Variations on Δ11\Delta^1_1 Determinacy and ℵω1\aleph_{\omega_1}

We consider a weaker form of $\Delta^1_1$ Turing determinacy. Let $2 \leqslant \rho < \omega_{1}^{\mathsf{CK}}$, $\textrm{Weak-Turing-Det}_{\rho}(\Delta^1_1)$ is the statement: Every $\Delta^1_1$ set of reals cofinal in the Turing degrees contains two Turing distinct $\Delta^0_\rho$-equivalent reals. We show in $\mathsf{ZF}^-$:$~~$ $\textrm{Weak-Turing-Det}_{\rho}(\Delta^1_1)$ implies for every $\nu < \omega_{1}^{\mathsf{CK}}$ there is a transitive model: $M \models \textrm{ZF}^- + "\aleph_\nu \textrm{ exists}"$. As a corollary: If every cofinal $\Delta^1_1$ set of Turing degrees contains both a degree and its jump, then for every $\nu < \omega_{1}^{\mathsf{CK}}$, there is a transitive model: $M \models \mathsf{ZF}^- + "\aleph_\nu \textrm{ exists}"$. $\bullet$ With a simple proof, this improves upon a well-known result of Harvey Friedman on the strength of Borel determinacy (though not assessed level-by-level). $\bullet$ Invoking Tony Martin's proof of Borel determinacy, $\textrm{Weak-Turing-Det}_{\rho}(\Delta^1_1)$ implies $\Delta^1_1$ determinacy. $\bullet$ We show further that, assuming $\Delta^1_1$ Turing determinacy, or Borel Turing determinacy, as needed: $-$ Every cofinal $\Sigma^1_1$ set of Turing degrees contains a "hyp-Turing cone": $\{x \in \mathcal{D} \mid d_0 \leqslant_T x \leqslant_h d_0 \}$. $-$ For a sequence $(A_{k})_{k < \omega}$ of analytic sets of Turing degrees, cofinal in $\mathcal{D}$, $\bigcap_{k} A_{k}$ is cofinal in $\mathcal{D}$.Comment: 10 page

Π^1_1 Borel Sets

The results in this paper were motivated by the following question of Sacks. Suppose T is a recursive theory with countably many countable models. What can you say about the least ordinal ɑ such that all models of T have Scott rank below ɑ? If Martin's conjecture is true for T then ɑ ≤ ω·2

Evaluation of a quality improvement intervention to reduce anastomotic leak following right colectomy (EAGLE): pragmatic, batched stepped-wedge, cluster-randomized trial in 64 countries

Background Anastomotic leak affects 8 per cent of patients after right colectomy with a 10-fold increased risk of postoperative death. The EAGLE study aimed to develop and test whether an international, standardized quality improvement intervention could reduce anastomotic leaks. Methods The internationally intended protocol, iteratively co-developed by a multistage Delphi process, comprised an online educational module introducing risk stratification, an intraoperative checklist, and harmonized surgical techniques. Clusters (hospital teams) were randomized to one of three arms with varied sequences of intervention/data collection by a derived stepped-wedge batch design (at least 18 hospital teams per batch). Patients were blinded to the study allocation. Low- and middle-income country enrolment was encouraged. The primary outcome (assessed by intention to treat) was anastomotic leak rate, and subgroup analyses by module completion (at least 80 per cent of surgeons, high engagement; less than 50 per cent, low engagement) were preplanned. Results A total 355 hospital teams registered, with 332 from 64 countries (39.2 per cent low and middle income) included in the final analysis. The online modules were completed by half of the surgeons (2143 of 4411). The primary analysis included 3039 of the 3268 patients recruited (206 patients had no anastomosis and 23 were lost to follow-up), with anastomotic leaks arising before and after the intervention in 10.1 and 9.6 per cent respectively (adjusted OR 0.87, 95 per cent c.i. 0.59 to 1.30; P = 0.498). The proportion of surgeons completing the educational modules was an influence: the leak rate decreased from 12.2 per cent (61 of 500) before intervention to 5.1 per cent (24 of 473) after intervention in high-engagement centres (adjusted OR 0.36, 0.20 to 0.64; P &lt; 0.001), but this was not observed in low-engagement hospitals (8.3 per cent (59 of 714) and 13.8 per cent (61 of 443) respectively; adjusted OR 2.09, 1.31 to 3.31). Conclusion Completion of globally available digital training by engaged teams can alter anastomotic leak rates. Registration number: NCT04270721 (http://www.clinicaltrials.gov)