Alphabeta Math
TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-14
How 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.

A Suslin line yields a Suslin tree

Statement

In ZFC, if a Suslin line exists, then a Suslin tree exists.

Facts & Assumptions

Given: A Suslin line S. Assume AC.

[F1]

The nowhere-separable quotient and exact completion of S is a dense no-endpoint ccc line with no separable nonempty interval. Nowhere-separable quotient of a Suslin line

[F2]

Reverse nesting of recursively selected closed intervals in such a line gives an ω1-height tree with countable levels, no cofinal branch, and no uncountable antichain. Nested intervals form a Suslin tree

[A1]

Both constructions declare their uses of AC. The Axiom of Choice

Proof

1.1

Apply F1 to S and obtain a dense no-endpoint ccc line L in which every nonempty open interval is nonseparable.

F1A1given
2.1

Apply F2 to L. Its recursively nested closed intervals form a tree of height exactly ω1 with countable levels, no cofinal branch, and no uncountable antichain; thus it is a Suslin tree. The source lemma derives the height and every forbidden-set conclusion, so this composition does not assume that construction stage equals tree level. Its inherited choice principle is AC, and no implication over ZF is asserted.

F2A1step 1.1∎

Depends on

Used by

Dependency tree · two levels

11 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