Modal Ω-Logic: Automata, Neo-Logicism, and Set-Theoretic Realism

This essay examines the philosophical significance of $\Omega$-logic in Zermelo-Fraenkel set theory with choice (ZFC). The duality between coalgebra and algebra permits Boolean-valued algebraic models of ZFC to be interpreted as coalgebras. The modal profile of $\Omega$-logical validity can then be countenanced within a coalgebraic logic, and $\Omega$-logical validity can be defined via deterministic automata. I argue that the philosophical significance of the foregoing is two-fold. First, because the epistemic and modal profiles of $\Omega$-logical validity correspond to those of second-order logical consequence, $\Omega$-logical validity is genuinely logical, and thus vindicates a neo-logicist conception of mathematical truth in the set-theoretic multiverse. Second, the foregoing provides a modal-computational account of the interpretation of mathematical vocabulary, adducing in favor of a realist conception of the cumulative hierarchy of sets

Complex Liver Resections for Intrahepatic Cholangiocarcinoma

The only curative treatment option for intrahepatic cholangiocarcinoma (iCCA) is liver resection. Due to central tumor localization and vascular invasion, complex liver resections play an important role in curative treatment. However, the long-term outcomes after complex liver resection are not known. Methods: A retrospective cohort study was conducted for all patients undergoing liver surgery for iCCA. Complex liver resections included ante situm resections, associating liver partition and portal vein ligation for staged hepatectomy (ALPPS) and major liver resection with vascular reconstructions. Results: Forty-nine patients (34%) received complex liver resection, 66 patients (46%) received conventional liver resection and 28 patients (20%) were not resectable during exploration. Preoperative characteristics were not different between the groups, except for Union for International Cancer Control (UICC) stages. The postoperative course for complex liver resections was associated with more complications and perioperative mortality. However, long-term survival was not different between complex and conventional resections. Independent risk factors for survival were R0 resections and UICC stage. Four patients underwent ante situm resection without any mortality. Conclusions: Complex liver resections are justified in selected patients and survival is comparable with conventional liver resections. Survival in iCCA is affected by UICC stage or resections margins and not by the complexity of the case

Interview with a set theorist

International audienceThe status of the set-theoretic independent statements is the main problem in the philosophy of set theory. We address this problem by presenting the perspective of a practising set theorist, and thus giving an authentic insight in the current state of thinking in set-theoretic practice, which is to a large extent determined by independence results. During several meetings, the second author has been asking the first author about the development of forcing, the use of new axioms and set-theoretic intuition on independence. Parts of these conversations are directly presented in this article. They are supplemented by important mathematical results as well as discussion sections. Finally, we present three hypotheses about set-theoretic practice: First that most set theorists were surprised by the introduction of the forcing method, second that most set theorists think that forcing is a natural part of contemporary set theory, and third that most set theorists prefer an answer to a problem with the help of a new axiom of lowest possible consistency strength, and for most set theorists, a difference in consistency strength weighs much more than the difference between forcing axiom and large cardinal axiom