Introduction to Permutation Models The Mostowski Model with an - - PowerPoint PPT Presentation

Introduction to Permutation Models Models of ZFA The Mostowski Model An Independence Result Root-Functions in Rings 2Cn Cn

Introduction to Permutation Models

with an Application to Ring Theory

• L. Halbeisen (ETH Z¨

urich) 11th Young Set Theory Workshop (Lausanne 2018)

Introduction to Permutation Models Models of ZFA The Mostowski Model An Independence Result Root-Functions in Rings 2Cn Cn

The Language of Set Theory with Atoms

Introduction to Permutation Models Models of ZFA The Mostowski Model An Independence Result Root-Functions in Rings 2Cn Cn

The Language of Set Theory with Atoms

◮ Atoms are objects which do not have any elements but

are distinct from the empty set.

Introduction to Permutation Models Models of ZFA The Mostowski Model An Independence Result Root-Functions in Rings 2Cn Cn

The Language of Set Theory with Atoms

◮ Atoms are objects which do not have any elements but

are distinct from the empty set.

◮ The collection of atoms is denoted by A, and we add

the constant symbol A to the language of Set Theory.

Introduction to Permutation Models Models of ZFA The Mostowski Model An Independence Result Root-Functions in Rings 2Cn Cn

The Language of Set Theory with Atoms

◮ Atoms are objects which do not have any elements but

are distinct from the empty set.

◮ The collection of atoms is denoted by A, and we add

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 Permutation Models Models of ZFA The Mostowski Model An Independence Result Root-Functions in Rings 2Cn Cn

The Axioms of Set Theory with Atoms

Introduction to Permutation Models Models of ZFA The Mostowski Model An Independence Result Root-Functions in Rings 2Cn Cn

The Axioms of Set Theory with Atoms

◮ The axioms of ZFA are essentially the axioms of ZF.

Introduction to Permutation Models Models of ZFA The Mostowski Model An Independence Result Root-Functions in Rings 2Cn Cn

The Axioms of Set Theory with Atoms

◮ The axioms of ZFA are essentially the axioms of ZF.

Exceptions:

◮ Axiom of Empty Set for ZFA

∃x

• x /

∈ A ∧ ∀z(z / ∈ x)

Introduction to Permutation Models Models of ZFA The Mostowski Model An Independence Result Root-Functions in Rings 2Cn Cn

The Axioms of Set Theory with Atoms

◮ The axioms of ZFA are essentially the axioms of ZF.

Exceptions:

◮ Axiom of Empty Set for ZFA

∃x

• x /

∈ A ∧ ∀z(z / ∈ x)

• ◮ Axiom of Extensionality for ZFA

∀x∀y

• (x /

∈ A ∧ y / ∈ A) →

• ∀z(z ∈ x ↔ z ∈ y) → x = y

Introduction to Permutation Models Models of ZFA The Mostowski Model An Independence Result Root-Functions in Rings 2Cn Cn

The Axioms of Set Theory with Atoms

◮ The axioms of ZFA are essentially the axioms of ZF.

Exceptions:

◮ Axiom of Empty Set for ZFA

∃x

• x /

∈ A ∧ ∀z(z / ∈ x)

• ◮ Axiom of Extensionality for ZFA

∀x∀y

• (x /

∈ A ∧ y / ∈ A) →

• ∀z(z ∈ x ↔ z ∈ y) → x = y
• ◮ Axiom of Atoms

∀x

• x ∈ A ↔
• x = ∅ ∧ ¬∃z(z ∈ x)

Introduction to Permutation Models Models of ZFA The Mostowski Model An Independence Result Root-Functions in Rings 2Cn Cn

A Model of ZFA + AC

M0 := A , Mα :=

• β∈α

Mβ if α is a limit ordinal , Mα+1 := P(Mα) , M :=

• α∈Ω

Mα .

Introduction to Permutation Models Models of ZFA The Mostowski Model An Independence Result Root-Functions in Rings 2Cn Cn

A Model of ZFA + AC

M0 := A , Mα :=

• β∈α

Mβ if α is a limit ordinal , Mα+1 := P(Mα) , M :=

• α∈Ω

Mα .

◮ The class M is a transitive model of ZFA.

Introduction to Permutation Models Models of ZFA The Mostowski Model An Independence Result Root-Functions in Rings 2Cn Cn

A Model of ZFA + AC

M0 := A , Mα :=

• β∈α

Mβ if α is a limit ordinal , Mα+1 := P(Mα) , M :=

• α∈Ω

Mα .

◮ The class M is a transitive model of ZFA. ◮ ˆ

V :=

α∈Ω Pα(∅) is a model of ZF and is called the

kernel of M.

Introduction to Permutation Models Models of ZFA The Mostowski Model An Independence Result Root-Functions in Rings 2Cn Cn

A Model of ZFA + AC

M0 := A , Mα :=

• β∈α

Mβ if α is a limit ordinal , Mα+1 := P(Mα) , M :=

• α∈Ω

Mα .

◮ The class M is a transitive model of ZFA. ◮ ˆ

V :=

α∈Ω Pα(∅) is a model of ZF and is called the

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 Models of ZFA The Mostowski Model An Independence Result Root-Functions in Rings 2Cn Cn

Permutation Models: normal filters

Introduction to Permutation Models Models of ZFA The Mostowski Model An Independence Result Root-Functions in Rings 2Cn Cn

Permutation Models: normal filters

◮ Let A be a set of atoms and let M = α∈Ω Mα be the

corresponding model of ZFA + AC.

Introduction to Permutation Models Models of ZFA The Mostowski Model An Independence Result Root-Functions in Rings 2Cn Cn

Permutation Models: normal filters

◮ Let A be a set of atoms and let M = α∈Ω Mα be the

corresponding model of ZFA + AC.

◮ In M, let G be a group of permutations of A.

Introduction to Permutation Models Models of ZFA The Mostowski Model An Independence Result Root-Functions in Rings 2Cn Cn

Permutation Models: normal filters

◮ Let A be a set of atoms and let M = α∈Ω Mα be the

corresponding model of ZFA + AC.

◮ In M, let G be a group of permutations of A. ◮ 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 Models of ZFA The Mostowski Model An Independence Result Root-Functions in Rings 2Cn Cn

Permutation Models: normal filters

◮ Let A be a set of atoms and let M = α∈Ω Mα be the

corresponding model of ZFA + AC.

◮ In M, let G be a group of permutations of A. ◮ 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 Models of ZFA The Mostowski Model An Independence Result Root-Functions in Rings 2Cn Cn

Permutation Models: normal filters

◮ Let A be a set of atoms and let M = α∈Ω Mα be the

corresponding model of ZFA + AC.

◮ In M, let G be a group of permutations of A. ◮ 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 Models of ZFA The Mostowski Model An Independence Result Root-Functions in Rings 2Cn Cn

Permutation Models: normal filters

◮ Let A be a set of atoms and let M = α∈Ω Mα be the

corresponding model of ZFA + AC.

◮ In M, let G be a group of permutations of A. ◮ 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 πHπ−1 ∈ F

Introduction to Permutation Models Models of ZFA The Mostowski Model An Independence Result Root-Functions in Rings 2Cn Cn

Permutation Models: normal filters

◮ Let A be a set of atoms and let M = α∈Ω Mα be the

corresponding model of ZFA + AC.

◮ In M, let G be a group of permutations of A. ◮ 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 πHπ−1 ∈ F (D) for each a ∈ A, {π ∈ G : πa = a} ∈ F

Introduction to Permutation Models Models of ZFA The Mostowski Model An Independence Result Root-Functions in Rings 2Cn Cn

Permutation Models: a simple normal filter

For each finite set E ⊆ A, let fixG(E) = {π ∈ G : πa = a for all a ∈ E}.

Introduction to Permutation Models Models of ZFA The Mostowski Model An Independence Result Root-Functions in Rings 2Cn Cn

Permutation Models: a simple normal filter

For each finite set E ⊆ A, let fixG(E) = {π ∈ G : πa = a for all a ∈ E}. Then the filter F on G generated by the subgroups fixG(E), where E is a finite subset of A, is a normal filter.

Introduction to Permutation Models Models of ZFA The Mostowski Model An Independence Result Root-Functions in Rings 2Cn Cn

Permutation Models: symmetric sets

For every π ∈ G and for every set x ∈ M we can define πx by stipulating πx =      ∅ if x = ∅, πx if x ∈ A, {πy : y ∈ x}

• therwise.

Introduction to Permutation Models Models of ZFA The Mostowski Model An Independence Result Root-Functions in Rings 2Cn Cn

Permutation Models: symmetric sets

For every π ∈ G and for every set x ∈ M we can define πx by stipulating πx =      ∅ if x = ∅, πx if x ∈ A, {πy : y ∈ x}

• therwise.

For x ∈ M, the symmetry group of x, denoted symG(x), is defined by symG(x) = {π ∈ G : πx = x}.

Introduction to Permutation Models Models of ZFA The Mostowski Model An Independence Result Root-Functions in Rings 2Cn Cn

Permutation Models: symmetric sets

For every π ∈ G and for every set x ∈ M we can define πx by stipulating πx =      ∅ if x = ∅, πx if x ∈ A, {πy : y ∈ x}

• therwise.

For x ∈ M, the symmetry group of x, denoted symG(x), is defined by symG(x) = {π ∈ G : πx = x}. A set x is symmetric (with respect to F) if symG(x) ∈ F.

Introduction to Permutation Models Models of ZFA The Mostowski Model An Independence Result Root-Functions in Rings 2Cn Cn

Permutation Models: symmetric sets

◮ By (D) we have that every atom a ∈ A is symmetric.

Introduction to Permutation Models Models of ZFA The Mostowski Model An Independence Result Root-Functions in Rings 2Cn Cn

Permutation Models: symmetric sets

◮ By (D) we have that every atom a ∈ A is symmetric. ◮ A set x is called hereditarily symmetric if x as well as

each element of its transitive closure is symmetric.

Introduction to Permutation Models Models of ZFA The Mostowski Model An Independence Result Root-Functions in Rings 2Cn Cn

Permutation Models: symmetric sets

◮ By (D) we have that every atom a ∈ A is symmetric. ◮ A set x is called hereditarily symmetric if x as well as

each element of its transitive closure is symmetric.

◮ If V ⊆ M is the class of all hereditarily symmetric sets,

then V is a transitive model of ZFA, a so-called permutation model.

Introduction to Permutation Models Models of ZFA The Mostowski Model An Independence Result Root-Functions in Rings 2Cn Cn

Permutation Models: symmetric sets

◮ By (D) we have that every atom a ∈ A is symmetric. ◮ A set x is called hereditarily symmetric if x as well as

each element of its transitive closure is symmetric.

◮ If V ⊆ M is the class of all hereditarily symmetric sets,

then V is a transitive model of ZFA, a so-called permutation model.

◮ If F is the normal filter of finite sets, then a set x

belongs to V if and only if there exists a finite set of atoms Ex ⊆ A, called a support of x, such that fixG(Ex) ⊆ symG(x).

Introduction to Permutation Models Models of ZFA The Mostowski Model An Independence Result Root-Functions in Rings 2Cn Cn

The Ordered Mostowski Model

Introduction to Permutation Models Models of ZFA The Mostowski Model An Independence Result Root-Functions in Rings 2Cn Cn

The Ordered Mostowski Model

◮ The set of atoms is

A := Q (the rational numbers)

Introduction to Permutation Models Models of ZFA The Mostowski Model An Independence Result Root-Functions in Rings 2Cn Cn

The Ordered Mostowski Model

◮ The set of atoms is

A := Q (the rational numbers)

◮ On Q we have the natural order-relation “<”.

Introduction to Permutation Models Models of ZFA The Mostowski Model An Independence Result Root-Functions in Rings 2Cn Cn

The Ordered Mostowski Model

◮ The set of atoms is

A := Q (the rational numbers)

◮ On Q we have the natural order-relation “<”. ◮ G =

• π ∈ Aut(A) : ∀a, b ∈ A (a < b ⇐

⇒ πa < πb )

Introduction to Permutation Models Models of ZFA The Mostowski Model An Independence Result Root-Functions in Rings 2Cn Cn

The Ordered Mostowski Model

◮ The set of atoms is

A := Q (the rational numbers)

◮ On Q we have the natural order-relation “<”. ◮ G =

• π ∈ Aut(A) : ∀a, b ∈ A (a < b ⇐

⇒ πa < πb )

• ◮ Which sets x ⊆ A belong to the ordered Mostowski

permutation model?

Introduction to Permutation Models Models of ZFA The Mostowski Model An Independence Result Root-Functions in Rings 2Cn Cn

The Ordered Mostowski Model

◮ The set of atoms is

A := Q (the rational numbers)

◮ On Q we have the natural order-relation “<”. ◮ G =

• π ∈ Aut(A) : ∀a, b ∈ A (a < b ⇐

⇒ πa < πb )

• ◮ Which sets x ⊆ A belong to the ordered Mostowski

permutation model?

Introduction to Permutation Models Models of ZFA The Mostowski Model An Independence Result Root-Functions in Rings 2Cn Cn

The Ordered Mostowski Model

◮ The set of atoms is

A := Q (the rational numbers)

◮ On Q we have the natural order-relation “<”. ◮ G =

• π ∈ Aut(A) : ∀a, b ∈ A (a < b ⇐

⇒ πa < πb )

• ◮ Which sets x ⊆ A belong to the ordered Mostowski

permutation model?

bc bc bc

Introduction to Permutation Models Models of ZFA The Mostowski Model An Independence Result Root-Functions in Rings 2Cn Cn

The Ordered Mostowski Model

◮ The set of atoms is

A := Q (the rational numbers)

◮ On Q we have the natural order-relation “<”. ◮ G =

• π ∈ Aut(A) : ∀a, b ∈ A (a < b ⇐

⇒ πa < πb )

• ◮ Which sets x ⊆ A belong to the ordered Mostowski

permutation model?

bc

/ ∈x

b

∈x

bc

/ ∈x

Introduction to Permutation Models Models of ZFA The Mostowski Model An Independence Result Root-Functions in Rings 2Cn Cn

The Ordered Mostowski Model

◮ The set of atoms is

A := Q (the rational numbers)

◮ On Q we have the natural order-relation “<”. ◮ G =

• π ∈ Aut(A) : ∀a, b ∈ A (a < b ⇐

⇒ πa < πb )

• ◮ Which sets x ⊆ A belong to the ordered Mostowski

permutation model?

bc

/ ∈x

b

∈x

bc

/ ∈x

b

∈x

Introduction to Permutation Models Models of ZFA The Mostowski Model An Independence Result Root-Functions in Rings 2Cn Cn

The Ordered Mostowski Model

◮ The set of atoms is

A := Q (the rational numbers)

◮ On Q we have the natural order-relation “<”. ◮ G =

• π ∈ Aut(A) : ∀a, b ∈ A (a < b ⇐

⇒ πa < πb )

• ◮ Which sets x ⊆ A belong to the ordered Mostowski

permutation model?

bc

/ ∈x

b

∈x

bc

/ ∈x ⊆x

Introduction to Permutation Models Models of ZFA The Mostowski Model An Independence Result Root-Functions in Rings 2Cn Cn

The Ordered Mostowski Model

◮ The set of atoms is

A := Q (the rational numbers)

◮ On Q we have the natural order-relation “<”. ◮ G =

• π ∈ Aut(A) : ∀a, b ∈ A (a < b ⇐

⇒ πa < πb )

• ◮ Which sets x ⊆ A belong to the ordered Mostowski

permutation model?

bc

/ ∈x

b

∈x

bc

/ ∈x ⊆x ⊆x

Introduction to Permutation Models Models of ZFA The Mostowski Model An Independence Result Root-Functions in Rings 2Cn Cn

The Ordered Mostowski Model

◮ The set of atoms is

A := Q (the rational numbers)

◮ On Q we have the natural order-relation “<”. ◮ G =

• π ∈ Aut(A) : ∀a, b ∈ A (a < b ⇐

⇒ πa < πb )

• ◮ Which sets x ⊆ A belong to the ordered Mostowski

permutation model?

bc

/ ∈x

b

∈x

bc

/ ∈x ⊆x ⊆x ◮ For a given finite set E ⊂ A of size m, there are 22m+1

sets x ⊆ A with support E.

Introduction to Permutation Models Models of ZFA The Mostowski Model An Independence Result Root-Functions in Rings 2Cn Cn

An Independence Result

In Mostowski’s model there is a surjection fin(A) ։ P(A) .

Introduction to Permutation Models Models of ZFA The Mostowski Model An Independence Result Root-Functions in Rings 2Cn Cn

An Independence Result

In Mostowski’s model there is a surjection fin(A) ։ P(A) . This gives us just consistency with ZFA, but fortunately, the result can be transferred to models of ZF!

Introduction to Permutation Models Models of ZFA The Mostowski Model An Independence Result Root-Functions in Rings 2Cn Cn

An Independence Result

We have ZF ⊢ | fin(X)| < |P(X)| for every infinite set X .

Introduction to Permutation Models Models of ZFA The Mostowski Model An Independence Result Root-Functions in Rings 2Cn Cn

An Independence Result

We have ZF ⊢ | fin(X)| < |P(X)| for every infinite set X . It is consistent with ZF that there exists an infinite set X, such that

◮ there is an injection fin(X) ֒

→ P(X) [trivial],

Introduction to Permutation Models Models of ZFA The Mostowski Model An Independence Result Root-Functions in Rings 2Cn Cn

An Independence Result

We have ZF ⊢ | fin(X)| < |P(X)| for every infinite set X . It is consistent with ZF that there exists an infinite set X, such that

◮ there is an injection fin(X) ֒

→ P(X) [trivial],

◮ there is a surjection fin(X) ։ P(X) [Mostowski],

Introduction to Permutation Models Models of ZFA The Mostowski Model An Independence Result Root-Functions in Rings 2Cn Cn

An Independence Result

We have ZF ⊢ | fin(X)| < |P(X)| for every infinite set X . It is consistent with ZF that there exists an infinite set X, such that

◮ there is an injection fin(X) ֒

→ P(X) [trivial],

◮ there is a surjection fin(X) ։ P(X) [Mostowski], ◮ there is no bijection between fin(X) and P(X) [ZF].

Introduction to Permutation Models Models of ZFA The Mostowski Model An Independence Result Root-Functions in Rings 2Cn Cn

An Independence Result

We have ZF ⊢ | fin(X)| < |P(X)| for every infinite set X . It is consistent with ZF that there exists an infinite set X, such that

◮ there is an injection fin(X) ֒

→ P(X) [trivial],

◮ there is a surjection fin(X) ։ P(X) [Mostowski], ◮ there is no bijection between fin(X) and P(X) [ZF].

Such sets do not exist in models of ZFC.

Introduction to Permutation Models Models of ZFA The Mostowski Model An Independence Result Root-Functions in Rings 2Cn Cn

An Independence Result

We have ZF ⊢ | fin(X)| < |P(X)| for every infinite set X . It is consistent with ZF that there exists an infinite set X, such that

◮ there is an injection fin(X) ֒

→ P(X) [trivial],

◮ there is a surjection fin(X) ։ P(X) [Mostowski], ◮ there is no bijection between fin(X) and P(X) [ZF].

Such sets do not exist in models of ZFC. ⇒ The existence of such a set X is independent of ZF.

Introduction to Permutation Models Models of ZFA The Mostowski Model An Independence Result Root-Functions in Rings 2Cn Cn

Root-Functions in Rings

Introduction to Permutation Models Models of ZFA The Mostowski Model An Independence Result Root-Functions in Rings 2Cn Cn

Root-Functions in Rings

For a commutative ring R and a positive integer n, let R(n) :=

• y ∈ R : ∃x (xn = y)
• .

Introduction to Permutation Models Models of ZFA The Mostowski Model An Independence Result Root-Functions in Rings 2Cn Cn

Root-Functions in Rings

For a commutative ring R and a positive integer n, let R(n) :=

• y ∈ R : ∃x (xn = y)
• .

A function f : R(n) → R is called an nth-root function if for every y ∈ R(n) we have f (y)n = y.

Introduction to Permutation Models Models of ZFA The Mostowski Model An Independence Result Root-Functions in Rings 2Cn Cn

Root-Functions in Rings

For a commutative ring R and a positive integer n, let R(n) :=

• y ∈ R : ∃x (xn = y)
• .

A function f : R(n) → R is called an nth-root function if for every y ∈ R(n) we have f (y)n = y.

Theorem (H., Hungerb¨ uhler, Lazarovich, Lederle, Lischka, Schumacher)

The following statements are equivalent:

Introduction to Permutation Models Models of ZFA The Mostowski Model An Independence Result Root-Functions in Rings 2Cn Cn

Root-Functions in Rings

For a commutative ring R and a positive integer n, let R(n) :=

• y ∈ R : ∃x (xn = y)
• .

A function f : R(n) → R is called an nth-root function if for every y ∈ R(n) we have f (y)n = y.

Theorem (H., Hungerb¨ uhler, Lazarovich, Lederle, Lischka, Schumacher)

The following statements are equivalent:

◮ Every ring has a square root function.

Introduction to Permutation Models Models of ZFA The Mostowski Model An Independence Result Root-Functions in Rings 2Cn Cn

Root-Functions in Rings

For a commutative ring R and a positive integer n, let R(n) :=

• y ∈ R : ∃x (xn = y)
• .

A function f : R(n) → R is called an nth-root function if for every y ∈ R(n) we have f (y)n = y.

Theorem (H., Hungerb¨ uhler, Lazarovich, Lederle, Lischka, Schumacher)

The following statements are equivalent:

◮ Every ring has a square root function. ◮ Axiom of Choice.

Introduction to Permutation Models Models of ZFA The Mostowski Model An Independence Result Root-Functions in Rings 2Cn Cn

Integral Domains

Introduction to Permutation Models Models of ZFA The Mostowski Model An Independence Result Root-Functions in Rings 2Cn Cn

Integral Domains

Axiom of Choice for Families of n-element Sets Cn: Every family of n-element sets has a choice function.

Introduction to Permutation Models Models of ZFA The Mostowski Model An Independence Result Root-Functions in Rings 2Cn Cn

Integral Domains

Axiom of Choice for Families of n-element Sets Cn: Every family of n-element sets has a choice function.

Proposition (H., H., L., L., L., S.)

For n = 2, 3: Cn ⇐ ⇒ every integral domain has an nth-root function

Introduction to Permutation Models Models of ZFA The Mostowski Model An Independence Result Root-Functions in Rings 2Cn Cn

Integral Domains

Axiom of Choice for Families of n-element Sets Cn: Every family of n-element sets has a choice function.

Proposition (H., H., L., L., L., S.)

For n = 2, 3: Cn ⇐ ⇒ every integral domain has an nth-root function

Question

For every positive integer n: Cn

?

⇐ ⇒ every integral domain has an nth-root function

Introduction to Permutation Models Models of ZFA The Mostowski Model An Independence Result Root-Functions in Rings 2Cn Cn

Integral Domains

Pair Choice for Families of n-element Sets 2Cn: If F = {Yλ : λ ∈ Λ} is a family of n-element, then from each Yλ ∈ F we can choose a non-empty set with at most 2 elements.

Introduction to Permutation Models Models of ZFA The Mostowski Model An Independence Result Root-Functions in Rings 2Cn Cn

Integral Domains

Pair Choice for Families of n-element Sets 2Cn: If F = {Yλ : λ ∈ Λ} is a family of n-element, then from each Yλ ∈ F we can choose a non-empty set with at most 2 elements.

Proposition (H., H., L., L., L., S.)

For odd integers n ≥ 3, 2Cn ⇒ every integral domain has an nth-root function.

Introduction to Permutation Models Models of ZFA The Mostowski Model An Independence Result Root-Functions in Rings 2Cn Cn

2Cn Cn

Introduction to Permutation Models Models of ZFA The Mostowski Model An Independence Result Root-Functions in Rings 2Cn Cn

2Cn Cn

In order to show that the answer to the question is no, it is enough to prove 2Cn Cn (for some odd integer n).

Introduction to Permutation Models Models of ZFA The Mostowski Model An Independence Result Root-Functions in Rings 2Cn Cn

2Cn Cn

In order to show that the answer to the question is no, it is enough to prove 2Cn Cn (for some odd integer n). For this, we modify Mostowski’s model:

Introduction to Permutation Models Models of ZFA The Mostowski Model An Independence Result Root-Functions in Rings 2Cn Cn

2Cn Cn

In order to show that the answer to the question is no, it is enough to prove 2Cn Cn (for some odd integer n). For this, we modify Mostowski’s model:

◮ Each atom ai (i ∈ Q) of Mostowski’s model is replaced

with two disjoint sets Qi = {bi,0, bi,1} and Pi = {ai,0, . . . , ai,p−1}, where p > 2 is an arbitrary but fixed prime.

Introduction to Permutation Models Models of ZFA The Mostowski Model An Independence Result Root-Functions in Rings 2Cn Cn

2Cn Cn

In order to show that the answer to the question is no, it is enough to prove 2Cn Cn (for some odd integer n). For this, we modify Mostowski’s model:

◮ Each atom ai (i ∈ Q) of Mostowski’s model is replaced

with two disjoint sets Qi = {bi,0, bi,1} and Pi = {ai,0, . . . , ai,p−1}, where p > 2 is an arbitrary but fixed prime.

◮ The set of atoms A:

A =

• i∈Q

Qi ∪

• j∈Q

Pj.

Introduction to Permutation Models Models of ZFA The Mostowski Model An Independence Result Root-Functions in Rings 2Cn Cn

2Cn Cn

Let G be the group of permutations of A consisting of permutations π which satisfy

◮ for all i, j ∈ Q, π is cyclic permutation of Qi and Pj

respectively,

Introduction to Permutation Models Models of ZFA The Mostowski Model An Independence Result Root-Functions in Rings 2Cn Cn

2Cn Cn

Let G be the group of permutations of A consisting of permutations π which satisfy

◮ for all i, j ∈ Q, π is cyclic permutation of Qi and Pj

respectively, and

◮ π[Qi] = Qi′, π[Pj] = Pj′,

where π restricted to i, j ∈ Q is order-preserving.

Introduction to Permutation Models Models of ZFA The Mostowski Model An Independence Result Root-Functions in Rings 2Cn Cn

2Cn Cn

Let G be the group of permutations of A consisting of permutations π which satisfy

◮ for all i, j ∈ Q, π is cyclic permutation of Qi and Pj

respectively, and

◮ π[Qi] = Qi′, π[Pj] = Pj′,

where π restricted to i, j ∈ Q is order-preserving.

Proposition (H., H., L., L., L., S.)

2Cp+2 Cp+2 is consistent with ZF.

Introduction to Permutation Models Models of ZFA The Mostowski Model An Independence Result Root-Functions in Rings 2Cn Cn

2Cn Cn

Let G be the group of permutations of A consisting of permutations π which satisfy

◮ for all i, j ∈ Q, π is cyclic permutation of Qi and Pj

respectively, and

◮ π[Qi] = Qi′, π[Pj] = Pj′,

where π restricted to i, j ∈ Q is order-preserving.

Proposition (H., H., L., L., L., S.)

2Cp+2 Cp+2 is consistent with ZF.

• Proof. In the modified Mostowski model we have

¬Cp+2

Introduction to Permutation Models Models of ZFA The Mostowski Model An Independence Result Root-Functions in Rings 2Cn Cn

2Cn Cn

Let G be the group of permutations of A consisting of permutations π which satisfy

◮ for all i, j ∈ Q, π is cyclic permutation of Qi and Pj

respectively, and

◮ π[Qi] = Qi′, π[Pj] = Pj′,

where π restricted to i, j ∈ Q is order-preserving.

Proposition (H., H., L., L., L., S.)

2Cp+2 Cp+2 is consistent with ZF.

• Proof. In the modified Mostowski model we have

¬Cp+2 ∧ 2Cp+2. ⊣

Introduction to Permutation Models Models of ZFA The Mostowski Model An Independence Result Root-Functions in Rings 2Cn Cn

2Cn Cn

Corollary (H., H., L., L., L., S.)

every integral domain has an 5th-root function C5

Introduction to Permutation Models Models of ZFA The Mostowski Model An Independence Result Root-Functions in Rings 2Cn Cn

2Cn Cn

Corollary (H., H., L., L., L., S.)

every integral domain has an 5th-root function C5

• Proof. This follows from

◮ 2C5 ⇒ every integral domain has an 5th-root function,

Introduction to Permutation Models Models of ZFA The Mostowski Model An Independence Result Root-Functions in Rings 2Cn Cn

2Cn Cn

Corollary (H., H., L., L., L., S.)

every integral domain has an 5th-root function C5

• Proof. This follows from

◮ 2C5 ⇒ every integral domain has an 5th-root function, ◮ 2C5 C5.

⊣