Introduction to Permutation Models Models of ZFA Introduction to Permutation Models The Mostowski Model with an Application to Ring Theory An Independence Result Root-Functions in Rings 2C n � C n L. Halbeisen (ETH Z¨ urich) 11th Young Set Theory Workshop (Lausanne 2018)
Introduction to The Language of Set Theory with Atoms Permutation Models Models of ZFA The Mostowski Model An Independence Result Root-Functions in Rings 2C n � C n
Introduction to The Language of Set Theory with Atoms Permutation Models Models of ZFA The Mostowski ◮ Atoms are objects which do not have any elements but Model An Independence are distinct from the empty set. Result Root-Functions in Rings 2C n � C n
Introduction to The Language of Set Theory with Atoms Permutation Models Models of ZFA The Mostowski ◮ Atoms are objects which do not have any elements but Model An Independence are distinct from the empty set. Result Root-Functions in Rings ◮ The collection of atoms is denoted by A , and we add 2C n � C n the constant symbol A to the language of Set Theory.
Introduction to The Language of Set Theory with Atoms Permutation Models Models of ZFA The Mostowski ◮ Atoms are objects which do not have any elements but Model An Independence are distinct from the empty set. Result Root-Functions in Rings ◮ The collection of atoms is denoted by A , and we add 2C n � C n the constant symbol A to the language of Set Theory. ◮ The language of Set Theory with atoms, denoted ZFA, consists of the relation symbol “ ∈ ” and the constant symbol “ A ”.
Introduction to The Axioms of Set Theory with Atoms Permutation Models Models of ZFA The Mostowski Model An Independence Result Root-Functions in Rings 2C n � C n
Introduction to The Axioms of Set Theory with Atoms Permutation Models ◮ The axioms of ZFA are essentially the axioms of ZF. Models of ZFA The Mostowski Model An Independence Result Root-Functions in Rings 2C n � C n
Introduction to The Axioms of Set Theory with Atoms Permutation Models ◮ The axioms of ZFA are essentially the axioms of ZF. Models of ZFA The Mostowski Model Exceptions: An Independence Result ◮ Axiom of Empty Set for ZFA Root-Functions in Rings 2C n � C n � � ∃ x x / ∈ A ∧ ∀ z ( z / ∈ x )
Introduction to The Axioms of Set Theory with Atoms Permutation Models ◮ The axioms of ZFA are essentially the axioms of ZF. Models of ZFA The Mostowski Model Exceptions: An Independence Result ◮ Axiom of Empty Set for ZFA Root-Functions in Rings 2C n � C n � � ∃ x x / ∈ A ∧ ∀ z ( z / ∈ x ) ◮ Axiom of Extensionality for ZFA ∀ x ∀ y � ∈ A ∧ y / ∈ A ) → � ∀ z ( z ∈ x ↔ z ∈ y ) → x = y �� ( x /
Introduction to The Axioms of Set Theory with Atoms Permutation Models ◮ The axioms of ZFA are essentially the axioms of ZF. Models of ZFA The Mostowski Model Exceptions: An Independence Result ◮ Axiom of Empty Set for ZFA Root-Functions in Rings 2C n � C n � � ∃ x x / ∈ A ∧ ∀ z ( z / ∈ x ) ◮ Axiom of Extensionality for ZFA ∀ x ∀ y � ∈ A ∧ y / ∈ A ) → � ∀ z ( z ∈ x ↔ z ∈ y ) → x = y �� ( x / ◮ Axiom of Atoms � � �� ∀ x x ∈ A ↔ x � = ∅ ∧ ¬∃ z ( z ∈ x )
Introduction to A Model of ZFA + AC Permutation Models Models of ZFA := A , M 0 The Mostowski Model � := if α is a limit ordinal , M α M β An Independence Result β ∈ α Root-Functions in := P ( M α ) , M α +1 Rings 2C n � C n � := M α . M α ∈ Ω
Introduction to A Model of ZFA + AC Permutation Models Models of ZFA := A , M 0 The Mostowski Model � := if α is a limit ordinal , M α M β An Independence Result β ∈ α Root-Functions in := P ( M α ) , M α +1 Rings 2C n � C n � := M α . M α ∈ Ω ◮ The class M is a transitive model of ZFA.
Introduction to A Model of ZFA + AC Permutation Models Models of ZFA := A , M 0 The Mostowski Model � := if α is a limit ordinal , M α M β An Independence Result β ∈ α Root-Functions in := P ( M α ) , M α +1 Rings 2C n � C n � := M α . M α ∈ Ω ◮ The class M is a transitive model of ZFA. ◮ ˆ α ∈ Ω P α ( ∅ ) is a model of ZF and is called the V := � kernel of M .
Introduction to A Model of ZFA + AC Permutation Models Models of ZFA := A , M 0 The Mostowski Model � := if α is a limit ordinal , M α M β An Independence Result β ∈ α Root-Functions in := P ( M α ) , M α +1 Rings 2C n � C n � := M α . M α ∈ Ω ◮ The class M is a transitive model of ZFA. ◮ ˆ α ∈ Ω P α ( ∅ ) is a model of ZF and is called the V := � kernel of M . ◮ If the construction of M was carried out in a model of ZFC, then ˆ V | = ZFC and M | = ZFA + AC.
Introduction to Permutation Models: normal filters Permutation Models Models of ZFA The Mostowski Model An Independence Result Root-Functions in Rings 2C n � C n
Introduction to Permutation Models: normal filters Permutation Models Models of ZFA ◮ Let A be a set of atoms and let M = � α ∈ Ω M α be the The Mostowski Model corresponding model of ZFA + AC. An Independence Result Root-Functions in Rings 2C n � C n
Introduction to Permutation Models: normal filters Permutation Models Models of ZFA ◮ Let A be a set of atoms and let M = � α ∈ Ω M α be the The Mostowski Model corresponding model of ZFA + AC. An Independence Result ◮ In M , let G be a group of permutations of A . Root-Functions in Rings 2C n � C n
Introduction to Permutation Models: normal filters Permutation Models Models of ZFA ◮ Let A be a set of atoms and let M = � α ∈ Ω M α be the The Mostowski Model corresponding model of ZFA + AC. An Independence Result ◮ In M , let G be a group of permutations of A . Root-Functions in Rings 2C n � C n ◮ We say that a set F of subgroups of G is a normal filter on G , if G ∈ F and for all H , K ≤ G we have:
Introduction to Permutation Models: normal filters Permutation Models Models of ZFA ◮ Let A be a set of atoms and let M = � α ∈ Ω M α be the The Mostowski Model corresponding model of ZFA + AC. An Independence Result ◮ In M , let G be a group of permutations of A . Root-Functions in Rings 2C n � C n ◮ We say that a set F of subgroups of G is a normal filter on G , if G ∈ F and for all H , K ≤ G we have: (A) if H ∈ F and H ≤ K , then K ∈ F
Introduction to Permutation Models: normal filters Permutation Models Models of ZFA ◮ Let A be a set of atoms and let M = � α ∈ Ω M α be the The Mostowski Model corresponding model of ZFA + AC. An Independence Result ◮ In M , let G be a group of permutations of A . Root-Functions in Rings 2C n � C n ◮ We say that a set F of subgroups of G is a normal filter on G , if G ∈ F and for all H , K ≤ G we have: (A) if H ∈ F and H ≤ K , then K ∈ F (B) if H ∈ F and K ∈ F , then H ∩ K ∈ F
Introduction to Permutation Models: normal filters Permutation Models Models of ZFA ◮ Let A be a set of atoms and let M = � α ∈ Ω M α be the The Mostowski Model corresponding model of ZFA + AC. An Independence Result ◮ In M , let G be a group of permutations of A . Root-Functions in Rings 2C n � C n ◮ We say that a set F of subgroups of G is a normal filter on G , if G ∈ F and for all H , K ≤ G we have: (A) if H ∈ F and H ≤ K , then K ∈ F (B) if H ∈ F and K ∈ F , then H ∩ K ∈ F (C) if π ∈ G and H ∈ F , then π H π − 1 ∈ F
Introduction to Permutation Models: normal filters Permutation Models Models of ZFA ◮ Let A be a set of atoms and let M = � α ∈ Ω M α be the The Mostowski Model corresponding model of ZFA + AC. An Independence Result ◮ In M , let G be a group of permutations of A . Root-Functions in Rings 2C n � C n ◮ We say that a set F of subgroups of G is a normal filter on G , if G ∈ F and for all H , K ≤ G we have: (A) if H ∈ F and H ≤ K , then K ∈ F (B) if H ∈ F and K ∈ F , then H ∩ K ∈ F (C) if π ∈ G and H ∈ F , then π H π − 1 ∈ F (D) for each a ∈ A , { π ∈ G : π a = a } ∈ F
Introduction to Permutation Models: a simple normal filter Permutation Models Models of ZFA For each finite set E ⊆ A , let The Mostowski Model An Independence fix G ( E ) = { π ∈ G : π a = a for all a ∈ E } . Result Root-Functions in Rings 2C n � C n
Introduction to Permutation Models: a simple normal filter Permutation Models Models of ZFA For each finite set E ⊆ A , let The Mostowski Model An Independence fix G ( E ) = { π ∈ G : π a = a for all a ∈ E } . Result Root-Functions in Rings 2C n � C n Then the filter F on G generated by the subgroups fix G ( E ), where E is a finite subset of A , is a normal filter.
Introduction to Permutation Models: symmetric sets Permutation Models For every π ∈ G and for every set x ∈ M we can define π x by stipulating Models of ZFA The Mostowski Model ∅ if x = ∅ , An Independence Result π x = π x if x ∈ A , Root-Functions in { π y : y ∈ x } otherwise. Rings 2C n � C n
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