I'm somewhat familiar with the school of intuitionistic logic. I know that an intuitionistic logician thinks of infinity as constructive as apposed to complete. Thus a intuitionistic logician cannot justify principles like the law of excluded middle and double negation elimination. I'm wondering how infinite sets are defined from a constructive point of view?
It is my understanding that intuitionism was a response to the paradoxes in the foundations of mathematics. How does intuitionism address Russel's Paradox for example? Can we even define a set of all sets which do not contain itself in intuitionistic logic. (This is asked from a non-formal perspective similar to Brouwer's.)