Automorphisms of Models of Set Theory and NFU

Zach McKenzie I A Brief Introductuion to NFU

I The connection between models of ZFC that admit automorphism and NFU

I Some models that admit automorphism I The connection between models of ZFC that admit automorphism and NFU

I Some models that admit automorphism

I A Brief Introductuion to NFU I Some models that admit automorphism

I A Brief Introductuion to NFU

I The connection between models of ZFC that admit automorphism and NFU I A Brief Introductuion to NFU

I The connection between models of ZFC that admit automorphism and NFU

I Some models that admit automorphism Definition We say that an L-formula φ is stratified if and only if there is a function σ from the variables appearing φ to the natural numbers such that (i) if x ∈ y is a subformula of φ then σ(y) = σ(x) + 1, (ii) if x = y is a subformula of φ then σ(x) = σ(y).

I The class of stratified formulae is exactly the class of well-formed formulae of Russell’s Theory of Types with the type indices removed.

Stratified Formulae

I Let L = hS, ∈i where S is a unary predicate and ∈ a binary predicate. I The class of stratified formulae is exactly the class of well-formed formulae of Russell’s Theory of Types with the type indices removed.

Stratified Formulae

I Let L = hS, ∈i where S is a unary predicate and ∈ a binary predicate.

Definition We say that an L-formula φ is stratified if and only if there is a function σ from the variables appearing φ to the natural numbers such that (i) if x ∈ y is a subformula of φ then σ(y) = σ(x) + 1, (ii) if x = y is a subformula of φ then σ(x) = σ(y). Stratified Formulae

I Let L = hS, ∈i where S is a unary predicate and ∈ a binary predicate.

Definition We say that an L-formula φ is stratified if and only if there is a function σ from the variables appearing φ to the natural numbers such that (i) if x ∈ y is a subformula of φ then σ(y) = σ(x) + 1, (ii) if x = y is a subformula of φ then σ(x) = σ(y).

I The class of stratified formulae is exactly the class of well-formed formulae of Russell’s Theory of Types with the type indices removed. (Extensionality for Sets)

∀x∀y(S(x) ∧ S(y) ⇒ (x = y ⇐⇒ ∀z(z ∈ x ⇐⇒ z ∈ y))),

(Stratified Comprehension) for all stratified formulae φ(x,~z),

∀~z∃y∀x(x ∈ y ⇐⇒ φ(x,~z)).

The Axiom of Infinity and the Axiom of Choice.

I V = {x | x = x} is provably a set.

I P(V ) ⊆ V , and so Cantor’s Theorem fails!!

NFU

Definition NFU is the L-theory with axioms (Stratified Comprehension) for all stratified formulae φ(x,~z),

∀~z∃y∀x(x ∈ y ⇐⇒ φ(x,~z)).

The Axiom of Infinity and the Axiom of Choice.

I V = {x | x = x} is provably a set.

I P(V ) ⊆ V , and so Cantor’s Theorem fails!!

NFU

Definition NFU is the L-theory with axioms (Extensionality for Sets)

∀x∀y(S(x) ∧ S(y) ⇒ (x = y ⇐⇒ ∀z(z ∈ x ⇐⇒ z ∈ y))), The Axiom of Infinity and the Axiom of Choice.

I V = {x | x = x} is provably a set.

I P(V ) ⊆ V , and so Cantor’s Theorem fails!!

NFU

Definition NFU is the L-theory with axioms (Extensionality for Sets)

∀x∀y(S(x) ∧ S(y) ⇒ (x = y ⇐⇒ ∀z(z ∈ x ⇐⇒ z ∈ y))),

(Stratified Comprehension) for all stratified formulae φ(x,~z),

∀~z∃y∀x(x ∈ y ⇐⇒ φ(x,~z)). I V = {x | x = x} is provably a set.

I P(V ) ⊆ V , and so Cantor’s Theorem fails!!

NFU

Definition NFU is the L-theory with axioms (Extensionality for Sets)

∀x∀y(S(x) ∧ S(y) ⇒ (x = y ⇐⇒ ∀z(z ∈ x ⇐⇒ z ∈ y))),

(Stratified Comprehension) for all stratified formulae φ(x,~z),

∀~z∃y∀x(x ∈ y ⇐⇒ φ(x,~z)).

The Axiom of Infinity and the Axiom of Choice. I P(V ) ⊆ V , and so Cantor’s Theorem fails!!

NFU

Definition NFU is the L-theory with axioms (Extensionality for Sets)

∀x∀y(S(x) ∧ S(y) ⇒ (x = y ⇐⇒ ∀z(z ∈ x ⇐⇒ z ∈ y))),

(Stratified Comprehension) for all stratified formulae φ(x,~z),

∀~z∃y∀x(x ∈ y ⇐⇒ φ(x,~z)).

The Axiom of Infinity and the Axiom of Choice.

I V = {x | x = x} is provably a set. NFU

Definition NFU is the L-theory with axioms (Extensionality for Sets)

∀x∀y(S(x) ∧ S(y) ⇒ (x = y ⇐⇒ ∀z(z ∈ x ⇐⇒ z ∈ y))),

(Stratified Comprehension) for all stratified formulae φ(x,~z),

∀~z∃y∀x(x ∈ y ⇐⇒ φ(x,~z)).

The Axiom of Infinity and the Axiom of Choice.

I V = {x | x = x} is provably a set.

I P(V ) ⊆ V , and so Cantor’s Theorem fails!! I Ordinals can be represented as equivalence classes of isomorphic well-orderings.

Definition We define the T operation on ordinals and cardinals as follows: (i) if X is a cardinal and Y ∈ X then T (X ) is the cardinality if {{y} | y ∈ Y }, (ii) if X is an ordinal and R ∈ X the T (X ) is order-type of {h{x}, {y}i | hx, yi ∈ R}.

I We use N to denote the set of all finite cardinals.

Mathematics in NFU

I Cardinals can be represented as equivalence classes of equipolent sets. Definition We define the T operation on ordinals and cardinals as follows: (i) if X is a cardinal and Y ∈ X then T (X ) is the cardinality if {{y} | y ∈ Y }, (ii) if X is an ordinal and R ∈ X the T (X ) is order-type of {h{x}, {y}i | hx, yi ∈ R}.

I We use N to denote the set of all finite cardinals.

Mathematics in NFU

I Cardinals can be represented as equivalence classes of equipolent sets.

I Ordinals can be represented as equivalence classes of isomorphic well-orderings. I We use N to denote the set of all finite cardinals.

Mathematics in NFU

I Cardinals can be represented as equivalence classes of equipolent sets.

I Ordinals can be represented as equivalence classes of isomorphic well-orderings.

Definition We define the T operation on ordinals and cardinals as follows: (i) if X is a cardinal and Y ∈ X then T (X ) is the cardinality if {{y} | y ∈ Y }, (ii) if X is an ordinal and R ∈ X the T (X ) is order-type of {h{x}, {y}i | hx, yi ∈ R}. Mathematics in NFU

I Cardinals can be represented as equivalence classes of equipolent sets.

I Ordinals can be represented as equivalence classes of isomorphic well-orderings.

Definition We define the T operation on ordinals and cardinals as follows: (i) if X is a cardinal and Y ∈ X then T (X ) is the cardinality if {{y} | y ∈ Y }, (ii) if X is an ordinal and R ∈ X the T (X ) is order-type of {h{x}, {y}i | hx, yi ∈ R}.

I We use N to denote the set of all finite cardinals. (Axiom of Counting) (∀n ∈ N)T (n) = n, (AxCount≤)(∀n ∈ N)n ≤ T (n), (AxCount≥)(∀n ∈ N)n ≥ T (n),

Extensions of NFU

NFU is unable to prove that for every finite cardinal n, T (n) = n. This observation suggests the following axioms that can be added to NFU: (AxCount≤)(∀n ∈ N)n ≤ T (n), (AxCount≥)(∀n ∈ N)n ≥ T (n),

Extensions of NFU

NFU is unable to prove that for every finite cardinal n, T (n) = n. This observation suggests the following axioms that can be added to NFU: (Axiom of Counting) (∀n ∈ N)T (n) = n, (AxCount≥)(∀n ∈ N)n ≥ T (n),

Extensions of NFU

NFU is unable to prove that for every finite cardinal n, T (n) = n. This observation suggests the following axioms that can be added to NFU: (Axiom of Counting) (∀n ∈ N)T (n) = n, (AxCount≤)(∀n ∈ N)n ≤ T (n), Extensions of NFU

NFU is unable to prove that for every finite cardinal n, T (n) = n. This observation suggests the following axioms that can be added to NFU: (Axiom of Counting) (∀n ∈ N)T (n) = n, (AxCount≤)(∀n ∈ N)n ≤ T (n), (AxCount≥)(∀n ∈ N)n ≥ T (n), I Let M |= Mac and j : M −→ M be an automorphism such that there is a c with j(c) 6= c and P(c) ∪ c ⊆ j(c). n I Let cn = j (c), ∈n=∈ ∩(cn × P(cn)) and Sn+1 = P(cn). The structure h(cn)n∈ω, (∈n)n∈ω, (Sn+1)n∈ωi is model of Russell’s Type Theory with Urelements.

I Now, define ∈NFU on c0 by

x ∈NFU y if and only if x ∈ j(y) ∧ j(y) ⊆ c.

Now, the structure hc0, ∈NFUi |= NFU and hc0, ∈NFUi agrees with h(cn)n∈ω, (∈n)n∈ω, (Sn+1)n∈ωi on stratified sentences.

Models of NFU

I We use Mac to denote the subsystem of ZFC axiomatised by: Extensionality, Pair, Union, Powerset, Foundation, Choice and ∆0-separation. n I Let cn = j (c), ∈n=∈ ∩(cn × P(cn)) and Sn+1 = P(cn). The structure h(cn)n∈ω, (∈n)n∈ω, (Sn+1)n∈ωi is model of Russell’s Type Theory with Urelements.

I Now, define ∈NFU on c0 by

x ∈NFU y if and only if x ∈ j(y) ∧ j(y) ⊆ c.

Now, the structure hc0, ∈NFUi |= NFU and hc0, ∈NFUi agrees with h(cn)n∈ω, (∈n)n∈ω, (Sn+1)n∈ωi on stratified sentences.

Models of NFU

I We use Mac to denote the subsystem of ZFC axiomatised by: Extensionality, Pair, Union, Powerset, Foundation, Choice and ∆0-separation.

I Let M |= Mac and j : M −→ M be an automorphism such that there is a c with j(c) 6= c and P(c) ∪ c ⊆ j(c). I Now, define ∈NFU on c0 by

x ∈NFU y if and only if x ∈ j(y) ∧ j(y) ⊆ c.

Now, the structure hc0, ∈NFUi |= NFU and hc0, ∈NFUi agrees with h(cn)n∈ω, (∈n)n∈ω, (Sn+1)n∈ωi on stratified sentences.

Models of NFU

I We use Mac to denote the subsystem of ZFC axiomatised by: Extensionality, Pair, Union, Powerset, Foundation, Choice and ∆0-separation.

I Let M |= Mac and j : M −→ M be an automorphism such that there is a c with j(c) 6= c and P(c) ∪ c ⊆ j(c). n I Let cn = j (c), ∈n=∈ ∩(cn × P(cn)) and Sn+1 = P(cn). The structure h(cn)n∈ω, (∈n)n∈ω, (Sn+1)n∈ωi is model of Russell’s Type Theory with Urelements. Models of NFU

I We use Mac to denote the subsystem of ZFC axiomatised by: Extensionality, Pair, Union, Powerset, Foundation, Choice and ∆0-separation.

I Let M |= Mac and j : M −→ M be an automorphism such that there is a c with j(c) 6= c and P(c) ∪ c ⊆ j(c). n I Let cn = j (c), ∈n=∈ ∩(cn × P(cn)) and Sn+1 = P(cn). The structure h(cn)n∈ω, (∈n)n∈ω, (Sn+1)n∈ωi is model of Russell’s Type Theory with Urelements.

I Now, define ∈NFU on c0 by

x ∈NFU y if and only if x ∈ j(y) ∧ j(y) ⊆ c.

Now, the structure hc0, ∈NFUi |= NFU and hc0, ∈NFUi agrees with h(cn)n∈ω, (∈n)n∈ω, (Sn+1)n∈ωi on stratified sentences. I If M |= ZFC and j : M −→ M is an automorphism such that M I j(n) ≤ n for all n ∈ ω , I there exists an ordinal α ≥ ω such that j(α) > α.

Then hVα, ∈NFUi |= NFU + AxCount≤. I If M |= ZFC and j : M −→ M is an automorphism such that M I j(n) ≥ n for all n ∈ ω , I there exists an ordinal α ≥ ω such that j(α) > α.

Then hVα, ∈NFUi |= NFU + AxCount≥.

Models of extensions of NFU

I If M |= ZFC and j : M −→ M is an automorphism such that M I j(n) = n for all n ∈ ω , I there exists an ordinal α ≥ ω such that j(α) > α.

Then hVα, ∈NFUi |= NFU + Axiom of Counting. I If M |= ZFC and j : M −→ M is an automorphism such that M I j(n) ≥ n for all n ∈ ω , I there exists an ordinal α ≥ ω such that j(α) > α.

Then hVα, ∈NFUi |= NFU + AxCount≥.

Models of extensions of NFU

I If M |= ZFC and j : M −→ M is an automorphism such that M I j(n) = n for all n ∈ ω , I there exists an ordinal α ≥ ω such that j(α) > α.

Then hVα, ∈NFUi |= NFU + Axiom of Counting. I If M |= ZFC and j : M −→ M is an automorphism such that M I j(n) ≤ n for all n ∈ ω , I there exists an ordinal α ≥ ω such that j(α) > α.

Then hVα, ∈NFUi |= NFU + AxCount≤. Models of extensions of NFU

I If M |= ZFC and j : M −→ M is an automorphism such that M I j(n) = n for all n ∈ ω , I there exists an ordinal α ≥ ω such that j(α) > α.

Then hVα, ∈NFUi |= NFU + Axiom of Counting. I If M |= ZFC and j : M −→ M is an automorphism such that M I j(n) ≤ n for all n ∈ ω , I there exists an ordinal α ≥ ω such that j(α) > α.

Then hVα, ∈NFUi |= NFU + AxCount≤. I If M |= ZFC and j : M −→ M is an automorphism such that M I j(n) ≥ n for all n ∈ ω , I there exists an ordinal α ≥ ω such that j(α) > α.

Then hVα, ∈NFUi |= NFU + AxCount≥. I (Jensen, Holmes) Using the Erd˝os-Rado Theorem one can build a model M |= ZFC endowed with an automorphism j : M −→ M such that M I j(n) = n for all n ∈ ω , M I j(α) > α for some α ∈ ωω .

I In the context of NFU this means that

NFU + Axiom of Counting proves the existence of iin for every concrete n.

I (Holmes) NFU + Axiom of Counting is equiconsistent with

Zermelo Set Theory + {iin exists | n ∈ N}.

Automorphisms

I If M |= ZFC and j : M −→ M is and automorphism such M that j(n) = n for all n ∈ ω , then if α < ωn for some concrete n then j(α) = α. I In the context of NFU this means that

NFU + Axiom of Counting proves the existence of iin for every concrete n.

I (Holmes) NFU + Axiom of Counting is equiconsistent with

Zermelo Set Theory + {iin exists | n ∈ N}.

Automorphisms

I If M |= ZFC and j : M −→ M is and automorphism such M that j(n) = n for all n ∈ ω , then if α < ωn for some concrete n then j(α) = α.

I (Jensen, Holmes) Using the Erd˝os-Rado Theorem one can build a model M |= ZFC endowed with an automorphism j : M −→ M such that M I j(n) = n for all n ∈ ω , M I j(α) > α for some α ∈ ωω . I (Holmes) NFU + Axiom of Counting is equiconsistent with

Zermelo Set Theory + {iin exists | n ∈ N}.

Automorphisms

I If M |= ZFC and j : M −→ M is and automorphism such M that j(n) = n for all n ∈ ω , then if α < ωn for some concrete n then j(α) = α.

I (Jensen, Holmes) Using the Erd˝os-Rado Theorem one can build a model M |= ZFC endowed with an automorphism j : M −→ M such that M I j(n) = n for all n ∈ ω , M I j(α) > α for some α ∈ ωω .

I In the context of NFU this means that

NFU + Axiom of Counting proves the existence of iin for every concrete n. Automorphisms

I If M |= ZFC and j : M −→ M is and automorphism such M that j(n) = n for all n ∈ ω , then if α < ωn for some concrete n then j(α) = α.

I (Jensen, Holmes) Using the Erd˝os-Rado Theorem one can build a model M |= ZFC endowed with an automorphism j : M −→ M such that M I j(n) = n for all n ∈ ω , M I j(α) > α for some α ∈ ωω .

I In the context of NFU this means that

NFU + Axiom of Counting proves the existence of iin for every concrete n.

I (Holmes) NFU + Axiom of Counting is equiconsistent with

Zermelo Set Theory + {iin exists | n ∈ N}. Question Does NFU + Axiom of Counting ` Con(NFU + AxCount≤) and NFU + AxCount≤ ` Con(NFU + AxCount≥)? Question Is NFU + AxCount≥ equiconsistent with NFU?

AxCount≤ and AxCount≥

Question Is there a model of NFU + AxCount≤ with a countable ordinal α such that T (α) < α? Question Is NFU + AxCount≥ equiconsistent with NFU?

AxCount≤ and AxCount≥

Question Is there a model of NFU + AxCount≤ with a countable ordinal α such that T (α) < α? Question Does NFU + Axiom of Counting ` Con(NFU + AxCount≤) and NFU + AxCount≤ ` Con(NFU + AxCount≥)? AxCount≤ and AxCount≥

Question Is there a model of NFU + AxCount≤ with a countable ordinal α such that T (α) < α? Question Does NFU + Axiom of Counting ` Con(NFU + AxCount≤) and NFU + AxCount≤ ` Con(NFU + AxCount≥)? Question Is NFU + AxCount≥ equiconsistent with NFU? Corollary If there is a countable transitive model of ZFC then there is M |= ZFC endowed with an automorphism j : M −→ M such that (i) j(n) ≤ n for all n ∈ ωM, ck M (ii) j(α) > α for some α ∈ (ω1 ) .

Automorphisms and Non-Standard Models

Theorem If M = hMM, ∈M, β¯Mi is an ω-model of Mac with a ¯M non-standard ordinal below β then there is an Lβ¯-structure N ≡ M admitting an automorphism j : N −→ N with Lβ¯ (i) j(n) ≤ n for all n ∈ ωN , (ii) j(α) > α for some α ∈ β¯N . Automorphisms and Non-Standard Models

Theorem If M = hMM, ∈M, β¯Mi is an ω-model of Mac with a ¯M non-standard ordinal below β then there is an Lβ¯-structure N ≡ M admitting an automorphism j : N −→ N with Lβ¯ (i) j(n) ≤ n for all n ∈ ωN , (ii) j(α) > α for some α ∈ β¯N .

Corollary If there is a countable transitive model of ZFC then there is M |= ZFC endowed with an automorphism j : M −→ M such that (i) j(n) ≤ n for all n ∈ ωM, ck M (ii) j(α) > α for some α ∈ (ω1 ) . Automorphisms from Standard Models

Theorem If hM, ∈i is a transitive model of Mac then there is a model M ≡ hM, ∈i endowed with an automorphism j : M −→ M such that for all ordinals α in M, j(α) ≥ α. Questions

Question P Is NFU + AxCount≤ equiconsistent with KP ?