Relative Predicativity and dependent recursion in second-order set theory and higher-orders theories

This article reports that some robustness of the notions of predicativity and of autonomous progression is broken down if as the given infinite total entity we choose some mathematical entities other than the traditional ω. Namely, the equivalence between normal transfinite recursion scheme and new dependent transfinite recursion scheme, which does hold in the context of subsystems of second order number theory, does not hold in the context of subsystems of second order set theory where the universe V of sets is treated as the given totality (nor in the contexts of those of n+3-th order number or set theories, where the class of all n+2-th order objects is treated as the given totality).

Explicit mathematics and operational set theory: some ontological comparisons

We discuss several ontological properties of explicit mathematics and operational set theory: global choice, decidable classes, totality and extensionality of operations, function spaces, class and set formation via formulas that contain the definedness predicate and applications.

Applications of random set theory in econometrics

In recent years, the econometrics literature has shown a growing interest in the study of partially identified models, in which the object of economic and statistical interest is a set rather than a point. The characterization of this set and the development of consistent estimators and inference procedures for it with desirable properties are the main goals of partial identification analysis. This review introduces the fundamental tools of the theory of random sets, which brings together elements of topology, convex geometry, and probability theory to develop a coherent mathematical framework to analyze random elements whose realizations are sets. It then elucidates how these tools have been fruitfully applied in econometrics to reach the goals of partial identification analysis.

A new model construction by making a detour via intuitionistic theories I: Operational set theory without choice is Π1-equivalent to KP

We introduce a version of operational set theory, OST−, without a choice operation, which has a machinery for Δ0Δ0 separation based on truth functions and the separation operator, and a new kind of applicative set theory, so-called weak explicit set theory WEST, based on Gödel operations. We show that both the theories and Kripke–Platek set theory KPKP with infinity are pairwise Π1Π1 equivalent. We also show analogous assertions for subtheories with ∈-induction restricted in various ways and for supertheories extended by powerset, beta, limit and Mahlo operations. Whereas the upper bound is given by a refinement of inductive definition in KPKP, the lower bound is by a combination, in a specific way, of realisability, (intuitionistic) forcing and negative interpretations. Thus, despite interpretability between classical theories, we make “a detour via intuitionistic theories”. The combined interpretation, seen as a model construction in the sense of Visser's miniature model theory, is a new way of construction for classical theories and could be said the third kind of model construction ever used which is non-trivial on the logical connective level, after generic extension à la Cohen and Krivine's classical realisability model.

Workshop: ASSIP - Attempted Suicide Short Intervention Program. Theory, Practice, and 24 Months Follow-up RCT.

Background: ASSIP is a manualized brief therapy based on a model of suicide as goal-directed action, aimed at establishing a therapeutic alliance in a patient-oriented, collaborative approach. The main goals of the three-session program ASSIP are for patients to understand, from an observer’s position, patterns leading to a suicidal crisis, recognize triggers and warning signs, and to establish individual safety strategies for future suicidal crises. An ongoing therapeutic support is provided with regular letters over 24 months. Method: The study was conducted in a naturalistic setting. 120 Patients were randomly assigned to an intervention group (60 participants) treated with ASSIP combined with follow-up contact through letters, and a control group (60 participants) receiving a single session of clinical assessment. Both groups had treatment as usual. Patients completed a set of psychosocial and clinical questionnaires every six months over a period of 24 months. Results: In the ASSIP group 5 patients made a total of 5 reattempts, compared to 15 patients with 41 reattempts in the control group. The survival analysis yielded a significant difference with a Wald Chi2 of .000003. The ASSIP group had significantly lower suicidal ideation and fewer days of inpatient treatment compared to the control group. Higher scores in the Penn Helping Alliance Questionnaire were associated with lower suicidal ideation during follow-up. Conclusions: ASSIP is a highly effective brief therapy for patients with recent suicide attempts. Forming a strong therapeutic alliance is considered to be a major factor for outcome. ASSIP can be used with minimal training by experienced therapists. An English version of the manual will be published in May 2015.