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**On some classical problems in descriptive set theory.**
*(English.
Russian original)*
Zbl 1064.03031

Russ. Math. Surv. 58, No. 5, 839-927 (2003); translation from Usp. Mat. Nauk 58, No. 5, 3-88 (2003).

The paper was written on the occasion of the 100th anniversary of the birth of P. S. Novikov. The anniversary of descriptive set theory in which Novikov reached fundamental results can be dated to the same time. One of the directions of the work of Novikov in descriptive set theory is the study of regularity properties, namely, perfect set property, Lebesgue measurability and Baire property. Let \(\mathbb K\) be a class of sets of reals and let \(\text{PK}(\mathbb K)\), \(\text{LM}(\mathbb K)\), \(\text{BP}(\mathbb K)\) be the assertions stating, respectively, the mentioned regularity properties for the class \(\mathbb K\).

The paper presents from the current point of view final solutions with complete proofs of some classical problems. These problems concern mainly the assertions \(\text{PK}(\boldsymbol\Pi^1_1)\), \(\text{LM}(\boldsymbol\Delta^1_2)\), \(\text{BP}(\boldsymbol\Delta^1_2)\), \(\text{LM}(\boldsymbol\Sigma^1_2)\), \(\text{BP}(\boldsymbol\Sigma^1_2)\). The authors present proofs of the implications \(\text{PK}(\boldsymbol\Pi^1_1)\Rightarrow \text{LM}(\boldsymbol\Sigma^1_2)\Rightarrow \text{LM}(\boldsymbol\Delta^1_2)\), and \(\text{PK}(\boldsymbol\Pi^1_1)\Rightarrow \text{BP}(\boldsymbol\Sigma^1_2)\Rightarrow \text{BP}(\boldsymbol\Delta^1_2)\). Characterizations of \(\text{LM}(\boldsymbol\Sigma^1_2)\) and \(\text{BP}(\boldsymbol\Sigma^1_2)\) using eventual domination relations are proved and applied for the proof of the implication \(\text{LM}(\boldsymbol\Sigma^1_2)\Rightarrow \text{BP}(\boldsymbol\Sigma^1_2)\). The possibility of finding elementary proofs for the implications is discussed and for the implication \(\text{PK}(\boldsymbol\Pi^1_1)\Rightarrow \text{LM}(\boldsymbol\Sigma^1_2)\) such a proof is found. Forcing models of ZFC are used to prove that no other implications can be proved and, in particular, none of the implications can be reversed.

The authors prove that the existence of counterexamples to the regularity properties of projective classes does not imply the existence of effective (i.e., ordinal-definable) counterexamples. They prove also a theorem of Solovay that regularity for all projective sets is consistent. The paper is written to be self-contained and it explains some basic aspects of the theory of constructibility and forcing but it cannot serve as a textbook for neither of these two fields.

The paper presents from the current point of view final solutions with complete proofs of some classical problems. These problems concern mainly the assertions \(\text{PK}(\boldsymbol\Pi^1_1)\), \(\text{LM}(\boldsymbol\Delta^1_2)\), \(\text{BP}(\boldsymbol\Delta^1_2)\), \(\text{LM}(\boldsymbol\Sigma^1_2)\), \(\text{BP}(\boldsymbol\Sigma^1_2)\). The authors present proofs of the implications \(\text{PK}(\boldsymbol\Pi^1_1)\Rightarrow \text{LM}(\boldsymbol\Sigma^1_2)\Rightarrow \text{LM}(\boldsymbol\Delta^1_2)\), and \(\text{PK}(\boldsymbol\Pi^1_1)\Rightarrow \text{BP}(\boldsymbol\Sigma^1_2)\Rightarrow \text{BP}(\boldsymbol\Delta^1_2)\). Characterizations of \(\text{LM}(\boldsymbol\Sigma^1_2)\) and \(\text{BP}(\boldsymbol\Sigma^1_2)\) using eventual domination relations are proved and applied for the proof of the implication \(\text{LM}(\boldsymbol\Sigma^1_2)\Rightarrow \text{BP}(\boldsymbol\Sigma^1_2)\). The possibility of finding elementary proofs for the implications is discussed and for the implication \(\text{PK}(\boldsymbol\Pi^1_1)\Rightarrow \text{LM}(\boldsymbol\Sigma^1_2)\) such a proof is found. Forcing models of ZFC are used to prove that no other implications can be proved and, in particular, none of the implications can be reversed.

The authors prove that the existence of counterexamples to the regularity properties of projective classes does not imply the existence of effective (i.e., ordinal-definable) counterexamples. They prove also a theorem of Solovay that regularity for all projective sets is consistent. The paper is written to be self-contained and it explains some basic aspects of the theory of constructibility and forcing but it cannot serve as a textbook for neither of these two fields.

Reviewer: Miroslav Repický (Košice)

### MSC:

03E15 | Descriptive set theory |

03-02 | Research exposition (monographs, survey articles) pertaining to mathematical logic and foundations |

28A05 | Classes of sets (Borel fields, \(\sigma\)-rings, etc.), measurable sets, Suslin sets, analytic sets |

03E35 | Consistency and independence results |

03E45 | Inner models, including constructibility, ordinal definability, and core models |