Abstract
In the first half of this paper, we study the way that sets of real numbers closed under Turing equivalence sit inside the real line from the perspective of algebra, measure and orders. Afterwards, we combine the results from our study of these sets as orders with a classical construction from Avraham to obtain a restriction about how non trivial automorphism of the Turing degrees (if they exist) interact with 1-generic degrees.
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Notes
An order preserving function is a function such that whenever \(x<y\) then \(f(x)<f(y)\). On the other hand, an order reversing function is such that if \(x<y\) then \(f(y)<f(x)\).
Here \(B_{\frac{1}{n}}^{K}(s)\) is the ball of radius \(\frac{1}{n}\) with center s in K, a subset of \({\mathbb {R}}\).
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Ongay-Valverde, I. Sets of real numbers closed under Turing equivalence: applications to fields, orders and automorphisms. Arch. Math. Logic 62, 843–869 (2023). https://doi.org/10.1007/s00153-023-00865-7
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DOI: https://doi.org/10.1007/s00153-023-00865-7