Are the only permissible operations/actions in an axiomatic system those that the axioms permit explicitly or through logical inference from the axioms?
Example:
From Jech and Hrbacek's Introduction to Set Theory, Page 8:
Intuitively, sets are collections of objects sharing some common property, so we expect to have axioms expressing this fact. But, as demonstrated by the paradoxes in Section 1, not every property describes a set; properties "$X\not\in X$" or "$X=X$" are typical examples.
In both cases, the problem seems to be that in order to collect all objects having such a property into a set, we already have to be able to perceive all sets. The difficulty is avoided if we postulate the existence of a set of all objects with a given property only if there already exists some set to which they all belong.
The Axiom Schema of Comprehension Let P(x) be a property of $x$. For any set A, there is a set B such that $x\in B$ if and only if $x\in A$ and P(x).
As far as I can tell, the above schema allows the existence of $B$ under certain conditions. However, it does not explicitly forbid the existence of the proposed set from Russell's Paradox.