The remarks on computable and continuous functions can also be thought about as follows.<p>The law of the excluded middle is only one non-constructive aspect of classical mathematics. Dropping it leaves you with what is usually called "intuionistic logic", and the characterization of constructive mathematics as broader than classic mathematics is literally correct in that every intuionistic theorem implies a classical theorem, and every classical theorem is equivalent to an intuionistic theorem.<p>Another non-constructive aspect of classical mathematics, which remains non-constructive in intuionistic logic, are axioms whose conclusion involves the existential quantifier. In both classical and intuionistic logic, the inference rule, which governs how the existential quantifer may be used, looks as follows.<p>> Asserting that "the existence of x such that phi(x) holds entails psi" where x is a variable bound by the existential quantifer and not occurring in psi, is equivalent to assering that "phi(y) entails psi" where y is a free variable not ocurring in psi.<p>This inference rule (assembly language of logic) seems weird, so an example of its use might help.<p>> "there exists x such that phi(x) holds entails that there exists x such that phi(x) holds" by tautological inference, and by the existential quantifier inference we have that "phi(y) entails the existence of x such that phi(x) holds" where y is a free variable not appearing in "exists x such that phi(x) holds".<p>When doing classical or intuionistic mathematics, you would usually use axioms of the form "psi entails that there exists x such that phi(x) holds". But such axioms are non-constructive because the inference rule does not allow you to start with "there exists x such that phi(x) holds" and somehow construct an actual term t such that phi(t) holds.<p>Now what Mike says in his slides is that it is consistent to modify intuionistic mathematics into a "fully constructive mathematics" by strengthening the existential quantifier so that "exists x such that phi(x) holds" would imply that we can (computably) come up with a specific term t such that phi(t) holds.<p>As a consequence, any statement in fully constructive mathematics involving the existential quantifier as a conclusion is a stronger statement than such in classical mathematics, which is roughly why all functions end up being continuous: the definition of the notion of a function itself requires that the function be computable, i.e. that its values exist in the sense of the strengthened existential quantifier.