I know the history of how set theory came about and how Cantor showed the real numbers between (0,1) were non-denumerable. He did this by showing that they cant be put into a one-one correspondence with N (1, 2, 3...) ...So what does that really tell me? I know it tells me that the infinity of the reals is larger, but how does that tell me that N is countable itself? Did we just assume N is countable by putting a one-to-one correspondence from N to N itself? Why say "A set is countable if it can be put into a one-to-one correspondence with N." Why pick N for the role of determining the denumerability of other sets?