Alphabeta MathHow statement and proof provenance work
The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
- Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
- AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
- AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.
These labels describe origin, not correctness: citations and verification chips remain separate evidence.
FALSE: an unsolvable word problem means no individual word can be decided
Statement
Assume AC. If a finitely presented group has unsolvable word problem, then no individual word in that group can ever be proved trivial or nontrivial.
Facts & Assumptions
Given: AC and a finitely presented group with unsolvable word problem.
Assuming AC, some finitely presented group has unsolvable word problem (Novikov boone undecidability of the word problem, The Axiom of Choice).
Refutation
An unsolvable word problem means that no single algorithm decides triviality for all input words in that fixed group.
The empty word represents the identity in every group, and its triviality has a direct proof from the group axioms. In particular this individual word is decidable even in the group supplied by [L1].
This explicit word refutes the assertion that no individual word can be proved trivial or nontrivial.
Depends on
Used by
Nothing in the library uses this result yet.
Dependency tree · two levels
9 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- Charles F. Miller III, Decision Problems for Groups - Survey and Reflections (standard reference, not scraped)