Questions: Naive Set Theory

Question 1

Which collection, formable under naive set theory's unrestricted comprehension, leads to an outright contradiction?

• A. The set of all natural numbers
• B. The set of all sets that do not contain themselves
• C. The set of all prime numbers
• D. The set of all subsets of the real numbers

Answer: (B)

The set R = {x : x ∉ x} — all sets that do not contain themselves — is Russell's paradox set. If R ∈ R, then by definition R ∉ R; if R ∉ R, then by definition R ∈ R. Either way yields a contradiction. The other options are perfectly well-behaved sets. This is the specific construction that showed naive set theory is logically inconsistent.

Question 2

Naive set theory is just informal set theory — it lacks formal axioms but is otherwise consistent.

• T. True
• F. False

Answer: False

Naive set theory has a specific and precise axiom: unrestricted comprehension, which asserts that any predicate P(x) defines a set {x : P(x)}. This is not a vague informal practice but a definite (and provably inconsistent) formal principle. Its inconsistency is what motivated the development of axiomatic systems like ZFC, which restrict comprehension to avoid paradoxes.

Question 3

What is the unrestricted comprehension principle, and why does it generate contradictions?

Model answer: Unrestricted comprehension states that for any predicate P(x), the collection {x : P(x)} is a set. It generates contradictions because some predicates are self-referential in pathological ways. The predicate 'x does not contain itself' (x ∉ x) defines a set R that both must and cannot contain itself — a logical impossibility. No self-restriction on which predicates are allowed means no protection against such paradoxes.

The power of unrestricted comprehension is also its fatal flaw: it lets you define sets by any property whatsoever, including properties that refer back to set membership itself. Axiomatic set theories like ZFC replace this with restricted comprehension (separation), which only allows forming subsets of already-existing sets, blocking self-referential constructions.