CorollaryStatement: AI-adaptedProof: AI-adaptedPipeline-generated

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 BPI counterexample theory obstructs a proof of DMC

Statement

If $Z F + B P I + \neg U R Y$ is consistent, then $Z F + B P I$ does not prove DMC. Here BPI and DMC are the principles of The Boolean prime ideal principle and Dependent multiple choice in finite-level tree form, and $U R Y$ is Urysohn separation in normal spaces (Normal spaces and $T_{4}$ spaces, with the source disagreement over whether normality includes $T_{1}$ stated explicitly).

No consistency of the counterexample theory or existence of its model is asserted; the hypothesis is an additional consistency premise, not a consequence of $Con (Z F)$ established here.

Facts & Assumptions

Given: Consistency of $T = Z F + B P I + \neg U R Y$ .

[F1]

ZF proves DMC implies Urysohn's lemma (DMC implies Urysohn's lemma).

Proof

1.1givenF1

If $Z F + B P I$ proved DMC, the same finite derivation would be a $T$ derivation, since $T$ extends that theory. The ZF derivation F1 would then give $U R Y$ in $T$ .

2.1givenstep 1.1∎

The axiom $\neg U R Y$ of $T$ would combine with that derivation to give a contradiction, contrary to the given consistency. Thus $Z F + B P I$ cannot prove DMC under the stated hypothesis.

Depends on

Used by

Choice ledger for Baire, Urysohn, Stone, and Tychonoff Remark

Dependency tree · two levels

25 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

Norbert Brunner, Geordnete Läuchli Kontinuen (standard reference, not scraped)