Set theory and model theory: a symbiosis Jouko Vnnen Helsinki, - PowerPoint PPT Presentation

Symbiosis Reflection LST Sort logic Set theory and model theory: a symbiosis Jouko Väänänen Helsinki, Finland Montseny, November 2018 1 / 53

Symbiosis Reflection LST Sort logic Model theory Model theory (in this talk) is the study of classes of models, closed under isomorphisms, on the basis of the formal languages used to define the classes. 2 / 53

Symbiosis Reflection LST Sort logic The basic equivalence • A model M satisfies the sentence ϕ of a logic L ∗ iff a certain formula Φ( x ) of set theory is true for x = M . • M | = ϕ ⇐ ⇒ Φ( M ) 3 / 53

Symbiosis Reflection LST Sort logic The basic equivalence, spelled out • M | = ϕ if an only if there is a model N of set theory such that M , ϕ ∈ N , in N the first order sentence “ M | = ϕ ” is true, and N satisfies some absoluteness criteria. • Φ( M ) if and only if there is a binary predicate E on the universe of M , augmented perhaps with some new sorts, such that E satisfies some set theory, some absoluteness criteria, and in the sense of E the set-theoretical formula Φ( x ) is satisfied by an element x = m of the universe such that the structure that E thinks m is, is isomorphic to M . 4 / 53

Symbiosis Reflection LST Sort logic 5 / 53

Symbiosis Reflection LST Sort logic An early example: simple theory of types • x , y , z , . . . for individuals. • X , Y , Z , . . . for sets. • X , Y , Z , . . . for sets of sets. • Etc... • First order logic: x = y , R ( x 1 , . . . , x n ) • Second order logic: X ( y ) • Third order logic: X ( y ) , X ( Y ) • Etc... 6 / 53

Symbiosis Reflection LST Sort logic Gödel "On the present situation in the foundations of mathematics" (*1933o): 1 "...but it turns out that this system [set theory] is nothing else but a natural generalization of the theory of types, or rather, it is what becomes of the theory of types if certain superfluous restrictions are removed." (Hodges, "Tarski’s Theory of Definition", New Essays on Tarski and Philosophy, 2008, D. Patterson, ed.) "The deductive theories in question (such as RCF) are formulated in simple type theory; by 1935 the axioms for RCF is regarded as a definition within set theory." 1 Unpublished manuscript. 7 / 53

Symbiosis Reflection LST Sort logic Model theory Set theory Sort logic First order logic Second order logic ∆ 2 in Levy hierarchy First order logic with ∆ 1 in Levy hierarchy the game quantifier ∆ KP Infinitary logic L HYP in Levy hierarchy 1 ∆ KPU − First order logic in Levy hierarchy 1 M | = ϕ ⇐ ⇒ Φ( M ) 8 / 53

Symbiosis Reflection LST Sort logic • First order logic L ωω , • Infinitary logics L κλ , • Logic with the generalized quantifier L ωω ( Q α ) , • Härtig-quantifier logic L ωω ( I ) . • Second order logic L 2 . • Cofinality logic L ωω ( Q cf ω ) . • Stationary logic L ωω ( aa ) . 9 / 53

Symbiosis Reflection LST Sort logic The map of logics 10 / 53

Symbiosis Reflection LST Sort logic Two key auxiliary concepts Definition (V. 78) Suppose R is a finite set of predicates of set theory and L ∗ is a logic. 1. ∆ 1 ( R ) means ∆ 1 in the extended language {∈ , R} . 2. ∆( L ∗ ) is the logic the definable model classes of which are such K that both K and − K are reducts of L ∗ -definable model classes. 11 / 53

Symbiosis Reflection LST Sort logic Symbiosis Definition A (finite set of) n -ary predicates R and a logic L ∗ are symbiotic if the following conditions are satisfied: 1. Every L ∗ -definable model class is ∆ 1 ( R ) -definable. 2. Every 2 ∆ 1 ( R ) -definable model class is ∆( L ∗ ) -definable. 2 It is enough that a particular ∆ 1 ( R ) -definable model class i.e. a particular generalized quantifier , derived from R , is ∆( L ∗ ) -definable. 12 / 53

Symbiosis Reflection LST Sort logic The following pairs ( R , L ∗ ) are symbiotic. 1. R : Cd , i.e. the predicate “ x is a cardinal". L ∗ : L ωω ( I ) , where Ixy ϕ ( x ) ψ ( y ) ↔ | ϕ | = | ψ | is the Härtig quantifier. 2. R : Cd L ∗ : L ωω ( R ) , where Rxy ϕ ( x ) ψ ( y ) ↔ | ϕ | ≤ | ψ | is the Rescher quantifier. 3. R : Cd L ∗ : L ωω ( W Cd ) , where W Cd xy ϕ ( x , y ) ↔ ϕ ( · , · ) is a well-ordering of the order-type of a cardinal. 4. R : Cd , WI L ∗ : L ωω ( I , W WI ) , where W WI x ϕ ( x ) ↔ | ϕ ( · ) | is weakly-inaccessible. 5. R : Rg , i.e. the predicate “ x is a regular cardinal" L ∗ : L ωω ( I , W Rg ) , where W Rg xy ϕ ( x , y ) ↔ ϕ ( · , · ) has the order-type of a regular cardinal. 13 / 53

Symbiosis Reflection LST Sort logic 1. R : Cd , WC L ∗ : L ωω ( I , Q Br ) , where Q Br xy ϕ ( x , y ) ↔ ϕ ( · , · ) is a tree order of height some α and has no branch of length α . 2. R : Cd , WC L ∗ : L ωω ( I , ¯ Q Br ) , where ¯ Q Br xyuv ϕ ( x , y ) ψ ( u , v ) ↔ ϕ ( · , · ) is a partial order with a chain of order-type ψ ( · , · ) . 3. R : Pw i.e. the predicate { ( x , y ) : y = P ( x ) } L ∗ : The second order logic L 2 . 14 / 53

Symbiosis Reflection LST Sort logic Cd and L ωω ( I ) are symbiotic. Theorem TFAE: 1. K is ∆( L ωω ( I )) -definable. 2. K is ∆ 1 ( Cd ) -definable in set theory Note: If V = L (or L µ ), then ∆( L ωω ( I )) = ∆( L 2 ) . 15 / 53

Symbiosis Reflection LST Sort logic Question Is there a natural canonical inner model for set theory in the vocabulary {∈ , Cd } ? Is there a minimal transitive class model with true cardinals? 16 / 53

Symbiosis Reflection LST Sort logic Pw and the L 2 are symbiotic. Let Pw be the relation { ( x , y ) : y = P ( x ) } . ∆ 1 ( Pw ) = ∆ 2 . Theorem TFAE: 1. K is ∆( L 2 ) -definable. 2. K is ∆ 2 -definable in set theory 17 / 53

Symbiosis Reflection LST Sort logic Structural reflection Let R be a finite set of predicates or relations. Definition (Bagaria-V. 2016) ( SR ) R ( κ ) : If K is a Σ 1 ( R ) class of models, then for every A ∈ K , there exist B ∈ K of cardinality less than κ and an elementary embedding e : B � A . 18 / 53

Symbiosis Reflection LST Sort logic • The principle ( SR ) Cd implies 0 ♯ , and much more, e.g. failure of square at all cardinals from κ on (Magidor and V. 2011, see later). 19 / 53

Symbiosis Reflection LST Sort logic Weaker forms of structural reflection ( SR ) −− : If K is a non-empty Σ 1 ( R ) class of models, then R there exists A ∈ K of cardinality less than κ . Proposition (Hanf) ZFC ⊢ ∃ κ (( SR ) −− holds for κ ) . R Proof: Let {K n : n < ω } list all non-empty Σ 1 ( R ) model classes. Pick A n ∈ K n , for each n < ω . Let κ be the supremum of all the cardinalities of the A n , for n < ω . Then κ + is as required. 20 / 53

Symbiosis Reflection LST Sort logic ( SR ) − R : If K is a Σ 1 ( R ) class of models and A ∈ K has cardinality κ , then there exists B ∈ K of cardinality less than κ and an elementary embedding e : B � A . 21 / 53

Symbiosis Reflection LST Sort logic Theorem (Stavi-V. 2002) If ( SR ) − Cd holds for κ , then there exists a weakly inaccessible cardinal λ ≤ κ . 22 / 53

Symbiosis Reflection LST Sort logic Let Rg be the predicate “ x is a regular ordinal". Theorem (Bagaria-V. 2016) If κ satisfies ( SR ) − Rg , then there exists a weakly Mahlo cardinal λ ≤ κ . 23 / 53

Symbiosis Reflection LST Sort logic • We cannot hope to get from ( SR ) Rg more than one weakly Mahlo cardinal ≤ κ . Indeed, starting from a weakly Mahlo cardinal one obtains a model of set theory in which ( SR ) Rg holds for the least weakly Mahlo cardinal. • We cannot hope either to obtain from ( SR ) Rg that κ is strongly inaccessible, for it can be shown that one can have ( SR ) Rg for κ = 2 ℵ 0 . 24 / 53

Symbiosis Reflection LST Sort logic There is a condition between ( SR ) − Cd and ( SR ) − Rg , namely ( SR ) − Cd , WI , where WI is the predicate “ x is weakly inaccessible". Proposition (Bagaria-V. 2016) If κ satisfies ( SR ) − Cd , WI , then there exists a 2 -weakly inaccessible cardinal λ ≤ κ . 25 / 53

Symbiosis Reflection LST Sort logic • We may also consider predicates α - WI , for α an ordinal. That is, the predicate “ x is α -weakly inaccessible". • Then, similar arguments would show that the principle ( SR ) − Cd , α - WI implies that there is an ( α + 1 ) -weakly inaccessible cardinal ≤ κ . 26 / 53

Symbiosis Reflection LST Sort logic Let WC ( x , α ) be the Π 1 relation “ α is a limit ordinal and x is a partial ordering with no chain of order-type α ". Theorem (Bagaria-V. 2016) If κ satisfies ( SR ) − Cd , WC , then there exists a weakly compact cardinal λ ≤ κ . 27 / 53