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.
Forcing preserves ordinals
Statement
In ambient ZF, if M is a transitive ZF ground model and G is M-generic for a nonempty forcing preorder in M, then
This is preservation of ordinals as sets; no preservation of their cardinality or cofinality is asserted.
Facts & Assumptions
Given: The stated ground model and generic extension.
Generic extensions satisfy ZF and preserve ground-model Choice gives transitive ZF M[G] containing M; only its ZF branch is used.
Names for pairs, functions and ordinals gives check names for ground ordinals with their original values.
Absoluteness of names and their ranks identifies the name rank of a ground name as an ordinal belonging to M.
Proof
If , F3 gives a ground name whose value is gamma, so . Its being an actual ordinal is unchanged. Thus .
If , write for a name . By F4, is an actual ordinal in M. Since an ordinal has membership rank equal to itself, F2 gives . If it is in M directly; if , transitivity of M puts . This includes gamma zero.
The inclusions in steps 1.1 and 1.2 prove the equality. The upper-bound argument compares actual ordinal sets and uses no enumeration, cardinal arithmetic, cofinal map or AC; it therefore makes no claim that the extension has the same cardinals or cofinalities.