Prove You, Regardless of Your Existence: On the Absolute Equivalence of Extraterrestrial Intelligence and Human Capability
Author: Jia, Baolong Identity: Independent Researcher Email: seer@139.com
Destroy you, what does it have to do with you. —— "The Three-Body Problem" Prove you, regardless of your existence. —— This Paper
Abstract
This is a highly counter-intuitive but mathematically strictly established conclusion: any hypothetical extraterrestrial intelligence has a fundamental upper limit of logical and computational capability that is absolutely equivalent to that of humanity today. This paper proves through set theory and computability theory that in the universal set of all possible problems, unsolvable problems occupy an absolute 100% measure. According to the Church-Turing-Deutsch Thesis, the computational capability of any physical system (whether human or alien) cannot exceed that of a Turing machine. Therefore, the solve rate of humanity is 0, and the solve rate of extraterrestrial intelligence is also 0. When facing the ultimate truth of the universe, all intelligent agents are firmly locked within the same mathematical limit. The universe is not a slaughterhouse of infinite technological explosions, but an absolute mathematical wall of sighs faced collectively by all intelligent agents.
Part I: Formal Mathematical Proof
To completely eliminate the ambiguity that natural language might bring, this section first translates the core logic into strict mathematical and formal logic language for a complete proof.
Definitions and Assumptions: * Let $H$ be the set of human intelligent agents. * Let $A$ be the set of hypothetical extraterrestrial intelligent agents. * Let $\mathcal{P} = \{0,1\}^{\mathbb{N}}$ be the set of all decision problems (i.e., all mappings from natural numbers to $\{0,1\}$). * Let $\mathcal{C} \subset \mathcal{P}$ be the set of Turing-computable problems.
Lemma 1 (Physical Computation Upper Bound): According to the Church-Turing-Deutsch Thesis, the set of problems $\text{Solve}(X)$ that any physical system (regardless of its physical substrate or technological level) can solve must be a subset of the Turing-computable set $\mathcal{C}$. That is: $\forall X \in \{H, A\}, \text{Solve}(X) \subseteq \mathcal{C}$.
Lemma 2 (Zero Measure of Computable Sets): According to Cantor's theorem, $|\mathcal{P}| = 2^{\aleph_0}$ (cardinality of the continuum). According to Turing machine encoding theory, $|\mathcal{C}| = \aleph_0$ (countably infinite). Under the standard measure of the uncountable set $\mathcal{P}$, the measure of any countable subset is zero. Therefore, $\mu(\mathcal{C}) = 0$.
Lemma 3 (Solve Rate Definition): Define the solve rate measure $r(X)$ of an intelligent agent $X$ as:
Main Theorem (Absolute Equivalence): For humanity $H$: $\text{Solve}(H) \subseteq \mathcal{C} \implies r(H) = \mu(\text{Solve}(H)) \leq \mu(\mathcal{C}) = 0 \implies r(H) = 0$. For extraterrestrial intelligence $A$: $\text{Solve}(A) \subseteq \mathcal{C} \implies r(A) = \mu(\text{Solve}(A)) \leq \mu(\mathcal{C}) = 0 \implies r(A) = 0$.
In summary, in the metric space of the solve rate for the ultimate problems of the universe:
Part II: Philosophical Speculation and Epistemological Dead End
1. The Failure of Physical Metrics and the Dimensionality Reduction of Logical Space
When discussing extraterrestrial civilizations, humanity habitually falls into the illusion of technological worship, attempting to measure a hypothetical intelligence using energy scales (like the Kardashev scale) or spatial spans. However, such discussions ignore a fundamental epistemological dead end: Any metric relying on physical observation inevitably collapses when crossing the chasm of existence.
We cannot "weigh" the mass of an unknown intelligence, nor can we "measure" the speed of their spacecraft. When all physical information channels are strictly closed by the absolute barrier of logic, attempting to use physical parameters to compare two intelligent agents that might exist in completely different states is futile.
When physical metrics completely fail, the only metric that can cross the boundary between existence and non-existence, achieving a universal definition, is a purely abstract scalar: the Solve Rate.
This is not a compromised choice among many indicators, but a logical, inevitable convergence. In the ultimate epistemological dead end, the solve rate becomes the only legitimate metric.
2. The Wall of Sighs: Zero Score Across the Board
Once this unique metric is established, the contest escapes the noise of the physical universe and enters the silent realm of pure mathematics.
- The Vastness of All Problems: The set of all decision problems is essentially all mappings from natural numbers to $\{0,1\}$. According to Cantor's diagonal argument, the size of this set is uncountably infinite.
- The Insignificance of Solvable Problems: According to the Church-Turing-Deutsch Thesis, the computational capability upper bound of any physical system (whether its substrate is carbon, silicon, or some unknown dark matter structure) is strictly framed by a Turing machine. All possible Turing machines can be encoded as finite-length strings, so the set of computable problems is merely countably infinite.
- The Ruthlessness of Measure Theory: In the ocean of uncountable infinity, the measure of any countably infinite subset is strictly 0.
Deduction Conclusion: The solve rate of humanity is 0. The solve rate of any hypothetical extraterrestrial intelligence is also 0.
3. 0=0: Equality Before the Limit
This is perhaps where the proof is most likely to trigger an intuitive backlash.
In daily, finite experience, 0=0 is often seen as meaningless nonsense. If the flight speed of both a stone and a human is 0, we obviously cannot say a stone and a human are equivalent, because we have countless other dimensions of metrics like weight and volume to distinguish them.
But here, the situation is fundamentally reversed: The solve rate is the only surviving metric.
When the epistemological dead end erases the definability of all other dimensions, when the information set converges to only one dimension, and the value of that dimension is 0, according to Leibniz's "Identity of Indiscernibles," 0=0 is no longer nonsense, but complete equivalence in the sense of information theory.
No matter how many specific "god-level technologies" an intelligence has mastered, in the face of the uncountably infinite logical space, finite differences are completely flattened.
Conclusion: The Cold Inevitability
There is a famous line in "The Three-Body Problem": "Destroy you, what does it have to do with you." It depicts a cold and accidental cosmic hunt based on physical laws.
The proof in this paper reveals a deeper truth based on mathematical laws: "Prove you, regardless of your existence."
The universe is not a slaughterhouse that allows infinite technological explosions. The iron law of mathematics dictates that there is no "god-level civilization" capable of infinitely unlocking mysteries. When humanity invented the first Turing machine, when we proved Gödel's Incompleteness Theorems and the Turing Halting Problem, we were already on equal footing in the ultimate upper limit of logical capability with any hypothetical intelligent entity.
They do not possess more magic than we do. When facing that 100% of unsolvable problems, the despair they feel is no different from ours.
This is a knot. Those who understand will naturally understand: In the face of absolute truth, all existences are equally powerless.