The weak choice principle WISC may fail in the category of sets

The set-theoretic axiom WISC states that for every set there is a set of surjections to it cofinal in all such surjections. By constructing an unbounded topos over the category of sets and using an extension of the internal logic of a topos due to Shulman, we show that WISC is independent of the rest of the axioms of the set theory given by a well-pointed topos. This also gives an example of a topos that is not a predicative topos as defined by van den Berg.Comment: v2 Change of title and abstract; v3 Almost completely rewritten after referee pointed out critical mistake. v4 Final version. Will be published in Studia Logica. License is CC-B

Constructing Initial Algebras Using Inflationary Iteration

An old theorem of Adámek constructs initial algebras for sufficiently cocontinuous endofunctors via transfinite iteration over ordinals in classical set theory. We prove a new version that works in constructive logic, using “inflationary” iteration over a notion of size that abstracts from limit ordinals just their transitive, directed and well-founded properties. Borrowing from Taylor’s constructive treatment of ordinals, we show that sizes exist with upper bounds for any given signature of indexes. From this it follows that there is a rich class of endofunctors to which the new theorem applies, provided one admits a weak form of choice (WISC) due to Streicher, Moerdijk, van den Berg and Palmgren, and which is known to hold in the internal constructive logic of many kinds of elementary topos.UK EPSRC PhD studentship 211980

Internal categories, anafunctors and localisations

In this article we review the theory of anafunctors introduced by Makkai and Bartels, and show that given a subcanonical site S, one can form a bicategorical localisation of various 2-categories of internal categories or groupoids at weak equivalences using anafunctors as 1-arrows. This unifies a number of proofs throughout the literature, using the fewest assumptions possible on S.Comment: 42 pages. Final version to appear in Theory and Applications of Categories. License is CC-B

Broad Infinity and Generation Principles

This paper introduces Broad Infinity, a new and arguably intuitive axiom scheme. It states that "broad numbers", which are three-dimensional trees whose growth is controlled, form a set. If the Axiom of Choice is assumed, then Broad Infinity is equivalent to the Ord-is-Mahlo scheme: every closed unbounded class of ordinals contains a regular ordinal. Whereas the axiom of Infinity leads to generation principles for sets and families and ordinals, Broad Infinity leads to more advanced versions of these principles. The paper relates these principles under various prior assumptions: the Axiom of Choice, the Law of Excluded Middle, and weaker assumptions.Comment: 52 page

The elementary construction of formal anafunctors

This article gives an elementary and formal 2-categorical construction of a bicategory of right fractions analogous to anafunctors, starting from a 2-category equipped with a family of covering maps that are fully faithful and co-fully faithful.Comment: v1: 20 pages, many diagrams. v2: updated after referee comments, 32 pages. v3: one minor terminology change, one proof streamlined after additional feedback, 32 page

Variations on Piaget\u27s Pre-number Development Tests Used as Learning Experiences

The effects of learning upon the rate of conservation attainment and its transference to other areas of performance were studied using 17 mentally retarded subjects. Subjects found to be non-conservers on pretests were taught conservation and correspondence using a variety of tasks modeled from Piaget\u27s experiments. They were also pretested on the WISC Information and Picture Arrangement Sub-tests and a number concept test. Following the learning experiences, the subjects were posttested using the same measures used for pretesting with the exception of the number test where an alternate form was used. Significant correlations were found between the conservation pretest scores and General Intelligence (r=.72), Chronological Age (r=.66), Mental Age (r=.91), Information sub-test (r=.76), Picture Arrangement sub-test (r=.83), and number concept test scores (r=.64). There were significant posttest gains on conservation (F=79.98, p Scores on an instrument designed to measure internalization of the concepts showed significant gains on posttest (F=15.97,