CorollaryStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-14

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Finite parameter sets admit disjoint clopen supports

Statement

Use the basic Cohen extension and $A$ from Schema of continuity in the basic Cohen model. Let $f$ be a finite tuple of distinct members of $A$ . Suppose that a Boolean algebra $B$ , a finite-arity map

$d : A_{\neq}^{r} ⟶ B,$

and finitely many sets used in assertions about its values are each ordinal-definable from $A, f$ and ordinal parameters. Here $A_{\neq}^{r}$ is the set of injective $r$ -tuples. Let $T \subseteq A_{\neq}^{r}$ be finite, let $F$ be the union of the coordinates occurring in $T$ , and assume $F \cap rng (f) = \emptyset$ .

Fix a finite conjunction $Φ$ of assertions about the values $d (t)$ for $t \in T$ , including assertions of the forms

$d (t) \in S, d (t) \notin S, \neg_{B} d (t) \in S, \neg_{B} d (t) \notin S,$

where every displayed $S$ is one of the fixed supported sets. If $Φ$ holds, then there are pairwise disjoint basic clopen neighbourhoods $(U_{a})_{a \in F}$ , all avoiding $rng (f)$ , such that every map $g : F \to A$ satisfying $g (a) \in U_{a}$ preserves $Φ$ after every tuple $t$ is replaced coherently by $g \circ t$ . The $U_{a}$ may be required to lie inside any previously prescribed basic clopen neighbourhoods of their respective $a$ .

In particular this applies with a supported ideal $I \subseteq B$ and $S = B ∖ I$ , where set difference has the convention of The difference $a ∖ b$ , the symmetric difference $a △ b$ , and the complement $X ∖ a$ relative to a set $X$ and Boolean complementation has the convention of Boolean ideals, filters, prime ideals and ultrafilters.

Facts & Assumptions

Given: The finite supported data, the finite family $T$ , the disjointness $F \cap rng (f) = \emptyset$ , and the true finite conjunction $Φ$ from the Statement.

[F1]

Schema of continuity in the basic Cohen model gives simultaneous clopen-box continuity for finitely many parameters ordinal-definable from $A$ and a fixed support tuple.

[F2]

The difference $a ∖ b$ , the symmetric difference $a △ b$ , and the complement $X ∖ a$ relative to a set $X$ defines $B ∖ I$ by membership in $B$ and nonmembership in $I$ .

[F3]

Boolean ideals, filters, prime ideals and ultrafilters supplies the Boolean complement operation and the ideal vocabulary.

Proof

technique · direct

1.1

If $F = \emptyset$ , then $T$ is empty unless $r = 0$ . In either case there are no coordinates to vary: take the empty clopen family, and the fixed sentence $Φ$ remains true. Assume henceforth that $F$ is nonempty, enumerate it without repetition as $⟨ a_{0}, \dots, a_{m - 1} ⟩$ , and concatenate this tuple after $f$ .

given

1.2

Choose the finitely many formulas and ordinal parameters which uniquely define $B, d$ , and the sets occurring in $Φ$ from $A, f$ . In one formula $ψ (A, f, a_{0}, \dots, a_{m - 1})$ , assert those unique definitions, reconstruct each member of $T$ by its finite list of coordinate positions, and assert the entire conjunction $Φ$ . If $S = B ∖ I$ occurs, replace it by the conjunction “belongs to $B$ and does not belong to $I$ ”; replace $\neg_{B} d (t)$ by the uniquely specified Boolean complement in $B$ . Thus $ψ$ is a single fixed membership formula and is true of the concatenated tuple.

F2F3construct

2.1

Apply F1 to $f^{⌢} ⟨ a_{0}, \dots, a_{m - 1} ⟩$ and the formula from step 1.2. Retain the clopens assigned to the $a_{i}$ and discard those assigned to $f$ . Pairwise disjointness of the larger family makes every retained clopen disjoint from every member of $rng (f)$ . If basic clopens $W_{a_{i}}$ were prescribed in advance, increase the finitely many defining prefix lengths so that the retained neighbourhood of $a_{i}$ lies in $W_{a_{i}}$ ; shrinking does not destroy the conclusion.

F1step 1.2

3.1

Let $g : F \to A$ select $g (a) \in U_{a}$ . Since the $U_{a}$ are pairwise disjoint, $g$ is automatically injective, so every $g \circ t$ remains in $A_{\neq}^{r}$ and repeated occurrences of one coordinate are replaced coherently. The continuity conclusion for $ψ$ keeps $f$ fixed and preserves all unique definitions and every conjunct of $Φ$ . This proves the simultaneous assertion, including the specialization to $B ∖ I$ . Only a finite tuple was enumerated and finitely many clopens were shrunk; no choice function on an arbitrary family was used.

F1F2F3step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

14 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

Miroslav Repický, A proof of the independence of the Axiom of Choice from the Boolean Prime Ideal Theorem, Corollary 3, p.545 (standard reference, not scraped)