Iterative forcing and hyperimmunity in reverse mathematics

The separation between two theorems in reverse mathematics is usually done by constructing a Turing ideal satisfying a theorem P and avoiding the solutions to a fixed instance of a theorem Q. Lerman, Solomon and Towsner introduced a forcing technique for iterating a computable non-reducibility in order to separate theorems over omega-models. In this paper, we present a modularized version of their framework in terms of preservation of hyperimmunity and show that it is powerful enough to obtain the same separations results as Wang did with his notion of preservation of definitions.Comment: 15 page

A Note on Computable Embeddings for Ordinals and Their Reverses

We continue the study of computable embeddings for pairs of structures, i.e. for classes containing precisely two non-isomorphic structures. Surprisingly, even for some pairs of simple linear orders, computable embeddings induce a non-trivial degree structure. Our main result shows that although $\{\omega \cdot 2, \omega^\star \cdot 2\}$ is computably embeddable in $\{\omega^2, {(\omega^2)}^\star\}$, the class $\{\omega \cdot k,\omega^\star \cdot k\}$ is \emph{not} computably embeddable in $\{\omega^2, {(\omega^2)}^\star\}$ for any natural number $k \geq 3$.Comment: 13 pages, accepted to CiE 202

Degrees of Non-computability of Homeomorphism Types of Polish Spaces

© 2020, Springer Nature Switzerland AG. There are continuum many homeomorphism types of Polish spaces. In particular, there is a Polish space which is not homeomorphic to any computably presented Polish space. We examine the details of degrees of non-computability of presenting homeomorphic copies of Polish spaces