Hi, I wonder if any of you can help me understand something that is considered 'basic', but I just can't get my head around it.

I was talking with a Canadian friend, who is a mathematician, and at some point we got to talk about chemical molecules (I am a chemist).

To cut a long story short, he was puzzled by me saying that the number of conceivable molecules is finite, so he made up a simplified example where all the molecules you can imagine are made of n repeating units M, joined together by a chemical bond. So your generic molecule would be

. This is what we chemists would call a polymer.

Here's the point that I didn't get. He told me that you can have an infinite set of such molecules

without the need for n to go to infinity. (Of course you can't have a molecule made of infinite units, because each unit has finite size, so this molecule would be infinite in size, which is absurd.)

So I kept telling him, how is it possible that n is finite but the sum of all the molecules in the set of

is infinite? And he kept telling me that this is a basic concept of set theory, and he couldn't see why it was so difficult for me to understand.

Then we didn't have time to talk about it any further, and I'm still left with the doubt.

I read something about sets, and if I got it right, a set as infinite if you can establish a correlation with the set of natural numbers. But there you go, back to my original objection: if we take n as the natural number, then if n is finite, the sum of all

molecules from 1 to n is a finite number, not infinite. Only if you allow n to go to infinity you have a complete map of your set to the natural numbers, don't you?

Does anyone know where I'm going wrong with my reasoning?

(I would only add that the number of conceivable chemical molecules is indeed finite - OK, it's a very big number, but it is finite; so this discussion is more about maths than the original problem itself).