The weakness of the pigeonhole principle under hyperarithmetical reductions

The infinite pigeonhole principle for 2-partitions ($\mathsf{RT}^1_2$) asserts the existence, for every set $A$, of an infinite subset of $A$ or of its complement. In this paper, we study the infinite pigeonhole principle from a computability-theoretic viewpoint. We prove in particular that $\mathsf{RT}^1_2$ admits strong cone avoidance for arithmetical and hyperarithmetical reductions. We also prove the existence, for every $\Delta^0_n$ set, of an infinite low${}_n$ subset of it or its complement. This answers a question of Wang. For this, we design a new notion of forcing which generalizes the first and second-jump control of Cholak, Jockusch and Slaman.Comment: 29 page

Pi01 encodability and omniscient reductions

A set of integers $A$ is computably encodable if every infinite set of integers has an infinite subset computing $A$. By a result of Solovay, the computably encodable sets are exactly the hyperarithmetic ones. In this paper, we extend this notion of computable encodability to subsets of the Baire space and we characterize the $\Pi^0_1$ encodable compact sets as those who admit a non-empty $\Sigma^1_1$ subset. Thanks to this equivalence, we prove that weak weak K\"onig's lemma is not strongly computably reducible to Ramsey's theorem. This answers a question of Hirschfeldt and Jockusch.Comment: 9 page

Higher randomness and forcing with closed sets

[Kechris, Trans. Amer. Math. Soc. 1975] showed that there exists a largest Pi_1^1 set of measure 0. An explicit construction of this largest Pi_1^1 nullset has later been given in [Hjorth and Nies, J. London Math. Soc. 2007]. Due to its universal nature, it was conjectured by many that this nullset has a high Borel rank (the question is explicitely mentioned by Chong and Yu, and in [Yu, Fund. Math. 2011]). In this paper, we refute this conjecture and show that this nullset is merely Sigma_3^0. Together with a result of Liang Yu, our result also implies that the exact Borel complexity of this set is Sigma_3^0. To do this proof, we develop the machinery of effective randomness and effective Solovay genericity, investigating the connections between those notions and effective domination properties

Another Characterization of the Higher K-Trivials

In algorithmic randomness, the class of K-trivial sets has proved itself to be remarkable, due to its numerous different characterizations. We pursue in this paper some work already initiated on K-trivials in the context of higher randomness. In particular we give here another characterization of the non hyperarithmetic higher K-trivial sets

Partition genericity and pigeonhole basis theorems

There exist two notions of typicality in computability theory, namely, genericity and randomness. In this article, we introduce a new notion of genericity, called partition genericity, which is at the intersection of these two notions of typicality, and show that many basis theorems apply to partition genericity. More precisely, we prove that every co-hyperimmune set and every Kurtz random is partition generic, and that every partition generic set admits weak infinite subsets. In particular, we answer a question of Kjos-Hanssen and Liu by showing that every Kurtz random admits an infinite subset which does not compute any set of positive Hausdorff dimension. Partition genericty is a partition regular notion, so these results imply many existing pigeonhole basis theorems.Comment: 23 page

Morality in Tele-immersive Environments

ABSTRACT Humans are spending an increasing amount of time in teleimmersive environments interacting with avatars or virtual human bodies. Additionally, human behavior and cognition are affected by experiences in tele-immersive environments. Although there is substantial psychological work surrounding the notion of morality, there is little work that examines the interplay of immersive digital environments and the moral identity of the digital medium user. We conducted a study to explore how participants' moral behaviors and self-ratings of morality changed after immersion in either a moral or immoral tele-immersive environment. Results revealed that participants who witnessed the immoral scenarios felt and acted more immoral than participants in the moral scenario condition. These findings have important implications for understanding the effects of digital media as well as for the study of the psychological construct of moral identity