Let's say you are a prospective mathematician with some addled ideas about cardinality.

If you assumed that the natural numbers were finite, you'd quickly vanish in a puff of logic. :)

If you thought that natural numbers and reals had the same cardinality - measure theory would almost surely break down, and your assumption would conflict with any number of "completeness theorems" in analysis (like the Baire Category Theorem for instance).

However, let's say you concluded that there were only three types of cardinality - finite, countably infinite, and uncountable.

Would this erroneous belief conflict with any major theorems in analysis, algebra or geometry ? Would any fields of math - outside set theory - be clearly incompatible with your assumption ?

PS: Apologies for the provocative title. Hope the question is clear.