Π₁¹ relations and paths through

Abstract

When bounds on complexity of some aspect of a structure are preserved under isomorphism, we refer to them as intrinsic. Here, building on work of Soskov [34], [33], we give syntactical conditions necessary and sufficient for a relation to be intrinsically on a structure. We consider some examples of computable structures and intrinsically relations R. We also consider a general family of examples of intrinsically relations arising in computable structures of maximum Scott rank.For three of the examples, the maximal well-ordered initial segment in a Harrison ordering, the superatomic part of a Harrison Boolean algebra, and the height-possessing part of a Harrison p-group, we show that the Turing degrees of images of the relation in computable copies of the structure are the same as the Turing degrees of paths through Kleene's . With this as motivation, we investigate the possible degrees of these paths. We show that there is a path in which ∅′ is not computable. In fact, there is one in which no noncomputable hyperarithmetical set is computable. There are paths that are Turing incomparable, or Turing incomparable over a given hyperarithmetical set. There is a pair of paths whose degrees form a minimal pair. However, there is no path of minimal degree.

Bi‐embeddability spectra and bases of spectra

“…and therefore every linear ordering of order type ω CK 1 can compute a linear ordering of order type r(L). Goncharov, Knight, Harizanov, and Shore characterized the degrees that compute maximal well ordered initial segments of the Harrison ordering which has order type ω CK 1 (1 + η) [12]. Let H be the family of these degrees.…”

Preserving Categoricity and Complexity of Relations

“…If A is relatively hyperarithmetically categorical, then R is relatively intrinsically hyperarithmetical on A * . By virtue of results in [18,19] (reworked in [20]), if R is intrinsically hyperarithmetical on A then it is relatively intrinsically hyperarithmetical on A.…”

COMPUTABLE STRUCTURES OF RANK $\omega_{1}^{{\rm CK}}$

“…We are ready to show that the Scott ranks of A and A * are related as in Theorem 2.1. If the orbit of a tuple in one of the structures is intrinsically hyperarithmetical, then by a result of Soskov [13], it is definable by a computable infinitary formula without parameters (see also [6]). Suppose one of the structures has computable rank.…”