Abstract
We study the proof strength of various second order logic principles that make statements about families of sets and functions. Usually, families of sets or functions are represented in a uniform way by a single object. In order to be able to go beyond the limitations imposed by this approach, we introduce the concept of weakly represented families of sets and functions. This allows us to study various types of families in the context of reverse mathematics that have been studied in set theory before. The results obtained witness that the concept of weakly represented families is a useful and robust tool in reverse mathematics.
R. Hölzl and F. Stephan were supported in part by MOE/NUS grants R146-000-181-112 and R146-000-184-112 (MOE2013-T2-1-062); D. Raghavan was supported in part by MOE/NUS grant R146-000-184-112 (MOE2013-T2-1-062). Work on this article begun as J. Zhang’s Undergraduate Research Opportunities Programme project while J. Zhang was an undergraduate students of NUS and R. Hölzl worked at NUS financed by MOE/NUS grant R146-000-184-112 (MOE2013-T2-1-062).
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- 1.
Note that to simplify notation, we do not explicitly define total characteristic functions of the sets \(A_e\), \(e \in \omega \), or the enumeration of a set that represents these functions as a weakly represented family. But since the elements of every \(A_e\) are enumerated in increasing order by the given procedure, it is easy to convert it into one defining the enumeration of such a set.
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Acknowledgments
The authors would like to thank C.T. Chong, Wei Li and Yue Yang for fruitful discussions and suggestions. They are also grateful to the anonymous referee for detailed and helpful comments, in particular for pointing out a simplification of the proof of Theorem 29.
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Hölzl, R., Raghavan, D., Stephan, F., Zhang, J. (2017). Weakly Represented Families in Reverse Mathematics. In: Day, A., Fellows, M., Greenberg, N., Khoussainov, B., Melnikov, A., Rosamond, F. (eds) Computability and Complexity. Lecture Notes in Computer Science(), vol 10010. Springer, Cham. https://doi.org/10.1007/978-3-319-50062-1_13
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