Jump Closure and Limit Uniformization in the Ideal Completion of the Turing Degrees

Abstract

The Turing jump has no fixed point on the Turing degrees: $\mathbf{a}{<}_T{\mathbf{a}}^{\prime }$ for every degree $\mathbf{a}$. After passing to the ideal completion, however, a natural fixed-point phenomenon appears. We study the Scott-continuous lifting $\Gamma :\mathrm{Idl}({\mathbf{D}}_T)\to \mathrm{Idl}({\mathbf{D}}_T)$, given by $\Gamma (I)=\downarrow \{{\mathbf{a}}^{\prime }:\mathbf{a}\in I\}$. Starting from the computable degree, Kleene iteration reaches its first fixed point at stage $\omega$, namely the Turing ideal of arithmetical degrees; more generally, above $\mathbf{a}$ the least fixed point is the ideal of degrees arithmetical in $\mathbf{a}$. To pass beyond this fixed point, we introduce a limit-uniformization operator. Although the ideal of finite jumps contains every $0^{(n)}$, it does not contain the uniform limit oracle $0^{(\omega )}={°}_T\left({⨁}_{n<\omega }{0}^{(n)}\right)$. The uniformization operator adjoins this oracle only when all finite jump degrees are present. It is monotone but not Scott-continuous. Composing jump closure with one such gate yields closure ordinal $\omega \cdot 2$; gates at $\omega ,2\omega ,3\omega ,\dots$ yield closure ordinal ${\omega }^2$. Thus non-uniform closure under relativized halting problems is Scott-continuous and reaches fixed ideals, while uniform coding of an entire prior hierarchy is infinitary, discontinuous, and reopens diagonalization. This gives a domain-theoretic semantics for the successor/limit distinction in transfinite Turing-jump hierarchies and links failures of Scott continuity with closure ordinals.