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Morse–Kelley set theory - Quiz
  • 1. What is the primary difference between Morse–Kelley set theory (MK) and von Neumann–Bernays–Gödel set theory (NBG)?
A) MK is a two-sorted theory while NBG is one-sorted.
B) MK does not include the axiom of foundation.
C) NBG includes the axiom of infinity while MK does not.
D) MK allows bound variables in class comprehension to range over proper classes.
  • 2. Who was one of the first people to suggest that bound variables in comprehension should range over proper classes?
A) Quine
B) Anthony Morse
C) Thoralf Skolem
D) John L. Kelley
  • 3. Can Morse–Kelley set theory be finitely axiomatized?
A) Yes, it is equivalent to NBG in that regard
B) No
C) Yes
D) Only if ZFC is inconsistent
  • 4. What is the relationship between MK and ZFC?
A) MK is a proper extension of ZFC.
B) MK is equivalent to ZFC.
C) MK and ZFC are mutually exclusive.
D) MK is a conservative extension of ZFC.
  • 5. In MK, what is a class that is not a set called?
A) A void class
B) A universe class
C) A proper class
D) A member class
  • 6. What does the monadic predicate Mx mean in MK?
A) The class x is the universal class.
B) The class x is empty.
C) The class x is a set.
D) The class x is a proper class.
  • 7. Which of these systems is a conservative extension of ZFC?
A) ML
B) NBG
C) The system of Quine and Morse
D) MK
  • 8. What is the definition of the universal class V in MK?
A) The class of all classes.
B) The class containing no members.
C) The class of all sets.
D) The set of all proper classes.
  • 9. What does the Axiom of Extensionality state in MK?
A) There exists a class containing all sets.
B) No set is a proper class.
C) Classes having the same members are the same class.
D) Every set is a member of a class.
  • 10. Is MK a two-sorted theory?
A) Yes, it distinguishes sets from classes syntactically.
B) It depends on the version being used.
C) Yes, it distinguishes proper classes from sets.
D) No, it is a one-sorted theory.
  • 11. Which axiom states that each nonempty class A is disjoint from at least one of its members?
A) Foundation
B) Extensionality
C) Class Comprehension
D) Limitation of Size
  • 12. According to the Limitation of Size axiom, when is a class C considered a proper class?
A) If it is a member of another class.
B) If it is an element of V.
C) If and only if V can be mapped one-to-one into C.
D) If it contains the empty set.
  • 13. In which book did Kelley first set out the MK system?
A) Foundations of Mathematics
B) A Theory of Sets
C) General Topology
D) Introduction to Topology
  • 14. How is the empty set ∅ defined in MK?
A) ∃x(x∈∅)
B) ∀x(x∉∅)
C) ∃x(Mx∧x∈∅)
D) ∀x(Mx→x∈∅)
  • 15. Which axiom in MK ensures the existence of a set with exactly two given set elements?
A) Union
B) Foundation
C) Power Set
D) Pairing
  • 16. How is Class Comprehension characterized in NBG?
A) Second-order
B) Restricted to proper classes
C) Predicative
D) Impredicative
  • 17. What does the axiom of Limitation of Size imply that Replacement does not?
A) Global choice
B) Empty set existence
C) Set existence
D) Well-ordering of ordinals
  • 18. What does Rubin's ontology include that others might not?
A) Strongly inaccessible cardinals
B) A countable first-order model
C) Second-order logic
D) Urelements
  • 19. Can the NBG axiom schema of Class Comprehension be replaced with finitely many instances?
A) Only in second-order logic
B) Yes
C) Only in ZFC
D) No
  • 20. What is the requirement for MK to be consistent relative to ZFC?
A) The absence of urelements
B) Augmentation by strongly inaccessible cardinals
C) The rejection of the empty set axiom
D) The use of Limitation of Size
  • 21. Who first set out the MK set theory?
A) Kelley
B) Morse
C) Wang
D) Quine
  • 22. Whose system is equivalent to Kelley's but uses an idiosyncratic formal language?
A) Mostowski
B) Wang
C) Quine
D) Anthony Morse
  • 23. What was the first set theory to include impredicative class comprehension?
A) ML
B) ZFC
C) MK
D) NBG
  • 24. Which theory did Quine's ML build upon?
A) ZFC
B) NBG
C) MK
D) New Foundations
  • 25. How many theorems and definitions are stated in the Appendix to Kelley's 1955 book?
A) 181
B) 100
C) 150
D) 200
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