I know I am jumping ahead, as I am still working through part 1 of Spivak's Calculus, and am absolutely not properly equipped to prove his properties (if they are indeed provable). However, as I am trying to work on my proofing skills, my interpretation thus far has been, that these 12 "rules" are givens...unprovable. If you accept these givens, everything else follows from them, provably. If you do not accept these givens, you reject the proofs drawn from them. When I get into the higher level maths one day in the future, will these become provable? The rules are: associative law for addition and multiplication, existence of an additive identitfy and inverse, commutative law for addition and multiplication, distributive law, existence of multiplicate inverse, existence of multiplicative identity, trichotomy law, closure under addition, closure under multiplication, least upper bound property No need to try to prove any of them to me as I likely am not in a position to understand, just curious. Also, is there a name for these properties of numbers? Clearly Spivak did not create these properties, so I am wondering what the popular name for these properties is. Thank you in advance.