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: $\mathbf a <_T \mathbf a'$ for every degree $\mathbf a$. After passing to the ideal completion, however, a natural fixed-point phenomenon appears. We study the Scott-continuous lifting $\Gamma:\operatorname{Idl}(\mathbf D_T)\to\operatorname{Idl}(\mathbf D_T)$, given by $\Gamma(I)=\downarrow\{\mathbf a':\mathbf a\in I\}$. 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 $\mathbf 0^{(n)}$, it does not contain the uniform limit oracle $\mathbf 0^{(\omega)}=\deg_T\!\left(\bigoplus_{n < \omega}0^{(n)}\right)$. 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 $\omega,2\omega,3\omega,\ldots$ 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.