Connectedness of the Continuum in Intuitionistic Mathematics
by Mark Bickford
Working in INT (Intuitionistic analysis) we prove a strong, constructive connectedness property of the continuum: for any non-empty sets, A and B, if R = A ∪ B then A ∩ B is non-empty. It is well known that the intuitionistic continuum is indecomposable: if R = A ∪ B and A ∩ B = ∅ then A = R or A = ∅, but this property is essentially negative — equivalent to if A, B are non-empty and R = A ∪ B then ¬(A ∩ B ≠ ∅). Our 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 BHK interpretation of quantifiers in constructive type theory). We also replace indecomposability by connectedness in some results of van Dalen that use additional intuitionistic axioms.
bibTex ref: RCB-764