Hi John Green;
I am afraid your conception of 16 year olds is not perfect.
I am 14 and I realise that f(n)=2n does a very good job of bijecting [0,1] to [0,2] which would pretty much imply the same cardinality.
I do not mean to say that there is no truth in your reasoning about kids. You are sort of correct about abstract reasoning but 16 is a little too old. 12 would be okay.
I'd have used the concepts of Aleph-Null and Aleph-One to explain my significant other on different sorts of infinities. Hilbert's hotel is even better.
By the way, do you think that there exists a set whose cadinality is strictly between that of natural numbers and reals.
That is what they say. The last person to say that was?
I agree with G.H. Hardy that mathematics is a young man's game (except for the "man" part), but "young" does not mean "under thirty", and not even "under forty". "Young" means, "young at heart", and willing to learn new things and new methodologies and master new technology. If you will follow my advice, I am sure that you would achieve much more than your already very impressive (albeit mostly boring) feats, and your future achievements will not only be technically challending, but also exciting, not just to you and to your thirty cronies, but to all of us common folks.
Hahahha, thanks! Professors are not the only ones having beards, though.
Mathematics is a young man's game.
Graph theory is a part of combinatorics.
Rewind your life back to Class XI. You would not have believed that solutions to x+y+z=10 is combinatorics. Would you? You'd have thought it is algebra.
In the same way, you probably do not see yet why graph theory is a part of combinatorics, but you'll later.