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In "A SPLITTING THEOREM FOR n−REA DEGREES" Shore and Slaman extend the following result of Sacks If $C$ is r.e., $D \leq_T C$ and $D, C \not\leq_T 0$ then there are sets $C_0, C_1$ such that $C \equ …

So a common method used to construct non-zero $\omega$-REA arithmetic degrees with various properties is to build an $\omega$-REA operator $J$ satisfying the constraints that (for all $X$) $$\tag{1} J …

So after some careful thought I'm pretty sure it is fully general. Let $A$ be a non-arithmetic $\omega$-REA set and $K(X)$ some $\omega$-REA operator satisfying 1 and 2 (see Odifreddi volume 2 XIII.3 …

Suppose that $A$ is an $\omega$-REA set (so $A^{[n]}$ is r.e. in the prior columns). It is a well-known result that if each column of $A$ is computable then $A \leq_T \emptyset^2$ ($\emptyset^2$ can …

Remember a degree $\mathbb{d}$ is the $n$-lub of $\mathbb{c}_j$ in the Turing degrees if it is the least element (not merely a minimal element) set of $\mathbb{c}^{(n)}$ such that $\mathbb{c}$ comput …

Ok, so unfortunately the answer is no. I've attached a proof that builds a $\omega$-REA set such that $A^{[\leq n]}$ is low for all $n$ yet $A$ computes $\emptyset'' \oplus X$ where $X$ is a natural …

Yes, it turns out that there is such an arithmetic (indeed 3-REA) non-trivial 2-lub. In fact, $\emptyset'''$ is such a degree. Consider the construction I give here in answer to this question. This …

Remember that an n-REA set is a set of the form $A_0 \oplus A_1 \oplus \cdots \oplus A_n$ with $A_n$ relatively r.e. in $A_m, m<n$ (so $A_0$ is r.e.) and that a degree is PA just if it computes a path …

The hop $H_e$ is defined by $H_e(X) = X \oplus W_e^{X}$. A 2-REA operator (or double hop) $J_{\langle e,i\rangle}$ is defined by $J_{\langle e,i\rangle}(X) = H_e(H_i(X))$ By a famous result from Pse …

Ohh, I think I was being dumb. There is a 2-REA operator $J$ such that $J(X) <_T X'$ isn't of degree r.e. in $X$. Since $0''$ is of r.e. degree in every $X < 0''$ with $X' \geq_T 0''$.

The existence of a high arithmetic degree (meaning the degrees induced by the notion of relative arithmetic definability) below $0^\omega$ can be established by using Harrington/Simpson's constructio …