CounterexampleConstruction: AI-adaptedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-07-31

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If the exclusion of $\varnothing$ is dropped, $\mathcal P(X)$ becomes the unique maximal improper filter

Statement refuted

Dropping the properness axiom $\emptyset\notin\mathcal F$ leaves the maximal filters, and therefore the notion of ultrafilter, unchanged.

Call a family a weak filter here if it satisfies the other filter axioms but may contain $\emptyset$ . Then $\mathcal P(X)$ is the unique improper weak filter on $X$ and the greatest weak filter under inclusion. Consequently it is the unique maximal weak filter, so maximality becomes degenerate if properness is not retained.

Facts & Assumptions

Given: A set $X$ and the enlarged convention in which a weak filter is a family $\mathcal G\subseteq\mathcal P(X)$ that contains $X$ , is closed under pairwise intersection, and is closed upward in $X$ , without requiring $\emptyset\notin\mathcal G$ .

[F1]

Under this library's convention a filter must also omit $\emptyset$ . The competing convention admits exactly the improper object $\mathcal P(X)$ (Filter on a set).

[F2]

An ultrafilter is maximal for inclusion among the proper filters on $X$ (Ultrafilter).

Counterexample

technique · direct

1.1

The family $\mathcal P(X)$ is a weak filter: it contains $X$ and $\emptyset$ , and it is closed under intersections and under taking supersets inside $X$ .

given

1.2

If a weak filter $\mathcal G$ contains $\emptyset$ , then for every $A\subseteq X$ the inclusions $\emptyset\subseteq A\subseteq X$ and upward closure give $A\in\mathcal G$ . Hence $\mathcal G=\mathcal P(X)$ .

given

2.1

Thus $\mathcal P(X)$ is the unique improper weak filter. Since every weak filter is a subfamily of $\mathcal P(X)$ , it is also the greatest and therefore the unique maximal weak filter.

step 1.1step 1.2

3.1

If maximality were taken in the enlarged class, no proper filter could be maximal because it would be strictly contained in $\mathcal P(X)$ . This differs from [F2] and refutes the claim that dropping properness leaves ultrafilters unchanged.

step 2.1F1F2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 14 results over 7 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

Filter (set theory) (Wikipedia) (standard reference, not scraped)
Ultrafilter (Wikipedia) (standard reference, not scraped)

If the exclusion of ∅\varnothing∅ is dropped, P(X)\mathcal P(X)P(X) becomes the unique maximal improper filter