With the cardinality of a set defined as the number of elements of a set, it might seem obvious that thee is no largest cardinal number. To find a larger cardinal number, just find a set with more elemts. However, many sets have an infinite number of elements. What we really mean in saying that the is no largest cardinal number is that there is no limit to the order of infinity. No matter how infinitely large our set is, there is always an infinitely larger set.
This is called Cantor's Theorem. More concisely, it states: for any set (orwhere is the power set, the set of all subsets ofIfhaselements, it haspossible subsets.
We can prove the theorem by induction on successive power sets
and so on
Now setis infinite so
and here is no largest cardinal number.