scholarly journals Lorenzen Between Gentzen and Schütte

2021 ◽  
pp. 63-76
Author(s):  
Reinhard Kahle ◽  
Isabel Oitavem
Keyword(s):  
Type Theory ◽  
Proof Theory ◽  
Historical Context ◽  
Consistency Proof

AbstractWe discuss Lorenzen’s consistency proof for ramified type theory without reducibility, published in 1951, in its historical context and highlight Lorenzen’s contribution to the development of modern proof theory, notably by the introduction of the $$\omega $$ ω -rule.

An intensional type theory: motivation and cut-elimination

2001 ◽  
Vol 66 (1) ◽  
pp. 383-400 ◽  
Author(s):  
Paul C Gilmore
Keyword(s):  
Type Theory ◽  
Proof Theory ◽  
Predicate Logic ◽  
Order Term ◽  
Higher Order ◽  
Cut Elimination ◽  
Consistency Proof ◽  
Completeness Proof ◽  
Higher Order Term

AbstractBy the theory TT is meant the higher order predicate logic with the following recursively defined types:(1) 1 is the type of individuals and [] is the type of the truth values:(2) [τ1…..τn] is the type of the predicates with arguments of the types τ1…..τn.The theory ITT described in this paper is an intensional version of TT. The types of ITT are the same as the types of TT, but the membership of the type 1 of individuals in ITT is an extension of the membership in TT. The extension consists of allowing any higher order term, in which only variables of type 1 have a free occurrence, to be a term of type 1. This feature of ITT is motivated by a nominalist interpretation of higher order predication.In ITT both well-founded and non-well-founded recursive predicates can be defined as abstraction terms from which all the properties of the predicates can be derived without the use of non-logical axioms.The elementary syntax, semantics, and proof theory for ITT are defined. A semantic consistency proof for ITT is provided and the completeness proof of Takahashi and Prawitz for a version of TT without cut is adapted for ITT: a consequence is the redundancy of cut.

Mathematical Logic: Proof Theory, Type Theory and Constructive Mathematics

2005 ◽  
pp. 779-813
Author(s):  
Helmut Schwichtenberg ◽  
Vladimir Keilis-Borok ◽  
Samuel Buss
Keyword(s):  
Mathematical Logic ◽  
Type Theory ◽  
Proof Theory ◽  
Constructive Mathematics

ETA-RULES IN MARTIN-LÖF TYPE THEORY

2019 ◽  
Vol 25 (03) ◽  
pp. 333-359
Author(s):  
ANSTEN KLEV
Keyword(s):  
Lower Order ◽  
Category Theory ◽  
Type Theory ◽  
Proof Theory ◽  
Arbitrary Element ◽  
Higher Order

AbstractThe eta rule for a set A says that an arbitrary element of A is judgementally identical to an element of constructor form. Eta rules are not part of what may be called canonical Martin-Löf type theory. They are, however, justified by the meaning explanations, and a higher order eta rule is part of that type theory. The main aim of this article is to clarify this somewhat puzzling situation. It will be argued that lower order eta rules do not, whereas the higher order eta rule does, accord with the understanding of judgemental identity as definitional identity. A subsidiary aim is to clarify precisely what an eta rule is. This will involve showing how such rules relate to various other notions of type theory, proof theory, and category theory.

Pure Proof Theory Aims, Methods and Results: Extended Version of Talks Given at Oberwolfach and Haifa

1996 ◽  
Vol 2 (2) ◽  
pp. 159-188 ◽  
Author(s):  
Wolfram Pohlers
Keyword(s):  
Mathematical Logic ◽  
Computer Science ◽  
Proof Theory ◽  
Axiom System ◽  
Theoretical Research ◽  
Cut Elimination ◽  
Extended Version ◽  
Consistency Proof ◽  
Exciting Field ◽  
Axiom Systems

Apologies. The purpose of the following talk is to give an overview of the present state of aims, methods and results in Pure Proof Theory. Shortage of time forces me to concentrate on my very personal views. This entails that I will emphasize the work which I know best, i.e., work that has been done in the triangle Stanford, Munich and Münster. I am of course well aware that there are as important results coming from outside this triangle and I apologize for not displaying these results as well.Moreover the audience should be aware that in some points I have to oversimplify matters. Those who complain about that are invited to consult the original papers.1.1. General. Proof theory startedwithHilbert's Programme which aimed at a finitistic consistency proof for mathematics.By Gödel's Theorems, however, we know that we can neither formalize all mathematics nor even prove the consistency of formalized fragments by finitistic means. Inspite of this fact I want to give some reasons why I consider proof theory in the style of Gentzen's work still as an important and exciting field of Mathematical Logic. I will not go into applications of Gentzen's cut-elimination technique to computer science problems—this may be considered as applied proof theory—but want to concentrate on metamathematical problems and results. In this sense I am talking about Pure Proof Theory.Mathematicians are interested in structures. There is only one way to find the theorems of a structure. Start with an axiom system for the structure and deduce the theorems logically. These axiom systems are the objects of proof-theoretical research. Studying axiom systems there is a series of more or less obvious questions.

The independence of Peano's fourth axiom from Martin-Löf's type theory without universes

1988 ◽  
Vol 53 (3) ◽  
pp. 840-845 ◽  
Author(s):  
Jan M. Smith
Keyword(s):  
Type Theory ◽  
Predicate Logic ◽  
Single Element ◽  
Simple Type ◽  
Consistency Proof ◽  
Computational System ◽  
First Order ◽  
Simple Type Theory ◽  
Individual Domain ◽  
Order Arithmetic

In Hilbert and Ackermann [2] there is a simple proof of the consistency of first order predicate logic by reducing it to propositional logic. Intuitively, the proof is based on interpreting predicate logic in a domain with only one element. Tarski [7] and Gentzen [1] have extended this method to simple type theory by starting with an individual domain consisting of a single element and then interpreting a higher type by the set of truth valued functions on the previous type.I will use the method of Hilbert and Ackermann on Martin-Löf's type theory without universes to show that ¬Eq(A, a, b) cannot be derived without universes for any type A and any objects a and b of type A. In particular, this proves the conjecture in Martin-Löf [5] that Peano's fourth axiom (∀x ϵ N)¬ Eq(N, 0, succ(x)) cannot be proved in type theory without universes. If by consistency we mean that there is no closed term of the empty type, then the construction will also give a consistency proof by finitary methods of Martin-Löf's type theory without universes. So, without universes, the logic obtained by interpreting propositions as types is surprisingly weak. This is in sharp contrast with type theory as a computational system, since, for instance, the proof that every object of a type can be computed to normal form cannot be formalized in first order arithmetic.

The equivalence of NF-style set theories with “tangled” type theories; the construction of ω-models of predicative NF (and more)

1995 ◽  
Vol 60 (1) ◽  
pp. 178-190 ◽  
Author(s):  
M. Randall Holmes
Keyword(s):  
Set Theory ◽  
Type Theory ◽  
The Other ◽  
Natural Numbers ◽  
Consistency Proof ◽  
Natural Behaviour ◽  
New Foundations ◽  
Consistency Problem

AbstractAn ω-model (a model in which all natural numbers are standard) of the predicative fragment of Quine's set theory “New Foundations” (NF) is constructed. Marcel Crabbé has shown that a theory NFI extending predicative NF is consistent, and the model constructed is actually a model of NFI as well. The construction follows the construction of ω-models of NFU (NF with urelements) by R. B. Jensen, and, like the construction of Jensen for NFU, it can be used to construct α-models for any ordinal α. The construction proceeds via a model of a type theory of a peculiar kind; we first discuss such “tangled type theories” in general, exhibiting a “tangled type theory” (and also an extension of Zermelo set theory with Δ0 comprehension) which is equiconsistent with NF (for which the consistency problem seems no easier than the corresponding problem for NF (still open)), and pointing out that “tangled type theory with urelements” has a quite natural interpretation, which seems to provide an explanation for the more natural behaviour of NFU relative to the other set theories of this kind, and can be seen anachronistically as underlying Jensen's consistency proof for NFU.

Sign in / Sign up

Export Citation Format

Share Document