Model Companion of Unstable Theories with an Automorphism Koushik Pal (joint with Chris Laskowski) University of Maryland College Park 2012 ASL North American Annual Meeting April 2, 2012 Koushik Pal, UMCP Unstable Theories with an Automorphism 1 / 12
Basic Set-up Let L be a first-order language, and T be an L -theory. Let σ be a “new” unary function symbol, and let L σ := L ∪ { σ } . Let T σ := T ∪ { “ σ is an L -automorphism” } . Question: Does T σ have a model companion in L σ ? (If it does, we denote the model companion by T A .) Koushik Pal, UMCP Unstable Theories with an Automorphism 2 / 12
Basic Set-up Let L be a first-order language, and T be an L -theory. Let σ be a “new” unary function symbol, and let L σ := L ∪ { σ } . Let T σ := T ∪ { “ σ is an L -automorphism” } . Question: Does T σ have a model companion in L σ ? (If it does, we denote the model companion by T A .) Koushik Pal, UMCP Unstable Theories with an Automorphism 2 / 12
Basic Set-up Let L be a first-order language, and T be an L -theory. Let σ be a “new” unary function symbol, and let L σ := L ∪ { σ } . Let T σ := T ∪ { “ σ is an L -automorphism” } . Question: Does T σ have a model companion in L σ ? (If it does, we denote the model companion by T A .) Koushik Pal, UMCP Unstable Theories with an Automorphism 2 / 12
Basic Set-up Let L be a first-order language, and T be an L -theory. Let σ be a “new” unary function symbol, and let L σ := L ∪ { σ } . Let T σ := T ∪ { “ σ is an L -automorphism” } . Question: Does T σ have a model companion in L σ ? (If it does, we denote the model companion by T A .) Koushik Pal, UMCP Unstable Theories with an Automorphism 2 / 12
Basic Set-up Let L be a first-order language, and T be an L -theory. Let σ be a “new” unary function symbol, and let L σ := L ∪ { σ } . Let T σ := T ∪ { “ σ is an L -automorphism” } . Question: Does T σ have a model companion in L σ ? (If it does, we denote the model companion by T A .) Koushik Pal, UMCP Unstable Theories with an Automorphism 2 / 12
History Theorem (Kikyo, 2000) If T is unstable without IP, then T A does not exist. Theorem (Kikyo-Shelah, 2002) If T has SOP, then T A does not exist. Theorem (Kudaibergenov, ????) If T is stable and has the fcp, then T A does not exist. Theorem (Baldwin-Shelah, 2003) If T is stable, then T A exists iff T does not admit obstructions. Open Problem: What happens if T has IP? Koushik Pal, UMCP Unstable Theories with an Automorphism 3 / 12
History Theorem (Kikyo, 2000) If T is unstable without IP, then T A does not exist. Theorem (Kikyo-Shelah, 2002) If T has SOP, then T A does not exist. Theorem (Kudaibergenov, ????) If T is stable and has the fcp, then T A does not exist. Theorem (Baldwin-Shelah, 2003) If T is stable, then T A exists iff T does not admit obstructions. Open Problem: What happens if T has IP? Koushik Pal, UMCP Unstable Theories with an Automorphism 3 / 12
History Theorem (Kikyo, 2000) If T is unstable without IP, then T A does not exist. Theorem (Kikyo-Shelah, 2002) If T has SOP, then T A does not exist. Theorem (Kudaibergenov, ????) If T is stable and has the fcp, then T A does not exist. Theorem (Baldwin-Shelah, 2003) If T is stable, then T A exists iff T does not admit obstructions. Open Problem: What happens if T has IP? Koushik Pal, UMCP Unstable Theories with an Automorphism 3 / 12
History Theorem (Kikyo, 2000) If T is unstable without IP, then T A does not exist. Theorem (Kikyo-Shelah, 2002) If T has SOP, then T A does not exist. Theorem (Kudaibergenov, ????) If T is stable and has the fcp, then T A does not exist. Theorem (Baldwin-Shelah, 2003) If T is stable, then T A exists iff T does not admit obstructions. Open Problem: What happens if T has IP? Koushik Pal, UMCP Unstable Theories with an Automorphism 3 / 12
History Theorem (Kikyo, 2000) If T is unstable without IP, then T A does not exist. Theorem (Kikyo-Shelah, 2002) If T has SOP, then T A does not exist. Theorem (Kudaibergenov, ????) If T is stable and has the fcp, then T A does not exist. Theorem (Baldwin-Shelah, 2003) If T is stable, then T A exists iff T does not admit obstructions. Open Problem: What happens if T has IP? Koushik Pal, UMCP Unstable Theories with an Automorphism 3 / 12
Kikyo-Shelah Theorem (Kikyo-Shelah, 2002) If T has SOP, then T A does not exist. Proof Sketch. Let ( M , σ ) | = T σ and � a i : i < ω � in M satisfy a i < a i +1 = σ ( a i ). Assuming T A exists, extend ( M , σ ) to a sufficiently saturated model ( N , σ ) of T A . Let p ( x ) := { x > a i : i < ω } and ψ ( x ) := ∃ y ( a 0 < σ ( y ) < y < x ). In ( N , σ ), 1 p ( x ) ⊢ ψ ( x ) 2 if q ( x ) is a finite subset of p ( x ), then q ( x ) �⊢ ψ ( x ). This is a contradiction to the saturation of ( N , σ ). Koushik Pal, UMCP Unstable Theories with an Automorphism 4 / 12
Kikyo-Shelah Theorem (Kikyo-Shelah, 2002) If T has SOP, then T A does not exist. Proof Sketch. Let ( M , σ ) | = T σ and � a i : i < ω � in M satisfy a i < a i +1 = σ ( a i ). Assuming T A exists, extend ( M , σ ) to a sufficiently saturated model ( N , σ ) of T A . Let p ( x ) := { x > a i : i < ω } and ψ ( x ) := ∃ y ( a 0 < σ ( y ) < y < x ). In ( N , σ ), 1 p ( x ) ⊢ ψ ( x ) 2 if q ( x ) is a finite subset of p ( x ), then q ( x ) �⊢ ψ ( x ). This is a contradiction to the saturation of ( N , σ ). Koushik Pal, UMCP Unstable Theories with an Automorphism 4 / 12
Kikyo-Shelah Theorem (Kikyo-Shelah, 2002) If T has SOP, then T A does not exist. Proof Sketch. Let ( M , σ ) | = T σ and � a i : i < ω � in M satisfy a i < a i +1 = σ ( a i ). Assuming T A exists, extend ( M , σ ) to a sufficiently saturated model ( N , σ ) of T A . Let p ( x ) := { x > a i : i < ω } and ψ ( x ) := ∃ y ( a 0 < σ ( y ) < y < x ). In ( N , σ ), 1 p ( x ) ⊢ ψ ( x ) 2 if q ( x ) is a finite subset of p ( x ), then q ( x ) �⊢ ψ ( x ). This is a contradiction to the saturation of ( N , σ ). Koushik Pal, UMCP Unstable Theories with an Automorphism 4 / 12
Kikyo-Shelah Theorem (Kikyo-Shelah, 2002) If T has SOP, then T A does not exist. Proof Sketch. Let ( M , σ ) | = T σ and � a i : i < ω � in M satisfy a i < a i +1 = σ ( a i ). Assuming T A exists, extend ( M , σ ) to a sufficiently saturated model ( N , σ ) of T A . Let p ( x ) := { x > a i : i < ω } and ψ ( x ) := ∃ y ( a 0 < σ ( y ) < y < x ). In ( N , σ ), 1 p ( x ) ⊢ ψ ( x ) 2 if q ( x ) is a finite subset of p ( x ), then q ( x ) �⊢ ψ ( x ). This is a contradiction to the saturation of ( N , σ ). Koushik Pal, UMCP Unstable Theories with an Automorphism 4 / 12
Kikyo-Shelah Theorem (Kikyo-Shelah, 2002) If T has SOP, then T A does not exist. Proof Sketch. Let ( M , σ ) | = T σ and � a i : i < ω � in M satisfy a i < a i +1 = σ ( a i ). Assuming T A exists, extend ( M , σ ) to a sufficiently saturated model ( N , σ ) of T A . Let p ( x ) := { x > a i : i < ω } and ψ ( x ) := ∃ y ( a 0 < σ ( y ) < y < x ). In ( N , σ ), 1 p ( x ) ⊢ ψ ( x ) 2 if q ( x ) is a finite subset of p ( x ), then q ( x ) �⊢ ψ ( x ). This is a contradiction to the saturation of ( N , σ ). Koushik Pal, UMCP Unstable Theories with an Automorphism 4 / 12
Kikyo-Shelah Theorem (Kikyo-Shelah, 2002) If T has SOP, then T A does not exist. Proof Sketch. Let ( M , σ ) | = T σ and � a i : i < ω � in M satisfy a i < a i +1 = σ ( a i ). Assuming T A exists, extend ( M , σ ) to a sufficiently saturated model ( N , σ ) of T A . Let p ( x ) := { x > a i : i < ω } and ψ ( x ) := ∃ y ( a 0 < σ ( y ) < y < x ). In ( N , σ ), 1 p ( x ) ⊢ ψ ( x ) 2 if q ( x ) is a finite subset of p ( x ), then q ( x ) �⊢ ψ ( x ). This is a contradiction to the saturation of ( N , σ ). Koushik Pal, UMCP Unstable Theories with an Automorphism 4 / 12
Linear Order Definition Let L be a linear order in the language L O := { < } . An L O -automorphism σ of L is called increasing if ∀ x ( x < σ ( x )). Definition Let LO + σ (DLO + σ ) denote the L O ,σ -theory of (dense) linear orders together with the axioms denoting “ σ is an increasing L O -automorphism”. Theorem (P.) LO + σ has a model companion (namely DLO + σ ) in L O ,σ . Moreover, DLO + σ eliminates quantifiers and is o-minimal. Koushik Pal, UMCP Unstable Theories with an Automorphism 5 / 12
Linear Order Definition Let L be a linear order in the language L O := { < } . An L O -automorphism σ of L is called increasing if ∀ x ( x < σ ( x )). Definition Let LO + σ (DLO + σ ) denote the L O ,σ -theory of (dense) linear orders together with the axioms denoting “ σ is an increasing L O -automorphism”. Theorem (P.) LO + σ has a model companion (namely DLO + σ ) in L O ,σ . Moreover, DLO + σ eliminates quantifiers and is o-minimal. Koushik Pal, UMCP Unstable Theories with an Automorphism 5 / 12
Linear Order Definition Let L be a linear order in the language L O := { < } . An L O -automorphism σ of L is called increasing if ∀ x ( x < σ ( x )). Definition Let LO + σ (DLO + σ ) denote the L O ,σ -theory of (dense) linear orders together with the axioms denoting “ σ is an increasing L O -automorphism”. Theorem (P.) LO + σ has a model companion (namely DLO + σ ) in L O ,σ . Moreover, DLO + σ eliminates quantifiers and is o-minimal. Koushik Pal, UMCP Unstable Theories with an Automorphism 5 / 12
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