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On Definability of Nonmeasurable Sets

Published online by Cambridge University Press:  20 November 2018

Harvey Friedman
Harvey Friedman*
Affiliation:
The Ohio State University, Columbus, Ohio
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In [1], Solovay constructed a model of ZFC in which every set of reals in OD (R) is Lebesgue measurable. Here we construct a model in which every equivalence class of sets of reals modulo null sets that is in OD(R) consists of Lebesgue measurable sets. This result immediately implies Solovay's, since the equivalence class of any set of reals in OD (R) is itself OD (R). As a consequence, one cannot provably explicitly define a nonmeasurable set modulo null sets within ZFC. We do not know whether this holds in the model Solovay uses (where an inaccessible cardinal is collapsed to ω 1). Instead, our model is a generic extension of his model.

In the model we construct, a somewhat stronger statement holds: every set in OD(R) of sets of reals which has < 2C inequivalent elements modulo null sets, consists entirely of Lebesgue measurable sets of reals.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 32 , Issue 3 , 01 June 1980 , pp. 653 - 656
Copyright
Copyright © Canadian Mathematical Society 1980

References

1. Solovay, R., A model of set theory in which every set of reals is Lebesgue measurable, Annals of Math. 92 (1970), 1–56.Google Scholar