If a set is a number of elements, what is meant by number here? What particular type of number (a natural number, real number, rational number, etc.) is meant? If it is a natural number, for instance, how is it possible to create a set of real numbers which has more elements than the set of natural numbers, thus making the construction of the set impossible insofar as there aren't enough natural numbers to provide positions for the real numbers (and also inversely, if you were to admit it as possible, wouldn't that imply that there is a one-to-one correspondance possible between the set of reals and naturals)? If it is a real number, wouldn't it be impossible to create the type of numbers with a cardinality higher than the cardinality of real numbers? (and so on)

Please help.