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: diagonalization requires a machine to read its physical source text
Statement
The diagonal contradiction needs a machine to read its own physical source text or manuscript.
Facts & Assumptions
Given: The chosen coding and the diagonal construction for .
The library has fixed an explicit binary code for each deterministic one-tape Turing machine, by A fixed effective binary encoding of deterministic one-tape Turing machines.
The diagonal machine against a hypothetical decider works by feeding that decider the coded self-application instance and then doing the opposite, by A hypothetical decider for yields a diagonal self-application machine.
A universal Turing machine exists for the chosen code syntax, so coded machine descriptions can be interpreted mechanically, by A universal Turing machine exists for the chosen coding.
Refutation
By [L1] and [L3], the data needed for self-reference are only a finite description string and a fixed interpreter for that string. Nothing in those items mentions any physical source file, manuscript, or external text artifact.
The actual diagonal step in [L2] uses the coded pair , not a literal reading of source text. So the contradiction is carried entirely by effective encoding and simulation.
Depends on
Used by
Nothing in the library uses this result yet.
Dependency tree · two levels
12 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
- A. M. Turing, On Computable Numbers, with an Application to the Entscheidungsproblem (standard reference, not scraped)