Without the theorem of the excluded middle, all kinds of difficulties arise. In fact, many mathematical proofs are based on ...

Peter Bruyns, Henry Rose, Varieties with Cofinal Sets: Examples and Amalgamation, Proceedings of the American Mathematical Society, Vol. 111, No. 3 (Mar., 1991), pp ...

We define an open symmetric two-place relation on the reals such that the reals cannot be covered by countably many sets of related elements, but there is no uncountable set of unrelated elements. The ...