Why doesn't the formula {x ∈ A : x ∉ x} generate Russell's paradox in ZFC?
ABecause 'x ∉ x' is not a valid first-order formula in ZFC's language
BBecause the axiom of regularity separately guarantees no set contains itself
CBecause you must specify an existing set A, and ZFC never asserts that a universal set exists to use as A
DBecause the axiom of separation only applies to sets with finitely many elements
The Russellian formula x ∉ x is perfectly valid in first-order logic, and ZFC allows it inside separation. The block is structural: you cannot form {x : x ∉ x} freely — you must write {x ∈ A : x ∉ x} for some specific set A. For this to produce the paradox, A would have to be 'the set of all sets' (a universal set). But ZFC never asserts any universal set exists — and in fact proves none can. Without a universal set to use as A, the Russellian construction cannot get started.
Question 2 Multiple Choice
Which of the following correctly uses the axiom of separation to construct the intersection A ∩ B?
A{x : x ∈ A and x ∈ B} — form the collection of all things satisfying both conditions
B{x ∈ A : x ∈ B} — carve from the existing set A exactly those elements also in B
C{x ∈ A ∪ B : x ∈ A and x ∈ B} — begin from the union and filter down
D{x ∈ A ∩ B : x ∈ A} — start from the intersection to define the intersection
Option A is naive comprehension — no starting set, just a bare property — which is exactly what ZFC prohibits. Option B is correct: choose A as the existing set to carve from, and φ(x) as the formula x ∈ B. Elements of A satisfying x ∈ B are precisely A ∩ B. Option C works formally (A ∪ B exists by the union axiom, and you can filter it) but is unnecessarily indirect. Option D is circular.
Question 3 True / False
The axiom schema of separation is technically an infinite collection of axioms — one instance for each first-order formula φ — because first-order logic cannot quantify over formulas directly.
TTrue
FFalse
Answer: True
To capture 'for any property φ, the separation axiom holds' in first-order logic, you cannot write a single quantified axiom over all φ (that would require second-order logic). Instead, ZFC includes the axiom schema as a template: for each specific formula φ you can write down, there is one axiom of the form '∀A ∃B ∀x (x ∈ B ↔ x ∈ A ∧ φ(x)).' This is an infinite but uniform family.
Question 4 True / False
The axiom of separation allows you to form the set {x : x = x} — the set of most self-identical things — because x = x is a valid first-order formula.
TTrue
FFalse
Answer: False
Separation requires you to start from an existing set A: you can form {x ∈ A : x = x}, which is just A itself (every element of A is self-identical). But {x : x = x} without a bounding set A would be the universal set — containing every object that exists. ZFC proves no such set exists, and separation's design is precisely to prevent this: any 'form from scratch using a property' construction is forbidden.
Question 5 Short Answer
Explain why the restriction 'x ∈ A' in {x ∈ A : φ(x)} is the fundamental fix for Russell's paradox, not merely a technical refinement.
Think about your answer, then reveal below.
Model answer: Without the restriction, you can freely form any collection satisfying any property — including {x : x ∉ x} — leading immediately to R ∈ R ↔ R ∉ R. The restriction forces every new set to be a subset of an already-existing set A. To form the Russellian R, you would need A to be a universal set (containing all sets), but ZFC has no axiom creating one. In fact, if a universal set V existed, separation would allow {x ∈ V : x ∉ x} — generating the paradox — which proves by contradiction that V cannot exist. The restriction is not cosmetic: it is the entire mechanism that makes bounded comprehension safe while proving the universal set's non-existence.
This is why the word 'restricted' in 'restricted comprehension' is doing heavy lifting. Unrestricted comprehension (Frege's Basic Law V) is what Russell's paradox destroyed. ZFC replaces it with a restricted version where every new set must be carved from a pre-existing one — ensuring the set-forming process never creates a set large enough to contain itself as an element.