Can someone help me understand the concept of Dedekind cut? I'm having trouble understanding it because it appears that in order to understand what Dedekind cuts are I already must have a very developed idea of real numbers. For example, the picture of a cut is the set of rationals smaller than a real number. Also, when I examine the definitions of addition and multiplication, they seem kind of like randomly constructed definitions, I don't think "yes, this definition really defines the addition". It seems I have to prove that the definition corresponds to some concept of addition and multiplication of real numbers that is already very understood. Thanks.