Absolute notions in model theory Syntactic and semantic notions - PowerPoint PPT Presentation

Absolute notions in Kinds of things one can add by forcing model theory Mirna Dˇ zamonja Introduction Call the model you start with (“the ground model”) V and In model theory the extension V [ G ] . We have V [ G ] ⊇ V . Syntactic and semantic notions This is the kind of objects that can appear in V [ G ] even if Absolutness from model theory in set they did not exists in V : theory (Non)-absolutness A new function from ω → 2, or ℵ V from set theory in 2 many of them (so model theory 2 = ℵ V [ G ] if ℵ V we violate CH) 2 A new branch to a tree of height ω V 1 (“killing” a Souslin tree, for example) A new surjection from ω to ω V 1 , (so we “collapse” ω 1 i.e. ω V [ G ] � = ω V 1 ). 1

Absolute notions in model theory Mirna Dˇ zamonja Introduction In model theory These observations ask for a reflection on the Syntactic and absoluteness of notions in model theory: for example, semantic notions Absolutness from can we make two models isomorphic if they were not model theory in set theory before? (Non)-absolutness from set theory in model theory

Absolute notions in model theory Mirna Dˇ zamonja Introduction In model theory These observations ask for a reflection on the Syntactic and absoluteness of notions in model theory: for example, semantic notions Absolutness from can we make two models isomorphic if they were not model theory in set theory before? Can we change the fact that some kind of “nice” (Non)-absolutness model exist? from set theory in model theory

Absolute notions in model theory Mirna Dˇ zamonja Introduction In model theory These observations ask for a reflection on the Syntactic and absoluteness of notions in model theory: for example, semantic notions Absolutness from can we make two models isomorphic if they were not model theory in set theory before? Can we change the fact that some kind of “nice” (Non)-absolutness model exist? Saturated, universal ... from set theory in model theory

Absolute notions in model theory Mirna Dˇ zamonja Introduction In model theory These observations ask for a reflection on the Syntactic and absoluteness of notions in model theory: for example, semantic notions Absolutness from can we make two models isomorphic if they were not model theory in set theory before? Can we change the fact that some kind of “nice” (Non)-absolutness model exist? Saturated, universal ... from set theory in model theory Here are some words on this from Gerald Sacks, the inventor of Sacks forcing (in 1971), from his book “Saturated Model Theory” in 1972.

Absolute notions in model theory Mirna Dˇ zamonja Introduction In model theory Syntactic and semantic notions Absolutness from model theory in set theory (Non)-absolutness from set theory in model theory

Absolute notions in model theory Łos conjecture was what is now known as Morley’s Mirna Dˇ zamonja theorem: Introduction In model theory Syntactic and semantic notions Absolutness from model theory in set theory (Non)-absolutness from set theory in model theory

Absolute notions in model theory Łos conjecture was what is now known as Morley’s Mirna Dˇ zamonja theorem: Introduction In model theory Theorem (Morley 1965) Syntactic and A (countable complete first order) theory T is categorical semantic notions Absolutness from in some uncountable power iff it is categorical in every model theory in set theory uncountable power. (Non)-absolutness from set theory in model theory

Absolute notions in model theory Łos conjecture was what is now known as Morley’s Mirna Dˇ zamonja theorem: Introduction In model theory Theorem (Morley 1965) Syntactic and A (countable complete first order) theory T is categorical semantic notions Absolutness from in some uncountable power iff it is categorical in every model theory in set theory uncountable power. (Non)-absolutness from set theory in It follows immediately that being uncountably categorical model theory cannot be changed by set forcing, since for such a forcing there will be a cardinal κ such that V and V [ G ] agree on all statements about object of size > κ .

Absolute notions in model theory Łos conjecture was what is now known as Morley’s Mirna Dˇ zamonja theorem: Introduction In model theory Theorem (Morley 1965) Syntactic and A (countable complete first order) theory T is categorical semantic notions Absolutness from in some uncountable power iff it is categorical in every model theory in set theory uncountable power. (Non)-absolutness from set theory in It follows immediately that being uncountably categorical model theory cannot be changed by set forcing, since for such a forcing there will be a cardinal κ such that V and V [ G ] agree on all statements about object of size > κ . To prove this theorem Morley introduced the idea of a rank , which is a measurable, absolute way to handle the formulas of a theory, and it is really because of ranks that categoricity is absolute, as we proceed to explain.

Absolute notions in Semantic versus Syntactic model theory Mirna Dˇ zamonja Morley’s theorem and subsequent huge amount of work Introduction by Shelah in his “Classification theory” is based upon a In model theory philosophy of connecting semantic notions with syntactic Syntactic and ones. semantic notions Absolutness from model theory in set theory (Non)-absolutness from set theory in model theory

Absolute notions in Semantic versus Syntactic model theory Mirna Dˇ zamonja Morley’s theorem and subsequent huge amount of work Introduction by Shelah in his “Classification theory” is based upon a In model theory philosophy of connecting semantic notions with syntactic Syntactic and ones. The number of pairwise non-isomorphic models of semantic notions Absolutness from a theory is a semantic notion, it talks about models. model theory in set theory (Non)-absolutness from set theory in model theory

Absolute notions in Semantic versus Syntactic model theory Mirna Dˇ zamonja Morley’s theorem and subsequent huge amount of work Introduction by Shelah in his “Classification theory” is based upon a In model theory philosophy of connecting semantic notions with syntactic Syntactic and ones. The number of pairwise non-isomorphic models of semantic notions Absolutness from a theory is a semantic notion, it talks about models. model theory in set theory Morley rank is a syntactic notion, it talks about formulas. (Non)-absolutness from set theory in model theory

Absolute notions in Semantic versus Syntactic model theory Mirna Dˇ zamonja Morley’s theorem and subsequent huge amount of work Introduction by Shelah in his “Classification theory” is based upon a In model theory philosophy of connecting semantic notions with syntactic Syntactic and ones. The number of pairwise non-isomorphic models of semantic notions Absolutness from a theory is a semantic notion, it talks about models. model theory in set theory Morley rank is a syntactic notion, it talks about formulas. (Non)-absolutness Morley’s original proof was based upon an analysis of from set theory in model theory Morley’s rank of formulas.

Absolute notions in Semantic versus Syntactic model theory Mirna Dˇ zamonja Morley’s theorem and subsequent huge amount of work Introduction by Shelah in his “Classification theory” is based upon a In model theory philosophy of connecting semantic notions with syntactic Syntactic and ones. The number of pairwise non-isomorphic models of semantic notions Absolutness from a theory is a semantic notion, it talks about models. model theory in set theory Morley rank is a syntactic notion, it talks about formulas. (Non)-absolutness Morley’s original proof was based upon an analysis of from set theory in model theory Morley’s rank of formulas. Observation : Syntactic notions in first order theories tend to be absolute because of the compactness theorem.

Absolute notions in Semantic versus Syntactic model theory Mirna Dˇ zamonja Morley’s theorem and subsequent huge amount of work Introduction by Shelah in his “Classification theory” is based upon a In model theory philosophy of connecting semantic notions with syntactic Syntactic and ones. The number of pairwise non-isomorphic models of semantic notions Absolutness from a theory is a semantic notion, it talks about models. model theory in set theory Morley rank is a syntactic notion, it talks about formulas. (Non)-absolutness Morley’s original proof was based upon an analysis of from set theory in model theory Morley’s rank of formulas. Observation : Syntactic notions in first order theories tend to be absolute because of the compactness theorem. We shall not go into the Morley rank, but let us give an example of a syntactic notion which will be relevant to us and show why it is absolute.

Absolute notions in Semantic versus Syntactic model theory Mirna Dˇ zamonja Morley’s theorem and subsequent huge amount of work Introduction by Shelah in his “Classification theory” is based upon a In model theory philosophy of connecting semantic notions with syntactic Syntactic and ones. The number of pairwise non-isomorphic models of semantic notions Absolutness from a theory is a semantic notion, it talks about models. model theory in set theory Morley rank is a syntactic notion, it talks about formulas. (Non)-absolutness Morley’s original proof was based upon an analysis of from set theory in model theory Morley’s rank of formulas. Observation : Syntactic notions in first order theories tend to be absolute because of the compactness theorem. We shall not go into the Morley rank, but let us give an example of a syntactic notion which will be relevant to us and show why it is absolute. In order to do this, we first have to talk about saturated models.

Absolute notions in Saturated models model theory Mirna Dˇ zamonja Introduction A model of a theory T is said to be κ -saturated if it In model theory realizes all (consistent) types of T of size < κ . Syntactic and semantic notions Absolutness from model theory in set theory (Non)-absolutness from set theory in model theory

Absolute notions in Saturated models model theory Mirna Dˇ zamonja Introduction A model of a theory T is said to be κ -saturated if it In model theory realizes all (consistent) types of T of size < κ . It is easy Syntactic and semantic notions to see that κ -saturated models are unique up to the Absolutness from cardinality (The Uniqueness Theorem) model theory in set theory (Non)-absolutness from set theory in model theory

Absolute notions in Saturated models model theory Mirna Dˇ zamonja Introduction A model of a theory T is said to be κ -saturated if it In model theory realizes all (consistent) types of T of size < κ . It is easy Syntactic and semantic notions to see that κ -saturated models are unique up to the Absolutness from cardinality (The Uniqueness Theorem) and using the model theory in set theory compactness theorem, we can prove that for every λ with λ = λ <κ there is a κ -saturated model. (Non)-absolutness from set theory in model theory

Absolute notions in Saturated models model theory Mirna Dˇ zamonja Introduction A model of a theory T is said to be κ -saturated if it In model theory realizes all (consistent) types of T of size < κ . It is easy Syntactic and semantic notions to see that κ -saturated models are unique up to the Absolutness from cardinality (The Uniqueness Theorem) and using the model theory in set theory compactness theorem, we can prove that for every λ with λ = λ <κ there is a κ -saturated model. (Non)-absolutness from set theory in model theory On the next slide we shall see a typical example of a syntactic notion defined using a saturated model.

Absolute notions in Saturated models model theory Mirna Dˇ zamonja Introduction A model of a theory T is said to be κ -saturated if it In model theory realizes all (consistent) types of T of size < κ . It is easy Syntactic and semantic notions to see that κ -saturated models are unique up to the Absolutness from cardinality (The Uniqueness Theorem) and using the model theory in set theory compactness theorem, we can prove that for every λ with λ = λ <κ there is a κ -saturated model. (Non)-absolutness from set theory in model theory On the next slide we shall see a typical example of a syntactic notion defined using a saturated model. Because the properties like this one do not depend on the choice of the sufficiently saturated model, we work with any fixed such model, which we denote by C T (monster model).

Absolute notions in SOP 2 model theory Mirna Dˇ zamonja Definition Introduction T has SOP 2 if there is a formula ϕ (¯ x , ¯ y ) which In model theory exemplifies this property in C = C T , which means: Syntactic and There are ¯ a η ∈ C for η ∈ ω> 2 such that semantic notions Absolutness from (a) for every ρ ∈ ω 2, the set model theory in set theory { ϕ (¯ x , ¯ a ρ ↾ n ) : n < ω } is consistent, (Non)-absolutness (b) if η, ν ∈ ω> 2 are incomparable, from set theory in model theory { ϕ (¯ x , ¯ a η ) , ϕ (¯ x , ¯ a ν ) } is inconsistent.

Absolute notions in SOP 2 model theory Mirna Dˇ zamonja Definition Introduction T has SOP 2 if there is a formula ϕ (¯ x , ¯ y ) which In model theory exemplifies this property in C = C T , which means: Syntactic and There are ¯ a η ∈ C for η ∈ ω> 2 such that semantic notions Absolutness from (a) for every ρ ∈ ω 2, the set model theory in set theory { ϕ (¯ x , ¯ a ρ ↾ n ) : n < ω } is consistent, (Non)-absolutness (b) if η, ν ∈ ω> 2 are incomparable, from set theory in model theory { ϕ (¯ x , ¯ a η ) , ϕ (¯ x , ¯ a ν ) } is inconsistent. Properties like this one (but not yet this one) have been shown to be equivalent to semantic notions, through the work in classification theory.

Absolute notions in SOP 2 model theory Mirna Dˇ zamonja Definition Introduction T has SOP 2 if there is a formula ϕ (¯ x , ¯ y ) which In model theory exemplifies this property in C = C T , which means: Syntactic and There are ¯ a η ∈ C for η ∈ ω> 2 such that semantic notions Absolutness from (a) for every ρ ∈ ω 2, the set model theory in set theory { ϕ (¯ x , ¯ a ρ ↾ n ) : n < ω } is consistent, (Non)-absolutness (b) if η, ν ∈ ω> 2 are incomparable, from set theory in model theory { ϕ (¯ x , ¯ a η ) , ϕ (¯ x , ¯ a ν ) } is inconsistent. Properties like this one (but not yet this one) have been shown to be equivalent to semantic notions, through the work in classification theory. For example, order properties, even as week as this one, imply the maximal possible number of non-isomorphic models at large enough cardinals.

Absolute notions in SOP 2 model theory Mirna Dˇ zamonja Definition Introduction T has SOP 2 if there is a formula ϕ (¯ x , ¯ y ) which In model theory exemplifies this property in C = C T , which means: Syntactic and There are ¯ a η ∈ C for η ∈ ω> 2 such that semantic notions Absolutness from (a) for every ρ ∈ ω 2, the set model theory in set theory { ϕ (¯ x , ¯ a ρ ↾ n ) : n < ω } is consistent, (Non)-absolutness (b) if η, ν ∈ ω> 2 are incomparable, from set theory in model theory { ϕ (¯ x , ¯ a η ) , ϕ (¯ x , ¯ a ν ) } is inconsistent. Properties like this one (but not yet this one) have been shown to be equivalent to semantic notions, through the work in classification theory. For example, order properties, even as week as this one, imply the maximal possible number of non-isomorphic models at large enough cardinals. We shall consider a semantic notion connected with SOP 2 .

Absolute notions in Keisler order model theory Mirna Dˇ zamonja Definition Introduction (1) For any cardinal λ , the Keisler order ⊳ λ among In model theory theories is defined as follows: Syntactic and semantic notions Absolutness from model theory in set theory (Non)-absolutness from set theory in model theory

Absolute notions in Keisler order model theory Mirna Dˇ zamonja Definition Introduction (1) For any cardinal λ , the Keisler order ⊳ λ among In model theory theories is defined as follows: Syntactic and semantic notions T 0 ⊳ λ T 1 if whenever M l is a model of T l ( l < 2 ) Absolutness from model theory in set theory (Non)-absolutness from set theory in model theory

Absolute notions in Keisler order model theory Mirna Dˇ zamonja Definition Introduction (1) For any cardinal λ , the Keisler order ⊳ λ among In model theory theories is defined as follows: Syntactic and semantic notions T 0 ⊳ λ T 1 if whenever M l is a model of T l ( l < 2 ) and D is Absolutness from model theory in set a regular ultrafilter over λ theory (Non)-absolutness from set theory in model theory

Absolute notions in Keisler order model theory Mirna Dˇ zamonja Definition Introduction (1) For any cardinal λ , the Keisler order ⊳ λ among In model theory theories is defined as follows: Syntactic and semantic notions T 0 ⊳ λ T 1 if whenever M l is a model of T l ( l < 2 ) and D is Absolutness from model theory in set a regular ultrafilter over λ then the λ + -saturation of M λ 1 / D theory implies the λ + -saturation of M λ 0 / D . (Non)-absolutness from set theory in model theory

Absolute notions in Keisler order model theory Mirna Dˇ zamonja Definition Introduction (1) For any cardinal λ , the Keisler order ⊳ λ among In model theory theories is defined as follows: Syntactic and semantic notions T 0 ⊳ λ T 1 if whenever M l is a model of T l ( l < 2 ) and D is Absolutness from model theory in set a regular ultrafilter over λ then the λ + -saturation of M λ 1 / D theory implies the λ + -saturation of M λ 0 / D . (Non)-absolutness from set theory in model theory (2) We say T 0 ⊳ T 1 if for all λ we have T 0 ⊳ λ T 1 .

Absolute notions in Keisler order model theory Mirna Dˇ zamonja Definition Introduction (1) For any cardinal λ , the Keisler order ⊳ λ among In model theory theories is defined as follows: Syntactic and semantic notions T 0 ⊳ λ T 1 if whenever M l is a model of T l ( l < 2 ) and D is Absolutness from model theory in set a regular ultrafilter over λ then the λ + -saturation of M λ 1 / D theory implies the λ + -saturation of M λ 0 / D . (Non)-absolutness from set theory in model theory (2) We say T 0 ⊳ T 1 if for all λ we have T 0 ⊳ λ T 1 . This order was introduced in the 1960s by Keisler and was later used by him and Shelah to complement the classification theory offered by stability.

Absolute notions in Keisler order model theory Mirna Dˇ zamonja Definition Introduction (1) For any cardinal λ , the Keisler order ⊳ λ among In model theory theories is defined as follows: Syntactic and semantic notions T 0 ⊳ λ T 1 if whenever M l is a model of T l ( l < 2 ) and D is Absolutness from model theory in set a regular ultrafilter over λ then the λ + -saturation of M λ 1 / D theory implies the λ + -saturation of M λ 0 / D . (Non)-absolutness from set theory in model theory (2) We say T 0 ⊳ T 1 if for all λ we have T 0 ⊳ λ T 1 . This order was introduced in the 1960s by Keisler and was later used by him and Shelah to complement the classification theory offered by stability. Keisler proved that having the strict order property implies that a theory is maximal in this order.

Absolute notions in Keisler order model theory Mirna Dˇ zamonja Definition Introduction (1) For any cardinal λ , the Keisler order ⊳ λ among In model theory theories is defined as follows: Syntactic and semantic notions T 0 ⊳ λ T 1 if whenever M l is a model of T l ( l < 2 ) and D is Absolutness from model theory in set a regular ultrafilter over λ then the λ + -saturation of M λ 1 / D theory implies the λ + -saturation of M λ 0 / D . (Non)-absolutness from set theory in model theory (2) We say T 0 ⊳ T 1 if for all λ we have T 0 ⊳ λ T 1 . This order was introduced in the 1960s by Keisler and was later used by him and Shelah to complement the classification theory offered by stability. Keisler proved that having the strict order property implies that a theory is maximal in this order. Note that this is semantic notion and there is no a priori reason why it should be absolute.

Absolute notions in model theory In a paper continuing a long list of work and in which they Mirna Dˇ zamonja used a completely original idea, Malliaris and Shelah (2013) showed that being SOP 2 suffices for maximality in Introduction In model theory Keisler’s order! Syntactic and semantic notions Absolutness from model theory in set theory (Non)-absolutness from set theory in model theory

Absolute notions in model theory In a paper continuing a long list of work and in which they Mirna Dˇ zamonja used a completely original idea, Malliaris and Shelah (2013) showed that being SOP 2 suffices for maximality in Introduction In model theory Keisler’s order! Syntactic and semantic notions In the same paper and using the same circle of ideas, Absolutness from Malliaris and Shelah solved a seemingly unrelated model theory in set theory problem, posed by Hausdorff in 1936: the equality (Non)-absolutness between two cardinal invariants of the continuum, namely from set theory in model theory p = t .

Absolute notions in model theory In a paper continuing a long list of work and in which they Mirna Dˇ zamonja used a completely original idea, Malliaris and Shelah (2013) showed that being SOP 2 suffices for maximality in Introduction In model theory Keisler’s order! Syntactic and semantic notions In the same paper and using the same circle of ideas, Absolutness from Malliaris and Shelah solved a seemingly unrelated model theory in set theory problem, posed by Hausdorff in 1936: the equality (Non)-absolutness between two cardinal invariants of the continuum, namely from set theory in model theory p = t . Although it is not necessary for us to have a definition of these invariants, the remarkable thing is that they are two out of at least 50 invariants known, each other pair having been shown independent by the method of forcing in the 1980s or so!

Absolute notions in model theory In a paper continuing a long list of work and in which they Mirna Dˇ zamonja used a completely original idea, Malliaris and Shelah (2013) showed that being SOP 2 suffices for maximality in Introduction In model theory Keisler’s order! Syntactic and semantic notions In the same paper and using the same circle of ideas, Absolutness from Malliaris and Shelah solved a seemingly unrelated model theory in set theory problem, posed by Hausdorff in 1936: the equality (Non)-absolutness between two cardinal invariants of the continuum, namely from set theory in model theory p = t . Although it is not necessary for us to have a definition of these invariants, the remarkable thing is that they are two out of at least 50 invariants known, each other pair having been shown independent by the method of forcing in the 1980s or so! This one was an unsolved puzzle and we now know why.

Absolute notions in model theory In a paper continuing a long list of work and in which they Mirna Dˇ zamonja used a completely original idea, Malliaris and Shelah (2013) showed that being SOP 2 suffices for maximality in Introduction In model theory Keisler’s order! Syntactic and semantic notions In the same paper and using the same circle of ideas, Absolutness from Malliaris and Shelah solved a seemingly unrelated model theory in set theory problem, posed by Hausdorff in 1936: the equality (Non)-absolutness between two cardinal invariants of the continuum, namely from set theory in model theory p = t . Although it is not necessary for us to have a definition of these invariants, the remarkable thing is that they are two out of at least 50 invariants known, each other pair having been shown independent by the method of forcing in the 1980s or so! This one was an unsolved puzzle and we now know why. p = t is absolute because it is connected to SOP 2 , which is absolute.

Absolute notions in Interpretrability model theory Mirna Dˇ zamonja How about the converse to the Malliaris-Shelah result? Introduction In model theory Syntactic and semantic notions Absolutness from model theory in set theory (Non)-absolutness from set theory in model theory

Absolute notions in Interpretrability model theory Mirna Dˇ zamonja How about the converse to the Malliaris-Shelah result? If Introduction a theory T is maximal in Keisler’s order, does it have In model theory SOP 2 ? Syntactic and semantic notions Absolutness from model theory in set theory (Non)-absolutness from set theory in model theory

Absolute notions in Interpretrability model theory Mirna Dˇ zamonja How about the converse to the Malliaris-Shelah result? If Introduction a theory T is maximal in Keisler’s order, does it have In model theory SOP 2 ? This would be the actual equivalence between Syntactic and semantic notions semantic and syntax and would prove the maximality in Absolutness from model theory in set Keisler’s order absolute. theory (Non)-absolutness from set theory in model theory

Absolute notions in Interpretrability model theory Mirna Dˇ zamonja How about the converse to the Malliaris-Shelah result? If Introduction a theory T is maximal in Keisler’s order, does it have In model theory SOP 2 ? This would be the actual equivalence between Syntactic and semantic notions semantic and syntax and would prove the maximality in Absolutness from model theory in set Keisler’s order absolute. theory This question is open . (Non)-absolutness from set theory in model theory

Absolute notions in Interpretrability model theory Mirna Dˇ zamonja How about the converse to the Malliaris-Shelah result? If Introduction a theory T is maximal in Keisler’s order, does it have In model theory SOP 2 ? This would be the actual equivalence between Syntactic and semantic notions semantic and syntax and would prove the maximality in Absolutness from model theory in set Keisler’s order absolute. theory This question is open . The best partial result known (Non)-absolutness from set theory in model theory comes from a combination of results in a paper by Dˇ z.-Shelah (2004) and a paper by Shelah and Usvyatsov (2008), as we shall now explain.

Absolute notions in Interpretrability model theory Mirna Dˇ zamonja How about the converse to the Malliaris-Shelah result? If Introduction a theory T is maximal in Keisler’s order, does it have In model theory SOP 2 ? This would be the actual equivalence between Syntactic and semantic notions semantic and syntax and would prove the maximality in Absolutness from model theory in set Keisler’s order absolute. theory This question is open . The best partial result known (Non)-absolutness from set theory in model theory comes from a combination of results in a paper by Dˇ z.-Shelah (2004) and a paper by Shelah and Usvyatsov (2008), as we shall now explain. This work concerns the interpretability order ⊳ ∗ , defined again using ultrapowers, and which is such that:

Absolute notions in Interpretrability model theory Mirna Dˇ zamonja How about the converse to the Malliaris-Shelah result? If Introduction a theory T is maximal in Keisler’s order, does it have In model theory SOP 2 ? This would be the actual equivalence between Syntactic and semantic notions semantic and syntax and would prove the maximality in Absolutness from model theory in set Keisler’s order absolute. theory This question is open . The best partial result known (Non)-absolutness from set theory in model theory comes from a combination of results in a paper by Dˇ z.-Shelah (2004) and a paper by Shelah and Usvyatsov (2008), as we shall now explain. This work concerns the interpretability order ⊳ ∗ , defined again using ultrapowers, and which is such that: Lemma A theory which is ⊳ ∗ -maximal is also ⊳ -maximal.

Absolute notions in model theory Mirna Dˇ zamonja Introduction In model theory Syntactic and semantic notions Absolutness from model theory in set theory (Non)-absolutness from set theory in model theory

Absolute notions in model theory Mirna Dˇ zamonja The theorem is then: Introduction Theorem (Dˇ z.-Shelah + Shelah-Usvyatsov) In model theory Syntactic and If a theory T is ⊳ ∗ -maximal in some model of GCH, then semantic notions it has SOP 2 . Absolutness from model theory in set theory (Non)-absolutness from set theory in model theory

Absolute notions in model theory Mirna Dˇ zamonja The theorem is then: Introduction Theorem (Dˇ z.-Shelah + Shelah-Usvyatsov) In model theory Syntactic and If a theory T is ⊳ ∗ -maximal in some model of GCH, then semantic notions it has SOP 2 . Absolutness from model theory in set theory This leaves open a (relatively minor) question of the (Non)-absolutness from set theory in connection between ⊳ and ⊳ ∗ -maximality (all indications model theory are that they are the same, see a recent paper by Malliaris and Shelah) and

Absolute notions in model theory Mirna Dˇ zamonja The theorem is then: Introduction Theorem (Dˇ z.-Shelah + Shelah-Usvyatsov) In model theory Syntactic and If a theory T is ⊳ ∗ -maximal in some model of GCH, then semantic notions it has SOP 2 . Absolutness from model theory in set theory This leaves open a (relatively minor) question of the (Non)-absolutness from set theory in connection between ⊳ and ⊳ ∗ -maximality (all indications model theory are that they are the same, see a recent paper by Malliaris and Shelah) and a major question: is GCH necessary?

Absolute notions in model theory Mirna Dˇ zamonja The theorem is then: Introduction Theorem (Dˇ z.-Shelah + Shelah-Usvyatsov) In model theory Syntactic and If a theory T is ⊳ ∗ -maximal in some model of GCH, then semantic notions it has SOP 2 . Absolutness from model theory in set theory This leaves open a (relatively minor) question of the (Non)-absolutness from set theory in connection between ⊳ and ⊳ ∗ -maximality (all indications model theory are that they are the same, see a recent paper by Malliaris and Shelah) and a major question: is GCH necessary? If we prove set theoretically that GCH was not necessary, then we will have a confirmation of it model-theoretically as SOP 2 is absolute, and vice versa.

Syntactic ⇐ ⇒ semantic is a fragile fact Absolute notions in model theory Mirna Dˇ zamonja Introduction In model theory The equivalence between semantic and syntactic notions Syntactic and semantic notions in first order model theory looks like a miracle (although Absolutness from many people take it for granted by looking only at model theory in set theory syntactic notions :-). (Non)-absolutness from set theory in model theory

Syntactic ⇐ ⇒ semantic is a fragile fact Absolute notions in model theory Mirna Dˇ zamonja Introduction In model theory The equivalence between semantic and syntactic notions Syntactic and semantic notions in first order model theory looks like a miracle (although Absolutness from many people take it for granted by looking only at model theory in set theory syntactic notions :-). That is because it is! (Non)-absolutness from set theory in model theory

Syntactic ⇐ ⇒ semantic is a fragile fact Absolute notions in model theory Mirna Dˇ zamonja Introduction In model theory The equivalence between semantic and syntactic notions Syntactic and semantic notions in first order model theory looks like a miracle (although Absolutness from many people take it for granted by looking only at model theory in set theory syntactic notions :-). That is because it is! Much of it (Non)-absolutness depends on the compactness and completeness of the from set theory in model theory first order logic, which is quite unique.

Syntactic ⇐ ⇒ semantic is a fragile fact Absolute notions in model theory Mirna Dˇ zamonja Introduction In model theory The equivalence between semantic and syntactic notions Syntactic and semantic notions in first order model theory looks like a miracle (although Absolutness from many people take it for granted by looking only at model theory in set theory syntactic notions :-). That is because it is! Much of it (Non)-absolutness depends on the compactness and completeness of the from set theory in model theory first order logic, which is quite unique. We shall now go through some examples of absolutness and non-absolutness in general model theory and finish by pointing out a (possible) example which if true, might be pleasing.

Absolute notions in Universal models and GCH model theory Let us start by contradicting Sacks :-) Mirna Dˇ zamonja Introduction In model theory Syntactic and semantic notions Absolutness from model theory in set theory (Non)-absolutness from set theory in model theory

Absolute notions in Universal models and GCH model theory Let us start by contradicting Sacks :-) and finding GCH Mirna Dˇ zamonja within the first order model theory. Introduction In model theory Syntactic and semantic notions Absolutness from model theory in set theory (Non)-absolutness from set theory in model theory

Absolute notions in Universal models and GCH model theory Let us start by contradicting Sacks :-) and finding GCH Mirna Dˇ zamonja within the first order model theory. Introduction M a model of T is universal in λ iff all models of T of In model theory power λ embed into M . Syntactic and semantic notions Absolutness from model theory in set theory (Non)-absolutness from set theory in model theory

Absolute notions in Universal models and GCH model theory Let us start by contradicting Sacks :-) and finding GCH Mirna Dˇ zamonja within the first order model theory. Introduction M a model of T is universal in λ iff all models of T of In model theory power λ embed into M . Syntactic and semantic notions Fact For countable f.o. T and λ <λ = λ > ℵ 0 , T has a Absolutness from model theory in set universal model in λ . theory (Non)-absolutness from set theory in model theory

Absolute notions in Universal models and GCH model theory Let us start by contradicting Sacks :-) and finding GCH Mirna Dˇ zamonja within the first order model theory. Introduction M a model of T is universal in λ iff all models of T of In model theory power λ embed into M . Syntactic and semantic notions Fact For countable f.o. T and λ <λ = λ > ℵ 0 , T has a Absolutness from model theory in set universal model in λ . theory We wish to classify theories by the class of cardinals λ for (Non)-absolutness from set theory in which there is a universal model at T independently of model theory the value of λ <λ .

Absolute notions in Universal models and GCH model theory Let us start by contradicting Sacks :-) and finding GCH Mirna Dˇ zamonja within the first order model theory. Introduction M a model of T is universal in λ iff all models of T of In model theory power λ embed into M . Syntactic and semantic notions Fact For countable f.o. T and λ <λ = λ > ℵ 0 , T has a Absolutness from model theory in set universal model in λ . theory We wish to classify theories by the class of cardinals λ for (Non)-absolutness from set theory in which there is a universal model at T independently of model theory the value of λ <λ . Long history, involving Shelah, Grossberg, Kojman, Dˇ z. and others.

Absolute notions in Universal models and GCH model theory Let us start by contradicting Sacks :-) and finding GCH Mirna Dˇ zamonja within the first order model theory. Introduction M a model of T is universal in λ iff all models of T of In model theory power λ embed into M . Syntactic and semantic notions Fact For countable f.o. T and λ <λ = λ > ℵ 0 , T has a Absolutness from model theory in set universal model in λ . theory We wish to classify theories by the class of cardinals λ for (Non)-absolutness from set theory in which there is a universal model at T independently of model theory the value of λ <λ . Long history, involving Shelah, Grossberg, Kojman, Dˇ z. and others. We quote a beautiful recent theorem by Shelah.

Absolute notions in Universal models and GCH model theory Let us start by contradicting Sacks :-) and finding GCH Mirna Dˇ zamonja within the first order model theory. Introduction M a model of T is universal in λ iff all models of T of In model theory power λ embed into M . Syntactic and semantic notions Fact For countable f.o. T and λ <λ = λ > ℵ 0 , T has a Absolutness from model theory in set universal model in λ . theory We wish to classify theories by the class of cardinals λ for (Non)-absolutness from set theory in which there is a universal model at T independently of model theory the value of λ <λ . Long history, involving Shelah, Grossberg, Kojman, Dˇ z. and others. We quote a beautiful recent theorem by Shelah. Theorem (Shelah, July 2017) There is a countable f.o. T ∗ such that T ∗ has a universal model in λ > ℵ 0 iff λ <λ = λ .

Absolute notions in Universal models and GCH model theory Let us start by contradicting Sacks :-) and finding GCH Mirna Dˇ zamonja within the first order model theory. Introduction M a model of T is universal in λ iff all models of T of In model theory power λ embed into M . Syntactic and semantic notions Fact For countable f.o. T and λ <λ = λ > ℵ 0 , T has a Absolutness from model theory in set universal model in λ . theory We wish to classify theories by the class of cardinals λ for (Non)-absolutness from set theory in which there is a universal model at T independently of model theory the value of λ <λ . Long history, involving Shelah, Grossberg, Kojman, Dˇ z. and others. We quote a beautiful recent theorem by Shelah. Theorem (Shelah, July 2017) There is a countable f.o. T ∗ such that T ∗ has a universal model in λ > ℵ 0 iff λ <λ = λ . So “GCH for uncountable cardinals” iff T ∗ has a universal model in every λ > ℵ 0 .

Absolute notions in L ω 1 ,ω model theory Mirna Dˇ zamonja Let now us consider L ω 1 ,ω , the logic which is like the first Introduction order logic but allows infinite ∧ . In model theory Syntactic and semantic notions Absolutness from model theory in set theory (Non)-absolutness from set theory in model theory

Absolute notions in L ω 1 ,ω model theory Mirna Dˇ zamonja Let now us consider L ω 1 ,ω , the logic which is like the first Introduction order logic but allows infinite ∧ . It can define well oder, so In model theory it is not compact. Syntactic and semantic notions Absolutness from model theory in set theory (Non)-absolutness from set theory in model theory

Absolute notions in L ω 1 ,ω model theory Mirna Dˇ zamonja Let now us consider L ω 1 ,ω , the logic which is like the first Introduction order logic but allows infinite ∧ . It can define well oder, so In model theory it is not compact. However, it is complete, by a proof of Syntactic and semantic notions Carol Karp from 1964. Absolutness from model theory in set theory (Non)-absolutness from set theory in model theory

Absolute notions in L ω 1 ,ω model theory Mirna Dˇ zamonja Let now us consider L ω 1 ,ω , the logic which is like the first Introduction order logic but allows infinite ∧ . It can define well oder, so In model theory it is not compact. However, it is complete, by a proof of Syntactic and semantic notions Carol Karp from 1964. Absolutness from model theory in set Question : Suppose that τ is some countable vocabulary theory and ϕ an L ω 1 ,ω -sentence in τ . Is the statement “ ϕ has an (Non)-absolutness from set theory in uncountable model” absolute? model theory

Absolute notions in L ω 1 ,ω model theory Mirna Dˇ zamonja Let now us consider L ω 1 ,ω , the logic which is like the first Introduction order logic but allows infinite ∧ . It can define well oder, so In model theory it is not compact. However, it is complete, by a proof of Syntactic and semantic notions Carol Karp from 1964. Absolutness from model theory in set Question : Suppose that τ is some countable vocabulary theory and ϕ an L ω 1 ,ω -sentence in τ . Is the statement “ ϕ has an (Non)-absolutness from set theory in uncountable model” absolute? model theory Recall that for the first order logic this follows by the Lowenheim-Skolem theorem, which implies that having any infinite model is equivalent to having models of any infinite cardinality, and having countable models is absolute.

Absolute notions in L ω 1 ,ω model theory Mirna Dˇ zamonja Let now us consider L ω 1 ,ω , the logic which is like the first Introduction order logic but allows infinite ∧ . It can define well oder, so In model theory it is not compact. However, it is complete, by a proof of Syntactic and semantic notions Carol Karp from 1964. Absolutness from model theory in set Question : Suppose that τ is some countable vocabulary theory and ϕ an L ω 1 ,ω -sentence in τ . Is the statement “ ϕ has an (Non)-absolutness from set theory in uncountable model” absolute? model theory Recall that for the first order logic this follows by the Lowenheim-Skolem theorem, which implies that having any infinite model is equivalent to having models of any infinite cardinality, and having countable models is absolute. No LS here (yes for downward LS for theories, but not for sentences in general).

Absolute notions in L ω 1 ,ω model theory Mirna Dˇ zamonja Let now us consider L ω 1 ,ω , the logic which is like the first Introduction order logic but allows infinite ∧ . It can define well oder, so In model theory it is not compact. However, it is complete, by a proof of Syntactic and semantic notions Carol Karp from 1964. Absolutness from model theory in set Question : Suppose that τ is some countable vocabulary theory and ϕ an L ω 1 ,ω -sentence in τ . Is the statement “ ϕ has an (Non)-absolutness from set theory in uncountable model” absolute? model theory Recall that for the first order logic this follows by the Lowenheim-Skolem theorem, which implies that having any infinite model is equivalent to having models of any infinite cardinality, and having countable models is absolute. No LS here (yes for downward LS for theories, but not for sentences in general). So, the completeness of the logic does not help.

Absolute notions in model theory The positive answer to the question was initiated by an Mirna Dˇ zamonja idea of Paul Larson in 2013, using set theory, and more general results in a joint paper by Baldwin-Larson-Shelah Introduction (2015). In model theory Syntactic and semantic notions Absolutness from model theory in set theory (Non)-absolutness from set theory in model theory

Absolute notions in model theory The positive answer to the question was initiated by an Mirna Dˇ zamonja idea of Paul Larson in 2013, using set theory, and more general results in a joint paper by Baldwin-Larson-Shelah Introduction (2015). We sketch the proof. In model theory Syntactic and semantic notions Absolutness from model theory in set theory (Non)-absolutness from set theory in model theory

Absolute notions in model theory The positive answer to the question was initiated by an Mirna Dˇ zamonja idea of Paul Larson in 2013, using set theory, and more general results in a joint paper by Baldwin-Larson-Shelah Introduction (2015). We sketch the proof. In model theory Syntactic and semantic notions Let ϕ be a L ω 1 ,ω -sentence in τ such that it is consistent Absolutness from that ϕ has an uncountable model. model theory in set theory (Non)-absolutness from set theory in model theory

Absolute notions in model theory The positive answer to the question was initiated by an Mirna Dˇ zamonja idea of Paul Larson in 2013, using set theory, and more general results in a joint paper by Baldwin-Larson-Shelah Introduction (2015). We sketch the proof. In model theory Syntactic and semantic notions Let ϕ be a L ω 1 ,ω -sentence in τ such that it is consistent Absolutness from that ϕ has an uncountable model. This can be stated in a model theory in set small fragment of ZFC, call it ZFC ∗ , and ZFC ∗ satisfies theory (Non)-absolutness the downward Lowenheim-Skolem. ( encoding from set theory in model theory technique of Shelah)

Absolute notions in model theory The positive answer to the question was initiated by an Mirna Dˇ zamonja idea of Paul Larson in 2013, using set theory, and more general results in a joint paper by Baldwin-Larson-Shelah Introduction (2015). We sketch the proof. In model theory Syntactic and semantic notions Let ϕ be a L ω 1 ,ω -sentence in τ such that it is consistent Absolutness from that ϕ has an uncountable model. This can be stated in a model theory in set small fragment of ZFC, call it ZFC ∗ , and ZFC ∗ satisfies theory (Non)-absolutness the downward Lowenheim-Skolem. ( encoding from set theory in model theory technique of Shelah) So let A be a countable model of ZFC ∗ containing τ that satisfies that ϕ has an uncountable model.

Absolute notions in model theory The positive answer to the question was initiated by an Mirna Dˇ zamonja idea of Paul Larson in 2013, using set theory, and more general results in a joint paper by Baldwin-Larson-Shelah Introduction (2015). We sketch the proof. In model theory Syntactic and semantic notions Let ϕ be a L ω 1 ,ω -sentence in τ such that it is consistent Absolutness from that ϕ has an uncountable model. This can be stated in a model theory in set small fragment of ZFC, call it ZFC ∗ , and ZFC ∗ satisfies theory (Non)-absolutness the downward Lowenheim-Skolem. ( encoding from set theory in model theory technique of Shelah) So let A be a countable model of ZFC ∗ containing τ that satisfies that ϕ has an uncountable model. In a highly non-trivial way, using nonstationary tower forcing

Absolute notions in model theory The positive answer to the question was initiated by an Mirna Dˇ zamonja idea of Paul Larson in 2013, using set theory, and more general results in a joint paper by Baldwin-Larson-Shelah Introduction (2015). We sketch the proof. In model theory Syntactic and semantic notions Let ϕ be a L ω 1 ,ω -sentence in τ such that it is consistent Absolutness from that ϕ has an uncountable model. This can be stated in a model theory in set small fragment of ZFC, call it ZFC ∗ , and ZFC ∗ satisfies theory (Non)-absolutness the downward Lowenheim-Skolem. ( encoding from set theory in model theory technique of Shelah) So let A be a countable model of ZFC ∗ containing τ that satisfies that ϕ has an uncountable model. In a highly non-trivial way, using nonstationary tower forcing construct B , an uncountable model of ZFC ∗ which is an elementary extension of A and such that B is correct about uncountability.

Absolute notions in model theory The positive answer to the question was initiated by an Mirna Dˇ zamonja idea of Paul Larson in 2013, using set theory, and more general results in a joint paper by Baldwin-Larson-Shelah Introduction (2015). We sketch the proof. In model theory Syntactic and semantic notions Let ϕ be a L ω 1 ,ω -sentence in τ such that it is consistent Absolutness from that ϕ has an uncountable model. This can be stated in a model theory in set small fragment of ZFC, call it ZFC ∗ , and ZFC ∗ satisfies theory (Non)-absolutness the downward Lowenheim-Skolem. ( encoding from set theory in model theory technique of Shelah) So let A be a countable model of ZFC ∗ containing τ that satisfies that ϕ has an uncountable model. In a highly non-trivial way, using nonstationary tower forcing construct B , an uncountable model of ZFC ∗ which is an elementary extension of A and such that B is correct about uncountability. Then the model of ϕ in B is actually an uncountable model of ϕ .

Absolute notions in Quasi-minimality model theory Here is a favourite question in model theory: Mirna Dˇ zamonja Question (Zilber 1996?) Is the complex field with Introduction exponentiation C exp quasi-minimal, In model theory Syntactic and semantic notions Absolutness from model theory in set theory (Non)-absolutness from set theory in model theory

Absolute notions in Quasi-minimality model theory Here is a favourite question in model theory: Mirna Dˇ zamonja Question (Zilber 1996?) Is the complex field with Introduction exponentiation C exp quasi-minimal, which means that In model theory every definable subset is either countable or Syntactic and semantic notions co-countable? Absolutness from model theory in set theory (Non)-absolutness from set theory in model theory

Absolute notions in Quasi-minimality model theory Here is a favourite question in model theory: Mirna Dˇ zamonja Question (Zilber 1996?) Is the complex field with Introduction exponentiation C exp quasi-minimal, which means that In model theory every definable subset is either countable or Syntactic and semantic notions co-countable? Absolutness from model theory in set Many excellent mathematicians have worked on this theory question: Wilkie, Zilber, then Bays, Kirby, Mantova and (Non)-absolutness from set theory in others trying for an (absolute) yes or no answer. model theory

Absolute notions in Quasi-minimality model theory Here is a favourite question in model theory: Mirna Dˇ zamonja Question (Zilber 1996?) Is the complex field with Introduction exponentiation C exp quasi-minimal, which means that In model theory every definable subset is either countable or Syntactic and semantic notions co-countable? Absolutness from model theory in set Many excellent mathematicians have worked on this theory question: Wilkie, Zilber, then Bays, Kirby, Mantova and (Non)-absolutness from set theory in others trying for an (absolute) yes or no answer. But the model theory question resists.