Unfortunately there is no way to get something from nothing, and so some basic assumptions are always needed as a starting point. These are called "Axioms".
In order to prove results involving Real numbers, we could express the reals and everything we want to say in terms of simpler objects - and maybe start with a set of axioms for just counting numbers, or even for the theory of sets. But another approach is to take as fundamental a set of basic properties of the Real numbers themselves. These are then referred to as Axioms for the Real Number System.
These ideas are elaborated more fully in an Analysis course.
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