On the Uniform Projection and Covering Problems in Descriptive Set Theory Under the Axiom of Constructibility
Abstract
:1. Introduction
- (a)
- (Uniform projection) any set is the projection of a uniform set
- (b)
- (Countable uniform non-covering) there is a set with countable cross-sections not covered by a union of countably many uniform sets.
2. Some References
Si est un ensemble plan uniforme relativement à l’axe , la projection E de sur cet axe est-elle nécessairement un , ou un ensemble de classe ? Un ensemble uniforme plan de classe a-t-il pour projection un , ou un ensemble de classe ?[Our italic here and below]
Or, dès que cette analogie est constatée, il est naturel de se poser les questions suivantes: peut-on trouver pour chaque ensemble une représentation paramétrique régulière?
Nous avons vu que chaque ensemble analytique uniforme est contenu dans une courbe uniforme mesurable et que chaque ensemble E mesurable qui est conpé par chaque parallèle à l’axe OY en un ensemble de points au plus dénombrables est composé d’une infinité dénombrable d’ensembles uniformes mesurables . Il est très naturel de se poser des questions analogues relativement aux ensembles projectifs et .
3. Preliminaries
- (ii)
- If , then there is a set such that if and a set belongs to , then there is an satisfying .
- (i)
- The binary relation , iff , on belongs to , where is defined by
- (ii)
- If , K is a class of the form , and is a set in K, then
- (i)
- If K is a class of the form , , , or , then every set in K is uniformizable by a set still in
- (ii)
- Any set is the projection of a uniform set;
- (iii)
- Any non-empty , resp., set contains a , resp., real
- (iv)
- If and , then .
4. Proof of the Uniform Projection Theorem
- (A)
- If , then exists and .
- (B)
- If , then .
- (C)
- If , then there is a such that .
- (D)
- If , then the equivalence holds.
- (E)
- .
- (F)
- If and a real belongs to , then there is an such that , where and .
- (G)
- .
5. Proof of the Uniform Covering Theorem
- (*)
- is the indicator function of a set
- (1)
- The set is .
- (2)
- If and , then
- (3)
- If is countable, then there is a with .
- (4)
- .
- 1°
- There exists a formula such that for all : .
- 2°
- For any , there is a well-ordering of of order type such that the ternary relation on is .
- 3°
- If , holds, , K is a class of the form , and is a set in K, then similarly to Proposition 4(ii), the setsare still sets in K. The same is true for and .
- 4°
- If and , then .
6. Alternative Proofs of the Main Results
- (A)
- and ;
- (B)
- ;
- (C)
- ;
- (D)
- .
- (I)
- ;
- (II)
- (a) If and , then ;(b) The set is equal to (see Definition 2 on ; we remove here to take care of the case when has to be the empty set);
- (III)
- —compared to (C)—class ;
- (IV)
- —compared to (D)—class ;
- (V)
- is the -least pair satisfying (II), (III), and (IV) for given .
7. Conclusions and Problems
- –
- –
- A generic model of , in which, for a given , there is a real coding the collapse of , whereas all reals are constructible, in [26];
- –
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
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Kanovei, V.; Lyubetsky, V. On the Uniform Projection and Covering Problems in Descriptive Set Theory Under the Axiom of Constructibility. Mathematics 2025, 13, 409. https://doi.org/10.3390/math13030409
Kanovei V, Lyubetsky V. On the Uniform Projection and Covering Problems in Descriptive Set Theory Under the Axiom of Constructibility. Mathematics. 2025; 13(3):409. https://doi.org/10.3390/math13030409
Chicago/Turabian StyleKanovei, Vladimir, and Vassily Lyubetsky. 2025. "On the Uniform Projection and Covering Problems in Descriptive Set Theory Under the Axiom of Constructibility" Mathematics 13, no. 3: 409. https://doi.org/10.3390/math13030409
APA StyleKanovei, V., & Lyubetsky, V. (2025). On the Uniform Projection and Covering Problems in Descriptive Set Theory Under the Axiom of Constructibility. Mathematics, 13(3), 409. https://doi.org/10.3390/math13030409