Intuitionism and rejection of standard logic postulates Intuitionism refuses the Cantor hypothesis about continuum as a hypothesis meainingless from the intuitionistic point of view Also, ' _the tertium non datur_ ' principle: $$A\vee\neg{A}$$ is rejected 'a priori' in the sense that we can only prove the validity of $A$ or $\neg{A}$. My question is: given the rejection of these principles, in particular the second, is it still possible to keep the validity of the mathematical analysis within the intuitionism? Or maybe, accepting the intuitionistic approach, you need to develop a new kind of analysis, because, for example, it's impossible to use the ' _reductio ad absurdum_ ' to prove a theorem? Thanks

There are indeed formal logical systems for intuitionism, just as there are formal logical systems for classical logic. These systems do have slightly different inference rules than classical logic. In the most common cases, though, a set of rules for intuitionistic logic becomes a complete set of rules for classical logic by simply adding the law of the excluded middle as one more rule. So there are not always large differences between the two systems.

By the way, "real numbers", "set of real numbers", "countable", and "bijection" all make sense as terms to intuitionists, and so the statement of CH as "any set of real numbers that is not countable has a bijection with the set of all real numbers" is not meaningless to them. It might be that the statement is neither proved nor disproved - so that they will not call it true or false - but that does not make it meaningless more than any other open problem.