Determine whether practical LLMs satisfy the expressiveness threshold

Determine whether a given large language model, represented as a finite syntactic system induced by its fixed weights, tokenizer, decoding procedure, and context window, satisfies the expressiveness condition required to encode its own Gödel numbering and formulate propositions about its derivability.

Background

The paper’s incompleteness theorem applies directly to an LLM only if the LLM-induced syntactic system is both coherent and sufficiently expressive. Sufficient expressiveness requires more than generating numerical or logical-looking token sequences: the system must internally represent the encoding of its own elements and reason about its own derivations.

The appendix notes that practical LLMs may lack an explicit symbolic representation of their rules and that their learned weights encode behavior implicitly and distributively. Consequently, whether an actual LLM crosses the formal expressiveness threshold is left unresolved as an empirical question; if it does not, the theorem does not apply directly to the LLM in isolation, although the paper argues that a composed LLM–parser or LLM–verifier system may still meet the threshold.