CorollaryStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passaudited 2026-09-14

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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Relative consistency of no free ultrafilter on omega over ZF

Statement

If ZF is consistent, then ZF is consistent with the assertion that every ultrafilter on $ω$ is principal and with $\neg B P I$ .

Facts & Assumptions

Given: $Con (Z F)$ for the fixed formal theories.

[F1]

The tail-flip symmetric model is finitely formalizable proves that ZFC+GCH proves a set model of every externally fixed finite fragment of the displayed target theory.

[F2]

Formal consistency of ZFC plus GCH relative to ZF transfers the given consistency hypothesis to $Con (Z F C + G C H)$ .

Proof

technique · contradiction using finite derivation support

1.1

F2 gives $Con (Z F C + G C H)$ . Suppose for contradiction that the target theory $T$ is inconsistent. One finite refutation uses only a finite list $Δ$ of ZF axiom instances together with the two additional sentences.

F2assume-contra

2.1

By F1, ZFC+GCH proves that a set structure satisfies every sentence in this exact $Δ$ . Formal first-order soundness for the alleged finite derivation then makes ZFC+GCH prove that this structure satisfies a contradiction, contrary to step 1.1.

F1step 1.1discharge-contradiction

3.1

Thus $T$ is consistent. Only the finite support of one hypothetical proof is used; the conclusion is conditional consistency and does not assert or require a countable transitive model of full ZF.

step 1.1step 2.1discharge-contradiction: step 1.1∎

Depends on

Used by

The Ultrafilter Lemma and BPI are not theorems of ZF Corollary
Strict relative placement of BPI between ZF and Choice Theorem

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

Solomon Feferman, Some applications of the notions of forcing and generic sets, Theorems 4.9 and 4.12, printed pp. 341, 343–344 (standard reference, not scraped)