Connectedness of the continuum in intuitionistic mathematics

Abstract

AbstractWorking in (Intuitionistic analysis) we prove a strong, constructive connectedness property of the continuum: for any non‐empty sets, A and B, if then is non‐empty. It is well known that the intuitionistic continuum is indecomposable: if and then or , but this property is essentially negative—equivalent to if A, B are non‐empty and then . Our connectedness property is positive; so, given , , and a witness to , to prove our theorem we must construct a real number . We can construct the needed real number using only Bishop's constructive mathematics () 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.