Abstract
Based on the definition of Beth-Kripke model by Dragalin, we describe Beth model from the topological point of view. We show the relation of the topological definition with more traditional relational definition of Beth model that is based on forcing. We apply the topological definition to construct a Beth model for a theory of intuitionistic functionals of high types and to prove its consistency
Highlights
The studies of metamathematical properties of nonclassical theories are based on a variety of models such as topological models, Beth model and Kripke model
In this study we describe the general concept of Beth model from the topological point of view
The following definitions of Beth frame, Beth algebra and Beth model are modified from the definitions of Beth-Kripke frame, Beth-Kripke algebra and BethKripke model given in the book (Dragalin, 1987)
Summary
The studies of metamathematical properties of nonclassical theories are based on a variety of models such as topological models, Beth model and Kripke model. Van Dalen (1978) constructed a Beth model for intuitionistic analysis. The applications of BK-model in (Dragalin, 1987) include different versions of intuitionistic arithmetic and analysis. In (Kachapova, 2014; 2015) we created a Beth model for intuitionistic functionals of high types: 1functionals (sequences of natural numbers), 2functionals (sequences of 1-functionals), ..., (n + 1)functionals (sequences of n-functionals). In this study we apply the topological version of Beth model to the intuitionistic theory SLP of high-order functionals from (Kachapova, 2015), including lawless functionals and the ”creating subject”. It can be seen that the topological version of Beth model simplifies some constructions and consistency proofs.
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