A model for the extended predicative Mahlo Universe Anton Setzer - PowerPoint PPT Presentation

A model for the extended predicative Mahlo Universe Anton Setzer (joint work with Reinhard Kahle, Lisbon) Proof Society Workshop Ghent 6 September 2018 Anton Setzer The extended predicative Mahlo Universe 1/ 31

Explicit Mathematics Extended Predicative Mahlo Model for Extended Predicative Mahlo Universe Anton Setzer The extended predicative Mahlo Universe 2/ 31

Explicit Mathematics Explicit Mathematics Extended Predicative Mahlo Model for Extended Predicative Mahlo Universe Anton Setzer The extended predicative Mahlo Universe 3/ 31

Explicit Mathematics Explicit Mathematics • Explicit mathematics based on term language where terms can denote elements of sets and sets. • No restriction on application, succ(nat) is a term. • a ˙ ∈ b for a is an element of the set denoted by b . • ℜ ( a ) for a is a name, i.e. denotes a set. Anton Setzer The extended predicative Mahlo Universe 4/ 31

Explicit Mathematics Inductive Generation • i( u , v ) denotes the accessible part of the relation v on domain u . • C losed i ( a , b , S ) := ∀ x ˙ ∈ a . ( ∀ y ˙ ∈ a . ( y , x ) ˙ ∈ b → y ∈ S ) → x ∈ S • ℜ ( a ) ∧ ℜ ( b ) → ∃ X . ℜ (i ( a , b ) , X ) ∧ C losed i ( a , b , X ) • ℜ ( a ) ∧ ℜ ( b ) ∧ C losed i ( a , b , φ ) → ∀ x ˙ ∈ i ( a , b ) .φ ( x ) Anton Setzer The extended predicative Mahlo Universe 5/ 31

Explicit Mathematics Universes • C los univ ( W , x ) expresses that x is formed using the above universe operations (excluding i) from elements in W . • U niv ( W ) := ( ∀ x ∈ W . ℜ ( x )) ∧ ∀ x . C los univ ( W , x ) → x ∈ W . • Univ( t ) := ∃ X . ℜ ( t , X ) ∧ U niv ( X ) . Anton Setzer The extended predicative Mahlo Universe 6/ 31

Extended Predicative Mahlo Explicit Mathematics Extended Predicative Mahlo Model for Extended Predicative Mahlo Universe Anton Setzer The extended predicative Mahlo Universe 7/ 31

Extended Predicative Mahlo Axiomatic Mahlo Universe • To deal with size problem, when working in type theory and explicit mathematics one needs large universes. • Allows as well to obtain proof theoretic stronger theories. • Axiomatic Mahlo universe is a universe M such as for every a ∈ M and f ∈ M → M there exists a subuniverse u( a , f ) of M which is closed under a and f and an element of M. Anton Setzer The extended predicative Mahlo Universe 8/ 31

Extended Predicative Mahlo Illustration of the Mahlo Universe M f Anton Setzer The extended predicative Mahlo Universe 9/ 31

Extended Predicative Mahlo Illustration of the Mahlo Universe M f Anton Setzer The extended predicative Mahlo Universe 9/ 31

Extended Predicative Mahlo Illustration of the Mahlo Universe M u( a , f ) f f Anton Setzer The extended predicative Mahlo Universe 9/ 31

Extended Predicative Mahlo Illustration of the Mahlo Universe M u( a , f ) u( a , f ) f f Anton Setzer The extended predicative Mahlo Universe 9/ 31

Extended Predicative Mahlo From Axiomatic to Extended Predicative Mahlo • Problem: introduction rule for u( a , f ) depends on total functions f : M → M, which is impredicative • Totality on M is not really needed, only that f is total on u. • Extended predicative Mahlo universe formalises this. Anton Setzer The extended predicative Mahlo Universe 10/ 31

Extended Predicative Mahlo Pre-Universe • Formula expressing that v is a relative preuniverse: ( ∀ x . C los univ ( u , x ) ∧ x ˙ ∈ v → x ˙ ∈ u ) ∧ RPU( a , f , v , u ) := ( a ˙ ∈ v → a ˙ ∈ u ) ∧ ( ∀ x ˙ ∈ u . f x ˙ ∈ v → f x ˙ ∈ u ) Anton Setzer The extended predicative Mahlo Universe 11/ 31

Extended Predicative Mahlo f c v c u f b b a Anton Setzer The extended predicative Mahlo Universe 12/ 31

Extended Predicative Mahlo Least pre-universes ℜ ℜ ( v ) → RPU( a , f , v , pre ( a , f , v )) . ℜ ℜ ( v ) ∧ RPU( a , f , v , φ ) → ∀ x ˙ ∈ pre ( a , f , v ) .φ ( x ) Anton Setzer The extended predicative Mahlo Universe 13/ 31

Extended Predicative Mahlo pre(a,f,v) f c v c pre ( a , f , v ) f b b Anton Setzer The extended predicative Mahlo Universe 14/ 31

Extended Predicative Mahlo Independence of pre(a,f,v) ( ∀ x . C los ‘ univ ( u , x ) → x ∈ v ) ∧ Indep ( a , f , v , u ) := a ˙ ∈ v ∧ ( ∀ x ˙ ∈ u . f x ˙ ∈ v ) Anton Setzer The extended predicative Mahlo Universe 15/ 31

Extended Predicative Mahlo Indep(a,f,v,u) f b b v f a u := pre ( a , f , v ) Indep ( a , f , v , u ) Anton Setzer The extended predicative Mahlo Universe 16/ 31

Extended Predicative Mahlo Axioms for M Univ( M ) ∧ i ∈ ( ∀ a , b ˙ ∈ M → i( a , b ) ˙ ∈ M ) Indep ( a , f , M , pre ( a , f , M )) → u ( a , f ) ˙ ∈ M ∧ u ( a , f ) ˙ = pre ( a , f , M ) Induction expressing M is least set with these closure properties can be added as well. Anton Setzer The extended predicative Mahlo Universe 17/ 31

Extended Predicative Mahlo Introduction Rule for M f b M b u ( a , f ) f a pre ( a , f , M ) Indep ( a , f , M , pre ( a , f , M )) Anton Setzer The extended predicative Mahlo Universe 18/ 31

Model for Extended Predicative Mahlo Universe Explicit Mathematics Extended Predicative Mahlo Model for Extended Predicative Mahlo Universe Anton Setzer The extended predicative Mahlo Universe 19/ 31

Model for Extended Predicative Mahlo Universe Model given by a Relation • Define codes for terms such as � int( a , b ) := � 3 , a , b � • Let predicates P ⊆ N 3 encode relations ℜ P , ∈ P , / ∈ P by ℜ P ( a ) := P ( a , 0 , 0) , b ∈ P a := P ( a , b , 1) , b �∈ P a ℜ P ( a ) ∧ ¬ ( b ∈ P a ) := Anton Setzer The extended predicative Mahlo Universe 20/ 31

Model for Extended Predicative Mahlo Universe Operator for Universe Constructions a = � ℜ int int( u , v ) ∧ ℜ P ( u ) ∧ ℜ P ( v ) P ( a , u , v ) := ℜ int ∃ u , v . ℜ int P ( a ) := P ( a , u , v ) b ∈ int ∃ u , v . ℜ int P ( a , u , v ) ∧ ( b ∈ P u ∧ b ∈ P v ) P a := similarly for other universe constructions Anton Setzer The extended predicative Mahlo Universe 21/ 31

Model for Extended Predicative Mahlo Universe Operator for Universe Constructions ℜ univ x = nat ∨ x = id ∨ ℜ int P ( x ) ∨ · · · ( x ) := P b ∨ a ∈ id b ∨ a ∈ int a ∈ univ a ∈ nat P b ∨ · · · b := P P Γ univ ( a , b ) := b = nat ∨ b = id P ∨ ( ∃ u , v . b = � int( u , v ) ∧ u ∈ P a ∧ v ∈ P a ) ∨ · · · Anton Setzer The extended predicative Mahlo Universe 22/ 31

Model for Extended Predicative Mahlo Universe Modelling Inductive Generation ℜ pre − i a = � ( a , u , v ) := i( u , v ) ∧ ℜ P ( u ) ∧ ℜ P ( v ) P Γ i , pot ∃ u , v . b = � ( a , b ) := i( u , v ) ∧ u ∈ P a ∧ v ∈ P a P ∃ u , v . a = � b ∈ i i( u , v ) ∧ b ∈ P u P a := ∧∀ x ∈ P u . � x , b � ∈ P v → x ∈ P a P � i( u , v ) . x ∈ P � C losed i ∀ x ∈ i P ( u , v ) := i( u , v ) ℜ pre − i ℜ i ( a , u , v ) ∧ C losed i P ( a , u , v ) := P ( u , v ) P ℜ i ∃ u , v . ℜ i P ( a ) := P ( a , u , v ) ∃ u , v . b = � i( u , v ) ∧ Γ i , pot ( a , b ) ∧ C losed i Γ i P ( a , b ) := P ( u , v ) p Anton Setzer The extended predicative Mahlo Universe 23/ 31

Model for Extended Predicative Mahlo Universe Modelling Pre universes ∃ a , f , v . a ′ = � b ∈ pre , pot a ′ := pre( a , f , v ) P ∧ ( b = a ∨ ( ∃ x ∈ P a ′ . b ≃ { f } ( x )) ∨ Γ univ ( � pre( a , f , v ) , b )) P a ′ ∧ b ∈ P v b ∈ pre b ∈ pre , pot a ′ := P P C losed pre ∀ b ∈ pre P ( a , f , v ) := pre( a , f , v ) . b ∈ P � � pre( a , f , v ) P ∃ a , f . a ′ = � Indep pre ( a ′ , v ) := pre( a , f , v ) ∧∀ b ∈ pre , pot a ′ . b ∈ P v P a ′ = � ℜ pre P ( a ′ , a , f , v ) := pre( a , f , v ) ∧ C losed pre P ( a , f , v ) ∧ ( ℜ P ( v ) ∨ Indep pre ( a ′ , v )) Anton Setzer The extended predicative Mahlo Universe 24/ 31

Model for Extended Predicative Mahlo Universe Modelling Pre universes ℜ pre ∃ a , f , v . ℜ pre P ( a ′ ) P ( a ′ , a , f , v ) := Anton Setzer The extended predicative Mahlo Universe 25/ 31

Model for Extended Predicative Mahlo Universe Modelling u(a,f) ℜ u , pot a ′ = � u( a , f ) ∧ Indep pre ( � ( a ′ , a , f ) := pre( a , f , M) , M) P ℜ u , pot ∃ a , f . ℜ u , pot ( a ′ ) ( a ′ , a , f ) := P P ℜ u , pot ℜ u P ( a ′ , a , f ) ( a ′ , a , f ) := P ∧ C losed pre ( � pre( a , f , M) , M) ℜ u ∃ a , f . ℜ u P ( a ′ ) P ( a ′ , a , f ) := b ∈ u P a ′ ∃ a , f . ℜ u P ( a ′ , a , f ) ∧ b ∈ P � := pre( a , f , M) Anton Setzer The extended predicative Mahlo Universe 26/ 31

Model for Extended Predicative Mahlo Universe Modelling u(a,f) b ∈ M , pot a := a = M P (M , b ) ∨ Γ i , pot (M , b ) ∨ ℜ u , pot ∧ (Γ univ ( b )) P P P b ∈ M P a := a = M ∧ (Γ univ (M , b ) ∨ Γ i P (M , b ) ∨ ℜ u P ( b )) P ∀ b ∈ M , pot C losed M := M . b ∈ P M P P a = M ∧ C losed M ℜ M P ( a ) := P Anton Setzer The extended predicative Mahlo Universe 27/ 31