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The Solovay model has no Vitali or Bernstein set
Statement
contains no Vitali selector modulo and no Bernstein subset of .
Facts & Assumptions
Given: The universal LM and PSP theorems above.
Every set of reals in the Solovay model is Lebesgue measurable: every alleged selector is measurable in .
Vitali set on and Lebesgue outer measure, Lebesgue measurability and Lebesgue measure are unchanged by translation: rational translates of a selector are disjoint and measurable with one common measure.
Measures on sigma-algebras, Finite and countable subadditivity of measures, A box in with parameters is Lebesgue measurable of measure , whichever of its faces are included, and is countably infinite: finite additivity handles disjoint finite families, subadditivity handles the null countable union, and the containing intervals have their stated finite positive measures.
Bernstein subset of and Every uncountable Solovay-model set of reals has a perfect subset: a Bernstein set and its complement meet every nonempty perfect set but contain no nonempty perfect set.
The Solovay inner model satisfies Dependent Choice, AC implies DC implies countable choice, Countable unions of at most countable sets, assuming , and is uncountable (Cantor's nested intervals, 1874): internal DC supplies countable choice, so the union of two countable sets is countable, whereas is uncountable.
Proof
Suppose were a Vitali selector. F1 makes it measurable. If , the countably many rational translates covering have null union, contradicting . If , finitely many pairwise disjoint translates inside have arbitrarily large total measure, contradicting . The selector and translation facts are F2, while F3 supplies subadditivity, finite additivity and the interval values.
Suppose were Bernstein. Both and contain no nonempty perfect subset. They cannot both be countable: F5 would make their two-term union countable, contrary to its uncountability. Therefore one is uncountable, and F4 gives it a nonempty perfect subset, a contradiction. This repairs the tempting but unsupported assertion that the definition alone makes uncountable.
The two contradictions exclude both supplied pathologies without using their ZFC existence constructions.
Depends on
- Every set of reals in the Solovay model is Lebesgue measurable
- Every uncountable Solovay-model set of reals has a perfect subset
- The Solovay inner model satisfies Dependent Choice
- AC implies DC implies countable choice
- Countable unions of at most countable sets, assuming $\mathrm{AC}_\omega$
- $\mathbb{R}$ is uncountable (Cantor's nested intervals, 1874)
- Vitali set on $[0,1]$
- Bernstein subset of $\mathbb{R}$
- Measures on sigma-algebras
- Lebesgue outer measure, Lebesgue measurability and Lebesgue measure are unchanged by translation
- Finite and countable subadditivity of measures
- A box in $\mathbb{R}^n$ with parameters $a_i\le b_i$ is Lebesgue measurable of measure $\prod_{i<n}(b_i-a_i)$, whichever of its faces are included
- $\mathbb{Q}$ is countably infinite
Used by
Dependency tree · two levels
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Sources
- Solovay, A model of set-theory in which every set of reals is Lebesgue measurable (standard reference, not scraped)