LemmaStatement: AI-adaptedProof: AI-generatedPipeline-generatedaudited 2026-09-09

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 consistent theory can decide one sentence

Statement

In ZF, for a consistent sentence theory $T$ and sentence $σ$ , at least one of $T \cup {σ}$ and $T \cup {\neg σ}$ is consistent. Moreover, for any sentence theory $T$ ,

$T \cup {\neg σ} is inconsistent ⟺ T ⊢ σ .$

Facts & Assumptions

Given: A sentence theory $T$ and sentence $σ$ ; consistency is additionally assumed for the first claim.

[F1]

Inconsistency means derivability of $⊥$ . (Consistency and syntactic completeness)

[F2]

Sentence deduction discharges a sentence premise as an implication. (Deduction theorem for sentence assumptions)

[F3]

The fixed calculus proves $\neg ⊥$ , double negation and Boolean explosion. (Derived propositional, quantifier and equality rules)

Proof

1.1

If $T \cup {\neg σ} ⊢ ⊥$ , deduction gives $T ⊢ \neg σ \to ⊥$ . Combining with the theorem $\neg ⊥$ gives $T ⊢ \neg \neg σ$ by Boolean contraposition, and hence $T ⊢ σ$ . Conversely a proof of $σ$ remains valid after adjoining $\neg σ$ ; explosion then proves $⊥$ . This proves both directions for arbitrary $T$ .

F1F2F3

2.1

If $T \cup {σ}$ is inconsistent, deduction and $\neg ⊥$ similarly give $T ⊢ \neg σ$ . If also $T \cup {\neg σ}$ were inconsistent, step 1.1 would give $T ⊢ σ$ ; the two conclusions would prove $⊥$ in $T$ . For consistent $T$ this is impossible. Thus if the positive extension is inconsistent the negative extension is consistent; otherwise the positive extension itself is consistent.

F1F2F3step 1.1∎

Depends on

Used by

Dependency tree · two levels

5 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

Moschovakis, Lemma 1H.12(3)–(4), printed p38; local deduction proof. (standard reference, not scraped)