No LLM generated, heavily assisted/co-written, or otherwise reliant work.
Difficult to evaluate, with potential yellow flags.
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If you ask five AI practitioners to define Artificial General Intelligence (AGI), you might get six answers. From “median human” to “generally capable” systems, there’s been a lot of hand-wavy vagueness. Some even reject the term entirely.
The definitions suggest the same North Star: AGI is a system that possesses human-level intelligence.
It is an intuitive idea. It is also, I believe, a logical impossibility.
I do not undermine the potential greatness of AI. But “human-level intelligence” is not a valid target. It’s important to bring in the lessons from Russell’s Paradox—and the crisis it caused in the mathematical community from 1901 to 1908—into our current AI discourse and research agenda. Specifically (no pun intended), we need to bring in the Axiom of Specification.
We need to discuss AI capabilities with precision instead of framing our technological goals around a paradox. Otherwise, we are creating confusion, fear, and hype, rather than clarity.
The Trap of Unrestricted Comprehension
In the early days of set theory, mathematicians operated under a principle called Unrestricted Comprehension. The idea was simple: If you can describe a property, you can create a set of all things that have that property.
For example, you can have a set of “all red things”, a set of “all even numbers”, or even a set of “all ideas.”
Bertrand Russell, however, found a fatal flaw in this freedom. He asked: What about the set of “all sets that do not contain themselves?”
If this set contains itself, it contradicts its own definition. If it does not contain itself, then it must be included, which again contradicts the definition. This was Russell’s Paradox. It proved that you cannot just define a collection loosely and expect it to exist logically.
It further revealed that the “set of all sets“ is an incoherent concept. It is a syntax error to define a set that contains all sets. Such an entity is not a well-defined set.
I propose that “human-level intelligence” is exactly this kind of syntax error and incoherent set. And we need to stop referring to it as our goal for AI.
“Human-Level Intelligence” is a “Set of all Sets”
We treat human intelligence as a fixed bar to be cleared. But strictly speaking, we have no definition for it.
Anecdotally, we know human intelligence operates like a "set of all sets." It is an unbounded, self-referential capacity that cannot be fully specified. For example, humans don't just solve problems; we think about how we solve problems. We don't just learn; we learn how to learn. This reflection itself becomes a new capability, and we can reflect on that too.
It is a self-reference all the way down.
We're essentially saying: "We will create a machine that can do all the things in a set that cannot be fully enumerated, which includes within it the capacity to transcend any enumeration we make."[1]
We are chasing a ghost. We are trying to define the undefined.
Debating when AGI (as human-level intelligence) will arrive is like debating the value of 10. The answer isn't "infinity," and the answer isn't "zero." The answer is undefined.
It is a syntax error in our thinking.
The Way Out: The Axiom of Specification
Mathematics resolved Russell’s Paradox by abandoning Unrestricted Comprehension. They replaced it with the Axiom of Specification (a.k.a. Restricted Comprehension). What it meant was: Instead of being able to define any set with a descriptive property, we have to first start with an existing set.
You could no longer say, "I want a set of everything that has property P."
Instead, you have to say: "From an existing, well-defined set A, I want to select the subset of items that have property P."
In formal notation, we moved from:
{x∣P(x)}
To:
{x∈A∣P(x)}
This shift saved mathematics. I believe it is also the way to save AI discourse.
Specifying the Set
If we stop trying to simulate the "human mind" (an undefined set) or start from "human-level intelligence" (an undefined set) but start applying the Axiom of Specification, the path becomes clear.
We start with domains that are actually defined.
For example, instead of "human intelligence," consider "knowledge work," which is the set of cognitively demanding tasks that drive modern economies. This is actually well-defined. We know what software engineers do, what researchers do, etc. We have job descriptions, and professional standards. Our economy is a massive, well-documented set of tasks. They form well-defined sets Ai.
When we build AI, we are not creating a new consciousness. We are building a machine that captures a growing subset of Set Ai.
In this framework, an AI is not a "mind." It is a union of specific, specified capabilities derived from the set of economic activity. The union becomes impressive without ever needing to be "everything human intelligence can do."
Using Non-Paradoxical Language Is Helpful
Some AI labs and commentators are increasingly changing their communications to reflect this, and I urge that we do this more broadly and consistently as a community.
This is not just semantic pedantry. Having the words to talk about things is the prerequisite for clear thinking.
The person who says “We achieved AGI” or “human-level intelligence” and the one who says “We haven’t” aren’t disagreeing on facts, but are using an undefined term. We know this. Somehow we keep throwing these undefined terms around.
Now, I am suggesting that AGI isn’t just undefined for a lack of consensus; it is undefined. It is a syntax error in our discourse. We are trying to talk about “the set of all sets” as if it is a meaningful target. It is akin to debating if 10 were "zero" or "infinity."
Let’s instead start talking about the union of subsets of well-defined sets Ai. In plain terms, it means to be precise about capabilities. Then, we can start having meaningful discussions.
Benefits of Precision Over Using Undefined Terms
We can finally debate whether a claim is true or not, and evaluate what its implications are. For example, “This AI can now perform 80% of tasks in software engineering roles" is verifiable, and is tractable in terms of what it means for the labor market.
Precision helps us identify and address real safety risks. Many risks from AI come not from “general intelligence” but from specificcapabilities: the ability to exploit security vulnerabilities, to build catastrophic weapons, or to optimize for goals misaligned with humans. Naming these specifically helps us assess and address them.
Precision helps us improve AI safety through clear thinking and discourse. Public delusion and mania are harmful to safety. “AGI” has become a placeholder for people’s hallucinations, and is actively harmful to our research and discourse. It’s like constructing a building with jello bricks, or developing a proof with syntax errors, or trying to push code that won’t compile. Whichever your preferred analogy, specification can help fix that.
"But we are building models that generalize..."
I can understand why we ended up in the trap of talking about AI in such ambiguous terms. We are building general models (as opposed to narrow intelligence) where the models are gaining capabilities beyond what the researchers have specified. For example, language models are excellent at translation without being specifically trained for it. The models have began to generalize. The can of worms has since opened for the unspecified and seemingly unbounded new capabilities that these models can have without our knowledge.
How do we speak with precision in this case?
Consider a simple proof of existence. Self-driving AI models today are models that generalize (the AI is able to handle many exceptional edge cases on the road outside of its training data and beyond what researchers can enumerate), and that doesn't stop it from being a capability that is an existing well-defined set (driving cars).
We can make a distinction between the fact that models generalize and the incoherent claim that the models have a set of capabilities models that is syntactically undefined. The former is the fuel for artificial intelligence, the latter is a recipe for getting lost due to loose language.
It is also fine that we end up measuring the capabilities after-the-fact when the models have been pre-trained or post-trained. We already do this to some extent with benchmarks. There is also a distinction between "we cannot predict the models' capacity upfront" and the syntax error that "the models have potentially 'human-level' capabilities." The former is a tractable problem; the latter is a statement that is contrary to speaking sense.
Conclusion
As long as we remain stuck in the "Unrestricted Comprehension" of AGI, we oscillate between delusion and terror. If we switch to "Specification," we can measure progress.[2] We can see a Venn diagram of well-defined tasks and AI capabilities.[3] We can watch the circles overlap. We can talk about efficiency, displacement, and wider implications without falling into syntax errors.
As the AI community and industry insiders, it makes sense to lead with clarity, instead of erroneous thinking.
This argument is not born of "human exceptionalism." I would argue that animal intelligence is also perhaps a "set of all sets": boundless, self-referential, open-ended, and ultimately resistant to simple computational definitions.
One possible hesitation might be that the list of well-defined tasks could be very long, and beyond the capacity of AI labs to communicate clearly. This is a useful opportunity to invite social participation. Concerned citizen or affected employees can contribute to building benchmarks that they are experts in. By speaking in precision, we establish a framework for the society at large to meaningfully contribute, to discover and establish together what the capabilities of AI are.
If you ask five AI practitioners to define Artificial General Intelligence (AGI), you might get six answers. From “median human” to “generally capable” systems, there’s been a lot of hand-wavy vagueness. Some even reject the term entirely.
The definitions suggest the same North Star: AGI is a system that possesses human-level intelligence.
It is an intuitive idea. It is also, I believe, a logical impossibility.
I do not undermine the potential greatness of AI. But “human-level intelligence” is not a valid target. It’s important to bring in the lessons from Russell’s Paradox—and the crisis it caused in the mathematical community from 1901 to 1908—into our current AI discourse and research agenda. Specifically (no pun intended), we need to bring in the Axiom of Specification.
We need to discuss AI capabilities with precision instead of framing our technological goals around a paradox. Otherwise, we are creating confusion, fear, and hype, rather than clarity.
The Trap of Unrestricted Comprehension
In the early days of set theory, mathematicians operated under a principle called Unrestricted Comprehension. The idea was simple: If you can describe a property, you can create a set of all things that have that property.
For example, you can have a set of “all red things”, a set of “all even numbers”, or even a set of “all ideas.”
Bertrand Russell, however, found a fatal flaw in this freedom. He asked: What about the set of “all sets that do not contain themselves?”
If this set contains itself, it contradicts its own definition. If it does not contain itself, then it must be included, which again contradicts the definition. This was Russell’s Paradox. It proved that you cannot just define a collection loosely and expect it to exist logically.
It further revealed that the “set of all sets“ is an incoherent concept. It is a syntax error to define a set that contains all sets. Such an entity is not a well-defined set.
I propose that “human-level intelligence” is exactly this kind of syntax error and incoherent set. And we need to stop referring to it as our goal for AI.
“Human-Level Intelligence” is a “Set of all Sets”
We treat human intelligence as a fixed bar to be cleared. But strictly speaking, we have no definition for it.
Anecdotally, we know human intelligence operates like a "set of all sets." It is an unbounded, self-referential capacity that cannot be fully specified. For example, humans don't just solve problems; we think about how we solve problems. We don't just learn; we learn how to learn. This reflection itself becomes a new capability, and we can reflect on that too.
It is a self-reference all the way down.
We're essentially saying: "We will create a machine that can do all the things in a set that cannot be fully enumerated, which includes within it the capacity to transcend any enumeration we make."[1]
We are chasing a ghost. We are trying to define the undefined.
Debating when AGI (as human-level intelligence) will arrive is like debating the value of 10. The answer isn't "infinity," and the answer isn't "zero." The answer is undefined.
It is a syntax error in our thinking.
The Way Out: The Axiom of Specification
Mathematics resolved Russell’s Paradox by abandoning Unrestricted Comprehension. They replaced it with the Axiom of Specification (a.k.a. Restricted Comprehension). What it meant was: Instead of being able to define any set with a descriptive property, we have to first start with an existing set.
You could no longer say, "I want a set of everything that has property P."
Instead, you have to say: "From an existing, well-defined set A, I want to select the subset of items that have property P."
In formal notation, we moved from:
{x∣P(x)}
To:
{x∈A∣P(x)}
This shift saved mathematics. I believe it is also the way to save AI discourse.
Specifying the Set
If we stop trying to simulate the "human mind" (an undefined set) or start from "human-level intelligence" (an undefined set) but start applying the Axiom of Specification, the path becomes clear.
We start with domains that are actually defined.
For example, instead of "human intelligence," consider "knowledge work," which is the set of cognitively demanding tasks that drive modern economies. This is actually well-defined. We know what software engineers do, what researchers do, etc. We have job descriptions, and professional standards. Our economy is a massive, well-documented set of tasks. They form well-defined sets Ai.
When we build AI, we are not creating a new consciousness. We are building a machine that captures a growing subset of Set Ai.
In this framework, an AI is not a "mind." It is a union of specific, specified capabilities derived from the set of economic activity. The union becomes impressive without ever needing to be "everything human intelligence can do."
Using Non-Paradoxical Language Is Helpful
Some AI labs and commentators are increasingly changing their communications to reflect this, and I urge that we do this more broadly and consistently as a community.
This is not just semantic pedantry. Having the words to talk about things is the prerequisite for clear thinking.
The person who says “We achieved AGI” or “human-level intelligence” and the one who says “We haven’t” aren’t disagreeing on facts, but are using an undefined term. We know this. Somehow we keep throwing these undefined terms around.
Now, I am suggesting that AGI isn’t just undefined for a lack of consensus; it is undefined. It is a syntax error in our discourse. We are trying to talk about “the set of all sets” as if it is a meaningful target. It is akin to debating if 10 were "zero" or "infinity."
Let’s instead start talking about the union of subsets of well-defined sets Ai. In plain terms, it means to be precise about capabilities. Then, we can start having meaningful discussions.
Benefits of Precision Over Using Undefined Terms
We can finally debate whether a claim is true or not, and evaluate what its implications are. For example, “This AI can now perform 80% of tasks in software engineering roles" is verifiable, and is tractable in terms of what it means for the labor market.
Precision helps us identify and address real safety risks. Many risks from AI come not from “general intelligence” but from specific capabilities: the ability to exploit security vulnerabilities, to build catastrophic weapons, or to optimize for goals misaligned with humans. Naming these specifically helps us assess and address them.
Precision helps us improve AI safety through clear thinking and discourse. Public delusion and mania are harmful to safety. “AGI” has become a placeholder for people’s hallucinations, and is actively harmful to our research and discourse. It’s like constructing a building with jello bricks, or developing a proof with syntax errors, or trying to push code that won’t compile. Whichever your preferred analogy, specification can help fix that.
"But we are building models that generalize..."
I can understand why we ended up in the trap of talking about AI in such ambiguous terms. We are building general models (as opposed to narrow intelligence) where the models are gaining capabilities beyond what the researchers have specified. For example, language models are excellent at translation without being specifically trained for it. The models have began to generalize. The can of worms has since opened for the unspecified and seemingly unbounded new capabilities that these models can have without our knowledge.
How do we speak with precision in this case?
Consider a simple proof of existence. Self-driving AI models today are models that generalize (the AI is able to handle many exceptional edge cases on the road outside of its training data and beyond what researchers can enumerate), and that doesn't stop it from being a capability that is an existing well-defined set (driving cars).
We can make a distinction between the fact that models generalize and the incoherent claim that the models have a set of capabilities models that is syntactically undefined. The former is the fuel for artificial intelligence, the latter is a recipe for getting lost due to loose language.
It is also fine that we end up measuring the capabilities after-the-fact when the models have been pre-trained or post-trained. We already do this to some extent with benchmarks. There is also a distinction between "we cannot predict the models' capacity upfront" and the syntax error that "the models have potentially 'human-level' capabilities." The former is a tractable problem; the latter is a statement that is contrary to speaking sense.
Conclusion
As long as we remain stuck in the "Unrestricted Comprehension" of AGI, we oscillate between delusion and terror. If we switch to "Specification," we can measure progress.[2] We can see a Venn diagram of well-defined tasks and AI capabilities.[3] We can watch the circles overlap. We can talk about efficiency, displacement, and wider implications without falling into syntax errors.
As the AI community and industry insiders, it makes sense to lead with clarity, instead of erroneous thinking.
This argument is not born of "human exceptionalism." I would argue that animal intelligence is also perhaps a "set of all sets": boundless, self-referential, open-ended, and ultimately resistant to simple computational definitions.
I find it gratifying that “Unrestricted Comprehension” and “Specification” make sense here both in natural language and as mathematical proper terms.
One possible hesitation might be that the list of well-defined tasks could be very long, and beyond the capacity of AI labs to communicate clearly. This is a useful opportunity to invite social participation. Concerned citizen or affected employees can contribute to building benchmarks that they are experts in. By speaking in precision, we establish a framework for the society at large to meaningfully contribute, to discover and establish together what the capabilities of AI are.