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.

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