Yet the set of all such infinite sets will be infinitely descending.
By proving that there are (infinitely) many possible sizes for infinite sets, Cantor established that set theory was not trivial, and it needed to be studied.
Cantor called the sizes of his infinite sets "transfinite cardinals".
To define infinite sets, Cantor used predicate formulas.
Brouwer denied that this dichotomy applied to infinite sets.
In the early 1900s a thorough theory of infinite sets was developed.
According to this view, which goes back to Aristotle, infinite sets do not exist, except potentially.
This is not so, however, for infinite sets, as is illustrated with the set N in the earlier example.
Cantor himself had given a way of defining real numbers as certain infinite sets of rational numbers.
This theory grew out of his investigations of some concrete problems regarding certain types of infinite sets of real numbers.
He demonstrated that there are many kinds of infinite sets, and some infinities are bigger than others.
