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