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.

Mapping Individual Variation in Male Mating Preference Space: Multiple Choice in a Color Polymorphic Cichlid Fish

Sexual selection theory largely rests on the assumption that populations contain individual variation in mating preferences and that individuals are consistent in their preferences. However, there are few empirical studies of within-population variation and even fewer have examined individual male mating preferences. Here, we studied a color polymorphic population of the Lake Victoria cichlid fish Neochromis omnicaeruleus, a species in which color morphs are associated with different sex-determining factors. Wild-caught males were tested in three-way choice trials with multiple combinations of different females belonging to the three color morphs. Compositional log-ratio techniques were applied to analyze individual male mating preferences. Large individual variation in consistency, strength, and direction of male mating preferences for female color morphs was found and hierarchical clustering of the compositional data revealed the presence of four distinct preference groups corresponding to the three color morphs in addition to a no-preference class. Consistency of individual male mating preferences was higher in males with strongest preferences. We discuss the implications of these findings for our understanding of the mechanisms underlying polymorphism in mating preferences.