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.
Martin's Axiom implies CH
False statement
Martin's Axiom implies the Continuum Hypothesis.
Facts & Assumptions
Given: The hypotheses, objects, and conventions in the Statement.
MA(aleph_0) and the implication from CH to MA proves the valid direction .
Externally fixed-fragment relative consistency of MA and not CH gives the external implication by fixed finite-fragment transfer.
Counterexample
Assuming ZFC is consistent, F2 gives the consistency of a theory in which MA holds and CH fails. Thus the implication is not a theorem of ZFC, on the same metatheoretic consistency assumption under which the false statement is posed.
F1 records that reversing the arrows would confuse the valid implication with its false converse. The refutation rests on the fixed-fragment model argument of F2; it does not assume a countable transitive model of full ZFC.
Depends on
Used by
Nothing in the library uses this result yet.
Dependency tree · two levels
6 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
- Karagila, Forcing & Symmetric Extensions, Theorem 7.10 (standard reference, not scraped)