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Zermelo–Fraenkel set theory - Quiz
Contributed by: Bull
  • 1. Which set theory axioms form ZF?
A) Axioms 1–8
B) Axioms 2–9
C) Axioms 1–9
D) Axioms 1–7
  • 2. What is the equivalent of the axiom of choice used by Kunen (1980) to turn ZF into ZFC?
A) The axiom of infinity
B) The well-ordering theorem
C) The axiom of extensionality
D) The axiom of regularity
  • 3. Why is it unnecessary to have a separate axiom asserting that at least one set exists?
A) It is a theorem of first-order logic that something exists
B) It is assumed that the domain of discourse is empty
C) The axiom of specification implies it
D) The axiom of extensionality implies it
  • 4. Under what condition does the axiom of infinity ensure that a set exists?
A) It is only guaranteed if the domain is finite
B) In free logic where first-order logic does not prove existence
C) It is never guaranteed by the axiom of infinity
D) It only exists in the axiom of pairing
  • 5. What does the axiom of extensionality state?
A) Two sets are equal if they have the same rank
B) Two sets are equal if they have the same elements
C) No set is an element of itself
D) Every set has a unique subset
  • 6. What is another name for the axiom of regularity?
A) The axiom of substitution
B) The axiom of specification
C) The axiom of foundation
D) The axiom of union
  • 7. What does the axiom of regularity imply about a non-empty set?
A) It contains an empty set
B) It is an element of itself
C) It contains a member disjoint from itself
D) It must be infinite
  • 8. With the axioms of pairing and union, what does the axiom of regularity imply?
A) All sets are finite
B) There exists an empty set
C) Every set is unique
D) No set is an element of itself
  • 9. What are other names for the axiom schema of specification?
A) Axiom of separation or restricted comprehension
B) Axiom of infinity or union
C) Axiom of identity or equality
D) Axiom of choice or foundation
  • 10. Why is the axiom schema of specification restricted to constructing subsets?
A) To force the existence of the empty set
B) To avoid Russell's paradox
C) Because sets cannot contain other sets
D) To increase the speed of proofs
  • 11. What is the result of applying unrestricted comprehension in naive set theory?
A) The uniqueness of the empty set
B) The well-ordering theorem
C) Russell's paradox
D) The existence of an ordinal rank
  • 12. How can the existence of the empty set be proven using the axiom schema of specification?
A) By the axiom of infinity
B) By the axiom of regularity
C) By the axiom of extensionality
D) By using a property that no set has
  • 13. What does the axiom of extensionality imply about the empty set?
A) It depends on the set w
B) It does not exist
C) It is unique
D) It contains itself
  • 14. According to the text, what does the axiom of infinity, replacement, and union imply together?
A) The axiom of choice is unnecessary
B) Every set has an ordinal rank
C) The empty set does not exist
D) Russell's paradox is resolved
  • 15. Who are the mathematicians that Zermelo–Fraenkel set theory is named after?
A) Georg Cantor and Gottlob Frege
B) Ernst Zermelo and Abraham Fraenkel
C) Kurt Gödel and John von Neumann
D) Bertrand Russell and Paul Bernays
  • 16. What is another name for the collection of all sets used to interpret the axioms?
A) The von Neumann universe
B) The Russellian hierarchy
C) The Zermelo collection
D) The Fraenkel universe
  • 17. What is the name of the cardinal number that Zermelo's theory could not prove existed?
A) aleph-null
B) aleph-omega
C) aleph-one
D) aleph-zero
  • 18. How is the equality sign usually treated in ZFC?
A) As a primitive logical symbol
B) As a non-logical predicate
C) As a set operator
D) As a quantifier
  • 19. The symbol ∈ represents what kind of relation?
A) An equality relation
B) A binary relation
C) A ternary relation
D) A unary relation
  • 20. How many variables are in the alphabet of the formal language?
A) An uncountable number
B) A countably infinite number
C) A finite number
D) Zero
  • 21. Which of the following is an atomic formula construction?
A) x ∧ y
B) x = y
C) ¬x
D) ∀xϕ
  • 22. How many well-formed formulae exist in the formal language L?
A) Uncountably infinitely many
B) Only one
C) Countably infinitely many
D) Finitely many
  • 23. Which symbol represents the interpretation of the membership relation in the structure S?
A) psi
B) I_in
C) phi
D) mu
  • 24. What is the function mu in the structure S?
A) Membership relation
B) Domain interpretation function
C) Variable assignment function
D) Truth value function
  • 25. What does the expression S |= phi represent?
A) S is a subset of phi
B) S satisfies phi
C) S is equivalent to phi
D) S contains phi
  • 26. What are phi and psi metavariables for?
A) Any variable
B) Any wff
C) Any domain element
D) Any constant
  • 27. The variable assignment function mu maps variables to elements of which set?
A) D
B) The set of wffs
C) A
D) The set of relations
  • 28. In the context of S |= phi, what does phi mean?
A) S assigns phi to the domain
B) phi is a member of D
C) phi is the interpretation of S
D) S makes phi true
  • 29. In the axiom schema of replacement, if φ represents a definable function f, what represents its domain?
A) A
B) φ
C) w1
D) B
  • 30. Which variable is restricted from being free in the formula φ for the axiom schema of replacement?
A) y
B) B
C) A
D) x
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