I sometimes try to convey mathematical concepts and results to children without going through proofs to varying degree of success. I have wondered if even the deepest mathematical concepts can be conveyed to anyone in an intuitive manner. I am intrigued by the notion of human intuition. What makes us understand and see things intuitively? When is it possible?

Consider the result that there are more reals than natural numbers but there are as many integers as natural numbers or odd numbers or even numbers or prime numbers. All are infinite but infinity comes in different sizes. Integers are referred to be countably infinite and reals as uncountably infinite. One can start by using analogy of matching girls with boys. Consider integers as girls and even numbers as boys and then each girl can indeed find a boy to marry and vice versa and therefore there are as many integers as even numbers.

But how to convey the idea behind the proof of the result that there are more real numbers than integers without first going through details of diagonalization? Assume here girls are real numbers and boys are integers. Each real number is an infinite sequence of digits. For example, the girl named Pi is:

Therefore, one could say that any girl has an infinite description. Irrespective of how you marry them, one can create a new girl who is different from each married girl and is single. You build this new girl from other married girls as a composite, an infinite montage, but *changing *each piece. Therefore, she will be a totally new girl different from everyone else. One could continue such an intuitive and sketchy exposition by gradually becoming more precise and rigorous, slowly adding details, till the child sees the basic idea.

In my view, such an intuitive exposition should be then followed by actual proof and then a combination of both will make something click in a young mind and enable it to grasp this fundamental concept of degrees of infinity.