CorollaryStatement: AI-adaptedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-13

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.

V equals L gives a Suslin tree

Statement

ZF proves that $V = L$ implies the existence of a normal splitting Suslin tree on $ω_{1}$ .

Facts & Assumptions

Given: Ambient ZF and the hypothesis $V = L$ .

[F1]

V equals L implies diamond derives a diamond sequence on $ω_{1}$ from $V = L$ in ZF.

[F2]

The constructible universe satisfies AC says that $L$ satisfies AC; under $V = L$ this is ambient AC.

[F3]

Diamond constructs a normal splitting Suslin tree proves in ZFC that diamond constructs a normal splitting Suslin tree with underlying set $ω_{1}$ .

[A1]

The Axiom of Choice names the hypothesis explicitly required by F3 and supplied here by F2.

Proof

1.1F1F2A1given

By F1, $V = L$ supplies a diamond sequence on $ω_{1}$ . By F2 and the equality $V = L$ , A1 holds in the ambient universe. This explicit use of Choice is essential to the cited tree proof: it supplies its countable-union, maximal-antichain, and deterministic well-ordering steps.

2.1F3step 1.1∎

Apply F3 with the diamond sequence and AC from step 1.1. The resulting tree is normal and splitting, has underlying set $ω_{1}$ , has countable levels, and has neither an uncountable antichain nor a cofinal branch; hence it is a Suslin tree. No tree is asserted at the degenerate heights zero or one, and no weakening from stationary guessing to merely unbounded guessing is made.

Depends on

Used by

Formal relative consistency of not SH Corollary
SH is not equivalent to CH False statement

Dependency tree · two levels

26 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

Lietz, Set Theory, Theorem 7.20, pp.60–61; published diamond-to-Suslin construction (standard reference, not scraped)