Constructing a universe for the setoid model

The setoid model is a model of intensional type theory that validates certain extensionality principles, like function extensionality and propositional extensionality, the latter being a limited form of univalence that equates logically equivalent propositions. The appeal of this model construction is that it can be constructed in a small, intensional, type theoretic metatheory, therefore giving a method to boostrap extensionality. The setoid model has been recently adapted into a formal system, namely Setoid Type Theory (SeTT). SeTT is an extension of intensional Martin-Löf type theory with constructs that give full access to the extensionality principles that hold in the setoid model.Although already a rich theory as currently defined, SeTT currently lacks a way to internalize the notion of type beyond propositions, hence we want to extend SeTT with a universe of setoids. To this aim, we present the construction of a (non-univalent) universe of setoids within the setoid model, first as an inductive-recursive definition, which is then translated to an inductive-inductive definition and finally to an inductive family. These translations from more powerful definition schemas to simpler ones ensure that our construction can still be defined in a relatively small metatheory which includes a proof-irrelevant identity type with a strong transport rule.

Inductive definitions and proofs

An inductive definition of a predicate R characterizes the Rs as the smallest class which satisfies a basis clause of the form (β(x)→Rx), telling us that certain things satisfy R, together with one or more closure clauses of the form (Φ(x,R)→Rx), which tell us that, if certain other things satisfy R, x satisfies R as well. ’Smallest’ here means that the class of Rs is included in every other class which satisfies the basis and closure clauses. Inductive definitions are useful because of inductive proofs. To show that every R has property P, show that the class of Rs that have P satisfies the basis and closure clauses. The closure clauses tell us that if certain things satisfy R, x satisfies R as well. Thus satisfaction of the condition Φ(x,R) should be ensured by positive information to the effect that certain things satisfy R, and not also require negative information that certain things fail to satisfy R. In other words, the condition Φ(x,R) should be monotone, so that, if R ⊆S and Φ(x,R), then Φ(x,S); otherwise, we would have no assurance of the existence of a smallest class satisfying the basis and closure conditions. While inductive definitions can take many forms, they have been studied most usefully in the special case in which the basis and closure clauses are formulated within the predicate calculus. Initiated by Yiannis Moschovakis, the study of such definitions has yielded an especially rich and elegant theory.

ABSTRACT INDUCTIVE AND CO-INDUCTIVE DEFINITIONS

In [G. Curi, On Tarski’s fixed point theorem. Proc. Amer. Math. Soc., 143 (2015), pp. 4439–4455], a notion of abstract inductive definition is formulated to extend Aczel’s theory of inductive definitions to the setting of complete lattices. In this article, after discussing a further extension of the theory to structures of much larger size than complete lattices, as the class of all sets or the class of ordinals, a similar generalization is carried out for the theory of co-inductive definitions on a set. As a corollary, a constructive version of the general form of Tarski’s fixed point theorem is derived.

Encountering the muse: An exploration of the relationship between inspiration and information in the museum context

How are information and inspiration connected? Answering this question can help information professionals facilitate the pathways to inspiration. Inspiration has previously been conceptualized as a goal or mode of information seeking, but this says little about the nature of inspiration or how it is experienced. In this study, we explore the connection between information and inspiration through a qualitative approach, using the museum as our setting; specifically, the researchers’ own visits to three separate museums. We use collaborative auto-hermeneutics, a methodology specifically suited to such a reflexive exploration, to document and analyze three individual museum visits. The following research questions were the main driver for this exploratory study: What is inspiration, and How are inspiration and information related? In answer, we present an inductive definition of inspiration as a kind of information, and we discuss how this definition fits in with the information science literature as well as offer some practical applications.

Motivating the Rules of Sequent Calculus

Parallelized elimination rules in natural deduction correspond to Left rules in the sequent calculus; and introduction rules correspond to Right rules. These rules may be construed as inductive clauses in the inductive definition of the notion of sequent proof. There is a natural isomorphism between natural deductions in Core Logic and the sequent proofs that correspond to them. We examine the relations, between sequents, of concentration and dilution; and describe what it is for one sequent to strengthen another. We examine some possible global restrictions on proof-formation, designed to prevent proofs from proving dilutions of sequents already proved by a subproof. We establish the important result that the sequent rules of Core Logic maintain concentration, and we explain its importance for automated proof-search.

Weak systems of determinacy and arithmetical quasi-inductive definitions

We locate winning strategies for various -games in the L-hierarchy in order to prove the following:Theorem 1. KP + Σ2-Comprehension -Determinacy.”Alternatively: “there is a β-model of -Determinacy.” The implication is not reversible. (The antecedent here may be replaced with instances of Comprehension with only -lightface definable parameters—or even weaker theories.)Theorem 2. KP + Δ2-Comprehension + Σ2-Replacement + -Determinacy.(Here AQI is the assertion that every arithmetical quasi-inductive definition converges.) Alternatively: -Determinacy.Hence the theories: , and are in strictly descending order of strength.

On a partial syntactical criterion for the left distributivity and the idempotency

We study here so called cuts of terms and their classes modulo the identities of the left distributivity and the idempotency. We give an inductive definition of such classes and this gives us a criterion that decides in some cases whether two terms are equivalent modulo both identities.

THE FINE STRUCTURE OF THE INTUITIONISTIC BOREL HIERARCHY

In intuitionistic analysis, a subset of a Polish space like ℝ or ${\cal N}$ is called positively Borel if and only if it is an open subset of the space or a closed subset of the space or the result of forming either the countable union or the countable intersection of an infinite sequence of (earlier constructed) positively Borel subsets of the space. The operation of taking the complement is absent from this inductive definition, and, in fact, the complement of a positively Borel set is not always positively Borel itself (see Veldman, 2008a). The main result of Veldman (2008a) is that, assuming Brouwer's Continuity Principle and an Axiom of Countable Choice, one may prove that the hierarchy formed by the positively Borel sets is genuinely growing: every level of the hierarchy contains sets that do not occur at any lower level. The purpose of the present paper is a different one: we want to explore the truly remarkable fine structure of the hierarchy. Brouwer's Continuity Principle again is our main tool. A second axiom proposed by Brouwer, his Thesis on Bars is also used, but only incidentally.