Sense, denotation and semantics natural deduction the Curry-Howard isomorphism the normalisation theorem Godel's system T coherence spaces denotational semantics of T sums in natural deduction system F coherence semantics of the sum cut elimination (Hauptsatz) strong normalisation for F representation theorem semantics of System F what is linear logic?

Proofs and types

Semantical Considerations on Modal Logic

• Constraint domains in which the possible values that a variable can take are restricted to a finite set. • Examples: Boolean constraints, or integer constraints in which each variable is constrained to lie in within a finite range of integers. • Widely used in constraint programming. • Many real problems can be easily represented using constraint domains, e.g.: scheduling, routing and timetabling. • They involve choosing amongst a finite number of possibilities. • Commercial importance to many businesses: e.g. deciding how air crews should be allocated to aircraft flights. • Developed methods by different research communities: Arc and node consistency techniques (artificial intelligence). Bounds propagation techniques (constraint programming). Integer programming (operations research).

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A computational logic

This book represents the first treatment of computable analysis at the graduate level within the tradition of classical mathematical reasoning. Among the topics dealt with are: classical analysis, Hilbert and Banach spaces, bounded and unbounded linear operators, eigenvalues, eigenvectors, and equations of mathematical physics. The book is self-contained, and yet sufficiently detailed to provide an introduction to research in this area.

Computability in analysis and physics

Admissible sets and structures

Publisher Summary This chapter demonstrates the informal rigor and completeness proofs in mathematical philosophy Informal rigor makes the analysis precise to eliminate the doubtful properties of the intuitive notions The principal emphasis is on intuitive notions, which do not occur in ordinary mathematical practice but leads to new axioms for current notions The difference between familiar independence results and the independence of the continuum hypothesis is discussed in the chapter; the difference is formulated in terms of higher order consequence The chapter discusses the relation between intuitive logical consequence on the one hand and so-called “semantic syntactic” consequence on the other Brouwer's empirical propositions and proof considers a striking use of a new primitive to derive a purely mathematical assertion The chapter explores the completeness questions for classical and intuitionist predicate logic The problems of completeness involve informal rigor, at least when deciding completeness with respect to an intuitive notion of consequence

Informal Rigour and Completeness Proofs

1. The purpose of the present paper is to formulate the problem of non-finitist proofs, and to solve it for certain extensions of the predicate calculus, and for analysis with the exclusion of the theory of sets of points. The corresponding problem for general formal systems is discussed in another publication [1]. To fix ideas we introduce the problem by examples from analysis. The general formulation is given in the text. Also, we shall use in the introduction the concepts decidable, verifiable, finitist without much formal explanation, because the reader is probably familiar with them, and they are defined early on in the text. The paper presupposes some knowledge of the methods and results in the theory of proofs. These enable one to state the general problem rather more precisely.

On the Interpretation of Non-Finitist Proofs--Part I

1. The purpose of the present paper is to formulate the problem of non-finitist proofs, and to solve it for certain extensions of the predicate calculus, and for analysis with the exclusion of the theory of sets of points.' The corresponding problem for general formal systems is discussed in another publication [1]. To fix ideas we introduce the problem by examples from analysis. The general formulation is given in the text. Also, we shall use in the introduction the concepts decidable, verifiable, finitist without much formal explanation, because the reader is probably familiar with them, and they are defined early on in the text. The paper presupposes some knowledge of the methods and results in the theory of proofs. These enable one to state the general problem rather more precisely.

On the Interpretation of Non-Finitist Proofs: Part II. Interpretation of Number Theory. Applications

Reflection Principles and Their Use for Establishing the Complexity of Axiomatic Systems

Suppose the ri-placed relation symbols Pi, 1 ≦ i ≦ k, are all the non-logical constants occurring in the closed formula , also written as , of Heyting's predicate calculus (HPC). Then HPC is called complete for provided , i.e.Here D ranges over arbitrary species, and over arbitrary (possibly incompletely defined) subspecies of ;

Georg Kreisel

Papers

Informal Rigour and Completeness Proofs

On the Interpretation of Non-Finitist Proofs--Part I

On the Interpretation of Non-Finitist Proofs: Part II. Interpretation of Number Theory. Applications

Reflection Principles and Their Use for Establishing the Complexity of Axiomatic Systems

On weak completeness of intuitionistic predicate logic