The Countable Cave of Language
Dr Charles T. Gray 
“Not everything that counts can be counted.”
But some things can. That’s the point here.
LLMs live in a countable cave
Let Σ be the set of tokens used in a language model.
It’s finite. The set of all finite-length sequences over Σ is denoted Σ*.
By basic set theory:
A countable union of finite sets is countable.
Since:
- Σ⁰ is finite (the empty string)
- Σ¹ is finite (all 1-token strings)
- Σ² is finite
- …
- Σ* = ⋃ₙ₌₀^∞ Σⁿ
Then Σ* is countably infinite.
No matter how large your model, how clever your architecture, how long your prompts—
if it operates over a finite vocabulary and generates finite-length strings,
its entire output space is countably infinite.
That’s not a limitation of scale.
It’s a foundational constraint.
Human language might not live in that cave
Do I think human language is uncountably infinite?
Maybe. Maybe not. It’s an open question. But I do think human expression—over time, across context, culture, ambiguity, and non-tokenised thought—may not be fully captured by a countable formal system.
That’s what this post is about.
The shape of truth claims.
The ontological structure of what LLMs cannot touch.
So what?
You can train on the entire internet and never leave Σ*.
You can hallucinate whole citations and still be inside a countable cave.
I don’t write this to diminish the systems we build.
I write it to remember that truth, expression, and human meaning
might cast shadows too rich for countable spaces.
And we should name that.
LLMs don’t reveal the world.
They render what’s visible from inside the cave.