arXiv subjects

Manuel Lerman

Publications and source records attributed to Manuel Lerman.

3 recordsLinked to original sources

arXiv2014

Self-embeddings of computable trees

We divide the class of infinite computable trees into three types. For the first and second types, $0'$ computes a nontrivial self-embedding while for the third type $0''$ computes a nontrivial self-embedding. These results are optimal and we obtain partial results concerning the complexity of nontrivial self-embeddings of infinite computable trees considered up to isomorphism. We show that every infinite computable tree must have either an infinite computable chain or an infinite $Π^0_1$ antichain. This result is optimal and has connections to the program of reverse mathematics.

math.LO↗

arXiv2014

On a conjecture of Dobrinen and Simpson concerning almost everywhere domination

The notions of almost everywhere (a.e.) domination and its uniform version were introduced and studied in reverse mathematics. This paper studies these notions from a recursion-theoretic point of view and explore their connections to notions such as randomness and genericity. It is shown that if $Z$ is a.e. dominating then each $1$-$Z$-random is $2$-random. In other words, $0'\leq_{\rm LR} Z$ for every a.e. dominating $Z$, where ${\rm LR}$ denotes low-for-random reducibility. Other results and corollaries are also given.

math.LO↗

arXiv2013

Separating principles below Ramsey's Theorem for Pairs

In recent years, there has been a substantial amount of work in reverse mathematics concerning natural mathematical principles that are provable from $\RT$, Ramsey's Theorem for Pairs. These principles tend to fall outside of the "big five" systems of reverse mathematics and a complicated picture of subsystems below $\RT$ has emerged. In this paper, we answer two open questions concerning these subsystems, specifically that $\ADS$ is not equivalent to $\CAC$ and that $\EM$ is not equivalent to $\RT$.

math.LO↗