Constructive set theoretic models of typed combinatory logic

1993 ◽  
Vol 58 (1) ◽  
pp. 99-118
Author(s):  
Andreas Knobel
Keyword(s):  
Function Space ◽  
Set Theory ◽  
Combinatory Logic ◽  
Common Features ◽  
Space Model ◽  
Constructive Set Theory

AbstractWe shall present two novel ways of deriving simply typed combinatory models. These are of interest in a constructive setting. First we look at extension models, which are certain subalgebras of full function space models. Then we shall show how the space of singletons of a combinatory model can itself be made into one. The two and the algebras in between will have many common features. We use these two constructions in proving:There is a model of constructive set theory in which every closed extensional theory of simply typed combinatory logic is the theory of a full function space model.

scholarly journals Homotopy type-theoretic interpretations of constructive set theories

2019 ◽  
pp. 1-32
Author(s):  
Cesare Gallozzi
Keyword(s):  
Comparative Analysis ◽  
Set Theory ◽  
Homotopy Type ◽  
Type Theory ◽  
Axiom Of Choice ◽  
Homotopy Type Theory ◽  
Constructive Set Theory

Abstract We introduce a family of (k, h)-interpretations for 2 ≤ k ≤ ∞ and 1 ≤ h ≤ ∞ of constructive set theory into type theory, in which sets and formulas are interpreted as types of homotopy level k and h, respectively. Depending on the values of the parameters k and h, we are able to interpret different theories, like Aczel’s CZF and Myhill’s CST. We also define a proposition-as-hproposition interpretation in the context of logic-enriched type theories. The rest of the paper is devoted to characterising and analysing the interpretations considered. The formulas valid in the prop-as-hprop interpretation are characterised in terms of the axiom of unique choice. We also analyse the interpretations of CST into homotopy type theory, providing a comparative analysis with Aczel’s interpretation. This is done by formulating in a logic-enriched type theory the key principles used in the proofs of the two interpretations. Finally, we characterise a class of sentences valid in the (k, ∞)-interpretations in terms of the ΠΣ axiom of choice.

Set Theory based on Combinatory Logic.

1970 ◽  
Vol 35 (1) ◽  
pp. 147
Author(s):  
Jonathan P. Seldin ◽  
Maarten Wicher Visser Bunder
Keyword(s):  
Set Theory ◽  
Combinatory Logic

Constructive set theory with operations

2014 ◽  
pp. 47-83 ◽  
Author(s):  
Andrea Cantini ◽  
Laura Crosilla
Keyword(s):  
Set Theory ◽  
Constructive Set Theory

VERking in constructive set theory

1981 ◽  
Vol 6 (3) ◽  
pp. 58-60
Author(s):  
Robert L. Constable
Keyword(s):  
Set Theory ◽  
Constructive Set Theory

A weak absolute consistency proof for some systems of illative combinatory logic

1983 ◽  
Vol 48 (3) ◽  
pp. 771-776 ◽  
Author(s):  
M.W. Bunder
Keyword(s):  
Set Theory ◽  
Combinatory Logic ◽  
Weak Consistency ◽  
Consistency Proof ◽  
First Order ◽  
Formal Systems ◽  
Elementary Methods ◽  
Order Arithmetic ◽  
Strongly Consistent ◽  
Order Predicate Calculus

A large number of formal systems based on combinatory logic or λ-calculus have been extended to include first order predicate calculus. Several of these however have been shown to be inconsistent, all, as far as the author knows, in the strong sense that all well formed formulas (which here include all strings of symbols) are provable. We will call the corresponding consistency notion—an arbitrary wff ⊥ is provable—weak consistency. We will say that a system is strongly consistent if no formula and its negation are provable.Now for some systems, such as that of Kuzichev [11], the strong and weak consistency notions are equivalent, but in the systems of [5] and [6], which we will be considering, they are not. Each of these systems is strong enough to have all of ZF set theory, except Grounding and Choice, interpretable in it, and the system of [5] can also encompass first order arithmetic (see [7]). It therefore seems unlikely that a strong consistency result could be proved for these systems using elementary methods. In this paper however, we prove the weak consistency of both these systems by means that could be formulated, at least within the theory of [5]. The method also applies to the typed systems of Curry, Hindley and Seldin [10] and to Seldin's generalised types [12].

Sign in / Sign up

Export Citation Format

Share Document