I decided to learn analysis over the summer for fun, but I'm really confused by the field properties. Why is the 2 element set of 0 and 1 a field? Addition wouldn't be satisfied, because 1 + 1 = 2, which isn't in the set.
Also, if there are a set of properties that completely determine the real number system, why does that imply that there is only one? My book says something about a one to one correspondence between the reals and some other real system that preserves the functions of + and *, but I don't understand this at all.