ϕ Equivalence between set theory and logic — and Russell’s paradox
When you formulate any logic predicate you automatically implicitly define a set of entities constrained by that predicate — i.e. a set of items for which the predicate is true. The set appears because predicate ‘works’ on a variable of variables, and variable is something that by definition can have multiple values, thus belongs to some set. So, when you define a predicate P(x) it is the same as to define a set {x| P(x)} and vice versa, union of set is disjunction of correspondent predicates and so on — equivalence between sets and predicates. Russell’s paradox Its original formulation is that a set of sets that don’t contain themselves is contradictory (indeed it can’t neither contain itself nor not contain). But that can be translated into a paradox of logic: Set of sets that don’t contain themselves is a predicate P(Q) = ¬Q(Q) — then P(P) = ¬P(P) — contradiction . _________ ..other articles..