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 2 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; 2i where S is a unary predicate and 2 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; 2i where S is a unary predicate and 2 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 2 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; 2i where S is a unary predicate and 2 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 2 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) 8x8y(S(x) ^ S(y) ) (x = y () 8z(z 2 x () z 2 y))); (Stratified Comprehension) for all stratified formulae φ(x;~z), 8~z9y8x(x 2 y () φ(x;~z)): The Axiom of Infinity and the Axiom of Choice. I V = fx j x = xg 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), 8~z9y8x(x 2 y () φ(x;~z)): The Axiom of Infinity and the Axiom of Choice. I V = fx j x = xg 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) 8x8y(S(x) ^ S(y) ) (x = y () 8z(z 2 x () z 2 y))); The Axiom of Infinity and the Axiom of Choice. I V = fx j x = xg 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) 8x8y(S(x) ^ S(y) ) (x = y () 8z(z 2 x () z 2 y))); (Stratified Comprehension) for all stratified formulae φ(x;~z), 8~z9y8x(x 2 y () φ(x;~z)): I V = fx j x = xg 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) 8x8y(S(x) ^ S(y) ) (x = y () 8z(z 2 x () z 2 y))); (Stratified Comprehension) for all stratified formulae φ(x;~z), 8~z9y8x(x 2 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) 8x8y(S(x) ^ S(y) ) (x = y () 8z(z 2 x () z 2 y))); (Stratified Comprehension) for all stratified formulae φ(x;~z), 8~z9y8x(x 2 y () φ(x;~z)): The Axiom of Infinity and the Axiom of Choice. I V = fx j x = xg is provably a set. NFU Definition NFU is the L-theory with axioms (Extensionality for Sets) 8x8y(S(x) ^ S(y) ) (x = y () 8z(z 2 x () z 2 y))); (Stratified Comprehension) for all stratified formulae φ(x;~z), 8~z9y8x(x 2 y () φ(x;~z)): The Axiom of Infinity and the Axiom of Choice. I V = fx j x = xg 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 2 X then T (X ) is the cardinality if ffyg j y 2 Y g, (ii) if X is an ordinal and R 2 X the T (X ) is order-type of fhfxg; fygi j hx; yi 2 Rg. 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 2 X then T (X ) is the cardinality if ffyg j y 2 Y g, (ii) if X is an ordinal and R 2 X the T (X ) is order-type of fhfxg; fygi j hx; yi 2 Rg. 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 2 X then T (X ) is the cardinality if ffyg j y 2 Y g, (ii) if X is an ordinal and R 2 X the T (X ) is order-type of fhfxg; fygi j hx; yi 2 Rg. 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 2 X then T (X ) is the cardinality if ffyg j y 2 Y g, (ii) if X is an ordinal and R 2 X the T (X ) is order-type of fhfxg; fygi j hx; yi 2 Rg. I We use N to denote the set of all finite cardinals. (Axiom of Counting) (8n 2 N)T (n) = n, (AxCount≤)(8n 2 N)n ≤ T (n), (AxCount≥)(8n 2 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≤)(8n 2 N)n ≤ T (n), (AxCount≥)(8n 2 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) (8n 2 N)T (n) = n, (AxCount≥)(8n 2 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) (8n 2 N)T (n) = n, (AxCount≤)(8n 2 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) (8n 2 N)T (n) = n, (AxCount≤)(8n 2 N)n ≤ T (n), (AxCount≥)(8n 2 N)n ≥ T (n), I Let M j= 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), 2n=2 \(cn × P(cn)) and Sn+1 = P(cn). The structure h(cn)n2!; (2n)n2!; (Sn+1)n2!i is model of Russell's Type Theory with Urelements. I Now, define 2NFU on c0 by x 2NFU y if and only if x 2 j(y) ^ j(y) ⊆ c: Now, the structure hc0; 2NFUi j= NFU and hc0; 2NFUi agrees with h(cn)n2!; (2n)n2!; (Sn+1)n2!i 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), 2n=2 \(cn × P(cn)) and Sn+1 = P(cn). The structure h(cn)n2!; (2n)n2!; (Sn+1)n2!i is model of Russell's Type Theory with Urelements.