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