is the inevitable consequence of incompleteness. We simply call the particles that survived "matter." If, by chance, antimatter had been the one to remain, we would have called that "matter" instead. This relative definition also suggests that existence itself is dependent on the context of the observer.
■ Black Holes and Incompleteness
Interestingly, this incompleteness may also be linked to the conditions for the birth of black holes. If the early universe were perfectly uniform, there would be no "seeds" for gravitational collapse, making black holes extremely rare. The fluctuations in density are the trigger for structure formation, and their source is also incompleteness.
■ Conclusion
Gödel's Incompleteness Theorem may not be limited to mathematics; it might prophesy the fundamental nature of the universe. Perfection is impossible, and incompleteness is the mother of existence.
Considering that this fated incompleteness slightly tipped the balance of matter and antimatter, allowing for the existence of galaxies, stars, and ourselves, it seems the universe was always, as a matter of logical necessity, a story where "something would remain."
