It is observed by Grishin that inconsistency of naive set theory can be avoided by restricting the logical law of contraction, as it is contraction that enables us to derive logical inconsistency from set-theoretic paradoxes such as Russell's paradox.
In this paper, we examine Grishin's contraction-free naive set theory to better understand Russell's paradox and the naive comprehension principle from a purely formal standpoint. We study both static-propositional and dynamic-procedural aspects of naive comprehension and argue that it could lead to an ideal formalization of (part of) mathematics, where both propositional knowledge (theorems) and procedural knowledge (algorithms) reside in harmony.
