# On Sets Not Belonging to Algebras of Subsets

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*L. Š. Grinblat*

The main results of this work can be formulated in such
an elementary way that it is likely to attract mathematicians from
a broad spectrum of specialties, though its main audience will likely
be combintorialists, set-theorists, and topologists. The central
question is this: Suppose one is given an at most countable family of
algebras of subsets of some fixed set such that, for each algebra, there
exists at least one set that is not a member of that algebra. Can one
then assert that there is a set that is not a member of

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A fascinating new angle that shows that non-measurable sets are here to stay.

-- The Bulletin of Mathematics Books and Computer Software

#### Table of Contents

# Table of Contents

## On Sets Not Belonging to Algebras of Subsets

- Table of Contents v6 free
- 1. Introduction 18 free
- 2. Main Results 613 free
- 3. Fundamental Idea 1118
- 4. Finite Sequences of Algebras (1) 1522
- 5. Countable Sequences of Algebras (1) 2734
- 6. Proof of Theorem II 3845
- 7. Improvement of Theorem II (Proof of Theorem II[sup(*)]) 4956
- 8. Proof of Theorems III and IV 6370
- 9. The Inverse Problem 6875
- 10. Finite Sequences of Algebras (2) 7279
- 11. Countable Sequences of Algebras (2) 8895
- 12. Improvement of some Main Results 97104
- 13. Sets not belonging to Semi-lattices of Subsets and not belonging to Lattices of Subsets 103110
- 14. Unsolved Problems 107114
- References 111118