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Extender (set theory) - Quiz
Contributed by: Casey
  • 1. What is the codomain of the elementary embedding j described in the text?
A) A transitive inner model M
B) The class of all sets
C) The universe V
D) A set of ordinals
  • 2. What is the critical point of the elementary embedding j denoted by?
A) κ
B) λ
C) E
D) V
  • 3. What is the range for the cardinal λ in relation to κ?
A) κ < λ < j(κ)
B) λ > j(κ)
C) λ < κ
D) κ ≤ λ ≤ j(κ)
  • 4. The definition of the extender E uses an elementary embedding from where to where?
A) V to M
B) V to V
C) κ to λ
D) M to V
  • 5. According to the text, the object E is referred to as what?
A) A (κ,λ)-extender
B) An elementary mapping
C) A transitive inner model
D) A critical point
  • 6. What does an extender represent in set theory?
A) A specific type of nonprincipal ultrafilter
B) A system of ultrafilters witnessing large cardinal properties
C) A type of set with no power set
D) A model of ZFC with no ordinals
  • 7. What is the most basic case of an extender?
A) A nonprincipal ultrafilter
B) A wellfounded ultrapower
C) A (kappa, lambda)-extender
D) A directed system of models
  • 8. In a (kappa, lambda)-extender definition, what is the critical point of the elementary embedding?
A) n
B) The cardinality of the model
C) lambda
D) kappa
  • 9. For a (kappa, lambda)-extender, what is the minimum value the embedding maps kappa to?
A) n
B) kappa
C) lambda
D) The cardinality of the power set
  • 10. In the formal definition, what kind of ultrafilters are the E_a?
A) kappa-incomplete nonprincipal ultrafilters
B) kappa-complete principal ultrafilters
C) kappa-complete nonprincipal ultrafilters
D) kappa-complete power set filters
  • 11. Which property regarding kappa-plus-completeness must hold for an extender?
A) At least one E_a is not kappa-plus-complete
B) All E_a must be non-kappa-complete
C) All E_a must be kappa-plus-complete
D) No E_a can be kappa-complete
  • 12. What is the result of the limit ultrapower Ult(V, E)?
A) It is a nonprincipal ultrafilter
B) It fails the coherence property
C) It is wellfounded
D) It maps kappa to an ordinal less than lambda
  • 13. What does coherence imply about the ultrapowers Ult(V, E_a)?
A) They form a directed system
B) They are all principal
C) They do not form a limit
D) They are all identical
  • 14. What is the domain of the indices for the collection of ultrafilters E_a?
A) Finite subsets of lambda
B) All subsets of lambda
C) Ordinal numbers less than kappa
D) All n-tuples of ordinals
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