This is an excerpt from Elements of Set theory by Enderton.(p. 11)
Our axiom system begins with two primitive notions, the concepts of "set" and "member." In terms of these concepts, we will define others, but the primitive notions remain undefined. Instead, we adopt a list of axioms concerning primitive notions. (The axioms can be thought of as divulging partial information regarding the meaning of the primitive notions.)
Having adopted a list of axioms, we will then proceed to derive sentences that are logical consequences (or theorems) of the axioms. Here a sentence $\sigma$ is said to be a logical consequence of the axioms if any assignment of meaning to the undefined notions of set and member making the axioms true also makes $\sigma$ true.
I do not understand what the highlighted sentences mean.
The axioms can be thought of as divulging partial information regarding the meaning of the primitive notions: Aren't the axioms giving all the meaning to primitive terms? I mean, 'set' and 'member' are not defined, but all that they mean, is conveyed using axioms, isn't it?
...any assignment of meaning to the undefined notions of set and member making the axioms true also makes $\sigma$ true: I have no idea what this means.
Is it that we're just not interested in the exact meaning of 'set' and 'member', but there are some properties that their meanings have, and it is these properties that we specify using axioms? (hence the use of "partial information" in Enderton's book).