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.
Positive relative consistency of CH and GCH
Statement
implies and ; consequently implies both positive consistency statements.
Facts & Assumptions
Given: The fixed effective presentations used by the preceding theorem; CH is the instance of its selected GCH sentence.
Formal consistency of ZFC plus GCH relative to ZF proves in PA that implies .
Proof
There is a fixed finite proof of CH: instantiate GCH at the first infinite initial ordinal and expand the selected cardinal notation. Hence appending this proof and replacing uses of the CH axiom gives a primitive-recursive map from any refutation to a refutation, and PA verifies the map by its finite line-prefix check. Therefore implies .
Combining step 1.1 with F1 gives both implications from . Separately, the inclusion of every certified ZF axiom in ZFC gives an identity-on-lines primitive-recursive map from ZF refutations to ZFC refutations. Thus PA proves . Composing this with the two implications already proved gives both conclusions from as claimed.