Connectedness of the continuum in intuitionistic mathematics

Abstract

Working in INT (Intuitionistic analysis) we prove a strong, constructive connectedness property of the continuum: for any non-empty sets, A andOur connectedness property is positive; so, given a ∈ A, b ∈ B, and a witness to R = A ∪ B, to prove our theorem we must construct a real number r ∈ A ∩ B. We can construct the needed real number using only Bishop's constructive mathematics (BISH) and a weak form of Brouwer's continuity principle (and the choice principles that come from the Brouwer-Heyting-Kolmogorov interpretation of quantifiers in constructive type theory). We also replace indecomposability by connectedness in some results of van Dalen that use additional intuitionistic axioms.

Cost-Aware Type Theory

“…NuPRL is a proof refinement framework developed in the 1980s by Constable et al[20]. It also refers to a family of computational type theories (which is based on) and has been fruitful in both mechanizing existing mathematics and proving new results[10].…”