But, how does a Mathematician reconcile itself with the idea of random? There's no axiom for "choice", no function for "random value", no explanation of what "chance" is.

Meanwhile I heard that someone spent 500 pages of logic to prove that 1+1 = 2 (or something like that), so how is it possible that mathematicians and logicians spend all that trouble to prove some really basic stuff, while at the same time just accept theories around probabilities and random numbers without (as least from my untrained point of view) an axiomatic foundation for choice?