Alphabeta Math
CorollaryStatement: AI-adaptedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-14
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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.

PFA implies MA(aleph-one) and the Suslin Hypothesis

Statement

In ZFC plus PFA, MA(ℵ1) holds and the Suslin Hypothesis holds.

Facts & Assumptions

Given: PFA.

[F1]

PFA supplies a filter meeting any family of at most ω1 dense sets in a proper forcing. The Proper Forcing Axiom

[F2]

Every ccc forcing is proper. Ccc and countably closed forcings are proper

[F3]

MA(ℵ1) rules out Suslin trees. MA(aleph-one) eliminates Suslin trees

[F4]

A Suslin line exists exactly when a Suslin tree exists; consequently SH is equivalent to nonexistence of a Suslin tree. Kurepa equivalence

[A1]

The supplier theorems work in ZFC and propagate their stated uses of AC. The Axiom of Choice

Proof

1.1

Let P be ccc and let D be a family of at most ω1 dense subsets of P. By F2, P is proper, so F1 gives a filter meeting all members of D. This is precisely MA(ℵ1).

F1F2A1Given
2.1

Applying F3 to step 1.1 shows that no Suslin tree exists.

F3A1step 1.1
3.1

By F4, nonexistence of Suslin trees is equivalent to nonexistence of Suslin lines, which is the Suslin Hypothesis. Thus PFA implies both asserted conclusions. No value of the continuum was used.

F4A1step 2.1∎

Depends on

Used by

Dependency tree · two levels

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Sources