TheoremStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (gpt-5.6-terra)audited 2026-08-26

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Assuming choice on the cosets of $Q$ in $R$ , a Vitali set in $[0, 1]$ exists

Statement

Assume the Axiom of Choice. Then there exists a Vitali set $V \subseteq [0, 1]$ in the sense of Vitali set on $[0, 1]$ .

Facts & Assumptions

Given: The Axiom of Choice.

[F1]

A Vitali set on $[0, 1]$ is a subset meeting every equivalence class of $x \sim y ⟺ x - y \in Q$ in exactly one point (Vitali set on $[0, 1]$ ).

[A1]

Every family of nonempty sets has a choice function (The Axiom of Choice).

[F2]

A choice function for a family $F$ is a function $g$ with domain $F$ such that $g (S) \in S$ for every $S \in F$ (Choice function).

Proof

technique · direct

1.1

Let $F$ be the family of all sets of the form $(x + Q) \cap [0, 1]$ with $x \in [0, 1]$ . Each member of $F$ is nonempty because it contains its defining point, and distinct members are exactly the equivalence classes of the relation in [F1].

F1construct

2.1

By [A1] and [F2] there is a choice function $g$ on $F$ . Put $V : = {g (S) : S \in F} \subseteq [0, 1]$ . Then $V$ meets every member of $F$ , and it meets each of them in exactly one point because $g$ assigns one value to each set in its domain.

step 1.1A1F2construct

3.1

Since the members of $F$ are exactly the equivalence classes of $x \sim y ⟺ x - y \in Q$ on $[0, 1]$ , step 2.1 says precisely that $V$ is a Vitali set on $[0, 1]$ .

step 1.1step 2.1F1∎

Remarks

The proof uses one simultaneous selector on the family of $Q$ -cosets meeting $[0, 1]$ . Nothing in the proof reduces that family to a countable one.

Depends on

Used by

Dependency tree · two levels

8 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

Jacek Cichoń, Aleksander Kharazishvili, and Bogdan Węglorz, Subsets of the Real Line, Chapter 8 (standard reference, not scraped)
Vitali set (Wikipedia) (standard reference, not scraped)

Assuming choice on the cosets of QQ in RR, a Vitali set in [0,1][0,1] exists