Cantor's Notion of Infinite

A student came to Dr. Cantor asking about infinity. Cantor replied, "A set is infinite if there is some portion of it -- a subset -- which is not the whole thing but nevertheless can be put in a one-to-one correspondence with the whole thing."

The student was puzzled. "Your definition seems odd to me. I always thought of an infinite set as one that went on and on without end. For example the counting numbers -- 1,2,3, and so on -- are an infinite set. I thought it was because you could keep counting on and on forever. By your definition, I don't even see why this set is considered infinite!"

Cantor took pity on the student. "Try taking each number and adding one to it."

"Ahhh...." The student thought for a moment. "Okay. I think I see how your definition works here. You're telling me to take each number on the one hand, and that number plus one on the other hand. So we could make two columns..." He started scribbling on a scrap of paper.

number	corresponds to
1	2
2	3
3	4
...	...

"It looks like we get a correspondence between the set of number 1,2,3,and so on forever, with subset consisting of 2,3,4, and so on. Since the number 1 isn't in the second set, this isn't the whole thing. But still we have a one-to-one correspondence between the two sets."

"Good." Cantor nodded approval.

"Your definition still seems too complicated," the student complained. Why can't we just say a set is infinite if it goes on forever?"

"Consider the set of all lengths shorter than one inch. Is this not also an infinite set?" Cantor sent the student away to puzzle over a response.