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Ordinary differential equations and descriptive set theory: uniqueness and globality of solutions of Cauchy problems in one dimension

100%
Andretta A., Marcone A.
Fundamenta Mathematicae
|
1997
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tom 153
|
nr 2
157-190
EN
We study some natural sets arising in the theory of ordinary differential equations in one variable from the point of view of descriptive set theory and in particular classify them within the Borel hierarchy. We prove that the set of Cauchy problems for ordinary differential equations which have a unique solution is $∏^0_2$-complete and that the set of Cauchy problems which locally have a unique solution is $∑^0_3$-complete. We prove that the set of Cauchy problems which have a global solution is $∑_0^4$-complete and that the set of ordinary differential equations which have a global solution for every initial condition is $∏^0_3$-complete. We prove that the set of Cauchy problems for which both uniqueness and globality hold is $∏^0_2$-complete.
2
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Borel-Wadge degrees

100%
Andretta A., Martin D.
Fundamenta Mathematicae
|
2003
|
tom 177
|
nr 2
175-192
EN
Two sets of reals are Borel equivalent if one is the Borel pre-image of the other, and a Borel-Wadge degree is a collection of pairwise Borel equivalent subsets of ℝ. In this note we investigate the structure of Borel-Wadge degrees under the assumption of the Axiom of Determinacy.
3
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Conjugacy equivalence relation on subgroups

81%
Andretta A., Camerlo R., Hjorth G.
Fundamenta Mathematicae
|
2001
|
tom 167
|
nr 3
189-212
EN
If G is a countable group containing a copy of F₂ then the conjugacy equivalence relation on subgroups of G attains the maximal possible complexity.
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