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

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].

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 Predicate functors revisited

1981 ◽  
Vol 46 (3) ◽  
pp. 649-652 ◽  
Author(s):  
W. V. Quine
Keyword(s):  
Set Theory ◽  
Predicate Logic ◽  
Combinatory Logic ◽  
Homogeneous Case ◽  
Quantification Theory ◽  
Present Scheme ◽  
First Order ◽  
Image Position ◽  
First Order Predicate Logic

Quantification theory, or first-order predicate logic, can be formulated in terms purely of predicate letters and a few predicate functors which attach to predicates to form further predicates. Apart from the predicate letters, which are schematic, there are no variables. On this score the plan is reminiscent of the combinatory logic of Schönfinkel and Curry. Theirs, however, had the whole of higher set theory as its domain; the present scheme stays within the bounds of predicate logic.In 1960 I published an apparatus to this effect, and an improved version in 1971. In both versions I assumed two inversion functors, major and minor; also a cropping functor and the obvious complement functor. The effects of these functors, when applied to an n-place predicate, are as follows:The variables here are explanatory only and no part of the final notation. Ultimately the predicate letters need exponents showing the number of places, but I omit them in these pages.A further functor-to continue now with the 1971 version-was padding:Finally there was a zero-place predicate functor, which is to say simply a constant predicate, namely the predicate ‘I’ of identity, and there was a two-place predicate functor ‘∩’ of intersection. The intersection ‘F ∩ G’ received a generalized interpretation, allowing ‘F’ and ‘G’ to be predicates with unlike numbers of places. However, Steven T. Kuhn has lately shown me that the generalization is unnecessary and reducible to the homogeneous case.

scholarly journals Various systems of set theory based on combinatory logic.

1974 ◽  
Vol 15 (2) ◽  
pp. 192-206 ◽  
Author(s):  
M. W. Bunder
Keyword(s):  
Set Theory ◽  
Combinatory Logic
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