Theory Of Inductive Definitions Research Articles

ABSTRACT INDUCTIVE AND CO-INDUCTIVE DEFINITIONS

AbstractIn [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.

Mathematical Proof Theory in the Light of Ordinal Analysis

We give an overview of recent results in ordinal analysis. Therefore,we discuss the different frameworks used in mathematical proof-theory, namely subsystem of analysis including reversemathematics, Kripke–Platek set theory, explicitmathematics, theories of inductive definitions,constructive set theory, and Martin-Lof’s typetheory.

The Gupta-Belnap systems ${\rm S}^\#$ and ${\rm S}^*$ are not axiomatisable.

Anil Gupta and Nuel Belnap's The Revision Theory of Truth presents revision theoretic systems of circular definitions. Part I of the present paper shows that the revision theories S# and S* are not axiomatis- able. Part II refines this result. Among other things, Part II shows that there is a strong relationship between revision theories and the theory of inductive definitions. This relationship is exploited to show that S# and S* (and all plausible revision theories of circular definitions) are of complexity at least Π^.

Coinductive formulas and a many-sorted interpolation theorem

AbstractWe use connections between conjunctive game formulas and the theory of inductive definitions to define the notions of a coinductive formula and its approximations. Corresponding to the theory of conjunctive game formulas we develop a theory of coinductive formulas, including a covering theorem and a normal form theorem for many sorted languages. Applying both theorems and the results on “model interpolation” obtained in this paper, we prove a many-sorted interpolation theorem for ω1ω-logic, which considers interpolation with respect to the language symbols, the quantifiers, the identity, and countably infinite conjunction and disjunction.

Addendum to “The truth is never simple”

The present note outlines an answer to a question listed as open in our recent survey article [1], familiarity with which is assumed.Let Γ be the class of all X ⊆ ω such that X is reducible to for some some arithmetical and some positive with Y ⊆ p(Y) for all Y.A.1. Theorem. is a complete set of class Γ.A.2. Corollary, (a) is-in-a--parameter.(b) (i) Every set-in-a--parameter is reducible to.(ii) Every set-in-a--parameter is reducible to.Remarks. A.1 answers 7.3 of [1]. A.2 says as much as can be said about in terms of the coarse classifications of the analytical hierarchy. A.2 follows from A.1 by general methods and results in the theory of inductive definitions (having nothing specifically to do with truth), and its proof will be omitted.Proof of A.1. We omit subscripts vF.Upper Bound. Define

Uniform inductive definability and infinitary languages

Introduction. The purpose of this paper is to show how results from the theory of inductive definitions can be used to obtain new compactness theorems for uncountable admissible languages. These will include improvements of the compactness theorem by J. Green [9].In [2] Barwise studies admissible sets satisfying the Σ1-compactness theorem. Our approach is to consider admissible sets satisfying what could be called the abstract extended completeness theorem, that is, sets where the consequence relation of the admissible fragment LA is Σ1-definable over A. We will call such sets Σ1-complete. For countable admissible sets, Σ1-completeness follows from the completeness theorem for LA.Having restricted our attention to Σ1-complete sets we are led to a stronger notion also true on countable admissible sets, namely what we shall call uniform Σ1-completeness. We will see that this notion can be viewed as extending to uncountable admissible sets, properties related to both the “recursion theory” and “proof theory” of countable admissible sets.By following Barwise's recent approach to admissible sets allowing “urelements,” we show that there is a natural connection between certain structures arising from the theory of inductive definability, and uniformly Σ1-complete admissible sets . The structures we have in mind are called uniform Kleene structures.

Uniform inductive definability and infinitary languages

Introduction. The purpose of this paper is to show how results from the theory of inductive definitions can be used to obtain new compactness theorems for uncountable admissible languages. These will include improvements of the compactness theorem by J. Green [9].In [2] Barwise studies admissible sets satisfying the Σ1-compactness theorem. Our approach is to consider admissible sets satisfying what could be called the abstract extended completeness theorem, that is, sets where the consequence relation of the admissible fragment LA is Σ1-definable over A. We will call such sets Σ1-complete. For countable admissible sets, Σ1-completeness follows from the completeness theorem for LA.Having restricted our attention to Σ1-complete sets we are led to a stronger notion also true on countable admissible sets, namely what we shall call uniform Σ1-completeness. We will see that this notion can be viewed as extending to uncountable admissible sets, properties related to both the “recursion theory” and “proof theory” of countable admissible sets.By following Barwise's recent approach to admissible sets allowing “urelements,” we show that there is a natural connection between certain structures arising from the theory of inductive definability, and uniformly Σ1-complete admissible sets . The structures we have in mind are called uniform Kleene structures.