I just thought of a relatively easy way in which to explain how two seemingly different sizes of infinity can be demonstrated to be of the same size (to the student struggling to get her mind around this part of set-theory). Let’s begin by choosing two sets: take Set1 to be the set of all natural even numbers from 0 to +∞, and take Set2 to be the set of all natural numbers from 0 to +∞. It would seem, at first blush, as though Set2 had twice as many members as Set1, since for every two members of one set, the corresponding set seems to have, within the same range of values, at least one value left over which cannot be mapped on to the other set. However, mathematicians, set theorists, and philosophers who specialize in philosophy of mathematics tell us that these two sets can indeed be bijected. Bijection is the operation in set theory of bringing two sets into one-to-one correspondence, such that each member of one set maps on to exactly one, and only one, member of some other set such that all members in both sets are mapped on to one and only one member of the opposite set (for any two sets).
What causes the confusion is that we are so used to seeing natural numbers as values, but as far as this operation in set theory is concerned the natural numbers, taken as set members, are not values, but names – The name of the first through to the fourth members of Set1 are: [2,4,6,8] and the name of the first through to the fourth members of Set2 are: [1,2,3,4]. Once one realizes that the numbers in either set are simply names for members which belong to that set, it becomes easier to see how Set1 and Set2 can be bijected. You could arbitrarily rename all of the members of each set with randomly selected letters or symbols, and then it would become clearer that what we have are two sets, each of which are infinitely large in such a way that those infinities are comparable, or rather equivalent.
Of course, there are still larger and smaller sizes of infinity; for example, Set1 or Set2 would be smaller than the infinite set of real numbers between 0 and 1. However, at least Set1 and Set2 are clearly not differently sized.