CorollaryStatement: AI-adaptedProof: AI-adaptedPipeline-generatedaudited 2026-09-22

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BPI does not imply DMC

Statement

Relative to $Con (Z F)$ , BPI (The Boolean prime ideal principle) does not imply DMC (Dependent multiple choice in finite-level tree form) over $Z F$ : there is a model of $Z F + B P I$ in which DMC fails.

Facts & Assumptions

Given: The assumed consistency of $Z F$ .

[F1]

Relative to $Con (Z F)$ , the theory $Z F + B P I + \neg U R Y$ is consistent, where $\neg U R Y$ asserts the existence of a normal space with two disjoint closed sets that admit no continuous separation (Relative consistency of BPI without Urysohn's lemma, Normal spaces and $T_{4}$ spaces, with the source disagreement over whether normality includes $T_{1}$ stated explicitly).

[F2]

Over $Z F$ , DMC implies Urysohn's lemma: in every normal space any two disjoint closed sets are separated by a continuous function (DMC implies Urysohn's lemma, Dependent multiple choice in finite-level tree form).

Proof

technique · contradiction

1.1

Assume $Con (Z F)$ and suppose, for the sake of contradiction, that $Z F + B P I$ proves DMC.

assume-contragiven

2.1

By [F2] the theory $Z F + B P I + \neg U R Y$ then proves DMC and hence proves $U R Y$ , since DMC implies Urysohn's lemma in ZF; but it also proves $\neg U R Y$ by its own axiom, so it is inconsistent.

step 1.1F2

3.1

This contradicts the consistency of $Z F + B P I + \neg U R Y$ given by [F1] under the assumption $Con (Z F)$ ; hence BPI does not imply DMC over $Z F$ , conditionally on the consistency of $Z F$ .

step 2.1F1discharge-contradiction∎

Depends on

Used by

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

Dependency tree · two levels

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Sources

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